forest insect population dynamics, outbreaks, and global warming effects

Forest Insect Population Dynamics, Outbreaks, and

Global Warming Eff ects

Forest Insect Population Dynamics, Outbreaks, and

Global Warming Eff ects

A. S. Isaev, V. G. Soukhovolsky, O. V. Tarasova, E. N. Palnikova

and A. V. Kovalev

Th is edition fi rst published 2017 by John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA and Scrivener Publishing LLC, 100 Cummings Center, Suite 541J, Beverly, MA 01915, USA© 2017 Scrivener Publishing LLCFor more information about Scrivener publications please visit www.scrivenerpublishing.com.

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Library of Congress Cataloging-in-Publication DataISBN 978-1-119-40646-4

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Set in size of 11pt and Minion Pro by Exeter Premedia Services Private Ltd., Chennai, India

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v

Contents

Authors ix

Introduction xi

1 Population Dynamics of Forest Insects: Outbreaks in Forest Ecosystems 11.1 Approaches to modeling population dynamics

of forest insects 11.2 Th e role of insects in the forest ecosystem 41.3 Th e phenomenological theory of forest insect

population dynamics: the principle of stability of fl exible ecologicalsystems 10

1.4 Classifi cation of the factors of forest insect population dynamics 12

1.5 Delayed and direct regulation mechanisms 14

2 Ways of Presenting Data on Forest Insect Population Dynamics 172.1 Representation of population dynamics data 172.2 Presenting the data on forest insect population

dynamics through changes in density over time 182.3 Presenting the data on populatiozn dynamics as

a phase portrait 242.4 Th e probability of the population leaving the

stability zone and reaching an outbreak density: a model of a one-dimensional potential well 40

2.5 Presenting the data on forest insect population dynamics as a potential function 47

3 Th e Eff ects of Weather Factors on Population Dynamics of Forest Defoliating Insects 533.1 Th e necessary and suffi cient weather conditions

for the development of outbreaks of defoliating insects in Siberia 53

vi Contents

3.2 Weather infl uence on the development of the pine looper Bupalus piniarius L. outbreaks 55

3.3 Siberian silk moth Dendrolimus sibiricus Tschetv. population dynamics as related to weather conditions 61

3.4 Synchronization of weather conditions on vast areas as a factor of the occurrence of pan-regional outbreaks 64

4 Spatial and Temporal Coherence of Forest Insect Population Dynamics 794.1 Coherence and synchronicity of population

dynamics 794.2 Spatiotemporal coherence of the population

dynamics of defoliating insects in pine forests of Middle Siberia 83

4.3 Spatiotemporal coherence of population dynamics of defoliating insects in the Alps 90

4.4 Global coherence of pine looper population dynamics in Eurasia 94

4.5 Synchronization of the time series of gypsy moth population dynamics in the South Urals 96

5 Interactions Between Phytophagous Insects and Th eir Natural Enemies and Population Dynamics of Phytophagous Insects During Outbreaks 1015.1 Entomophagous organisms as a regulating factor

in forest insect population dynamics 1015.2 A “phytophagous – entomophagous insect” model 106

6 Food Consumption by Forest Insects 1136.1 Energy balance of food consumption by insects:

an optimization model 1136.2 A population-energy model of insect outbreaks 127

7 AR- and ADL-Models of Forest Insect Population Dynamics 1397.1 An ADL-model (autoregressive distributed lag) of

insect population dynamics 1397.2 A model of population dynamics of the gypsy moth

in the South Urals 1457.3 Modeling population dynamics of the larch bud moth

in the Alps 155

Contents vii

7.4 Simulation models of population dynamics of defoliating insects in the Krasnoturansk pine forest 165

7.5 Modeling and predicting population dynamics of the European oak leaf-roller 172

7.6 Gain margin of the AR-models of forest insect population dynamics 176

8 Modeling of Population Dynamics and Outbreaks of Forest Insects as Phase Transitions 1838.1 Models of phase transitions for describing critical

events in complex systems 1838.2 Population buildup and development of an outbreak

of forest insects as a fi rst-order phase transition 1858.3 Possible mechanisms of the development of forest

insect outbreaks 1928.4 Colonization of the tree stands by forest insects as a

second-order phase transition 1948.5 Risks of elimination of the population from the

community 201

9 Forecasting Population Dynamics and Assessing the Risk of Damage to Tree Stands Caused by Outbreaks of Forest Defoliating Insects 2079.1 Methods of forecasting forest insect population dynamics 2079.2 Long-term forecast of population dynamics of

defoliating insects 2179.3 Assessment of the maximum risk of damage to tree

stands caused by insects 2239.4 Modeling and forecasting of eastern spruce

budworm population dynamics 225

10 Global Warming and Risks of Forest Insect Outbreaks 23310.1 Climate change and forest insect outbreaks in the

Siberian taiga 23310.2 Stress testing of insect impact on forest ecosystems

under diff erent scenarios of climate changes in the Siberian taiga 236

10.3 Risks of outbreaks of forest insect species with the stable type of population dynamics 244

Conclusion 251

References 255

Index 285

AuthorsAlexander S. Isaev, D.Sc. (Biology), Full Member of the Russian Academy of Sciences (RAS), Head of Research at the Centre for Problems of Ecology and Productivity of Forests RAS (CEPF RAS). Graduated from the Leningrad Forestry Engineering Academy. An expert in forest ento-mology and ecology. Director of the V. N. Sukachev Institute of Forest and Wood SB USSR AS (1976–1985), Head of the USSR Forest State Committee (1985–1991), Director of CEPF RAS. Th e author of more than 300 published studies, including over 20 monographs on forest ecology and forest entomology. Awards: Gold Medal of the International Union of Forest Research Organizations (IUFRO), V. N. Sukachev Medal of RAS, and IUFRO George Varley Award for Excellence in Forest Insect Research.

Vladislav G. Soukhovolsky, D.Sc. (Biology), Professor, Leading Researcher at the V. N. Sukachev Institute of Forest SB RAS. Graduated from the Faculty of Physics at the Krasnoyarsk State University. An expert in mathematical modeling of complex biological, ecological, social, and political systems. Th e author of over 500 published studies, including 16 monographs.

Olga V. Tarasova, D.Sc. (Agriculture), Professor of the Department of Ecology at the Siberian Federal University. Graduated from the Faculty of Biology at the Krasnoyarsk State University. Between 1978 and 1981, a graduate student at the Department of Ecology at the Krasnoyarsk State University (Academic Adviser – A. S. Isaev). An expert in forest ento mology. Th e author of over 150 published studies, including four monographs. Award: V. I. Vernadsky Award for Excellence in Ecological Education.

Elena N. Palnikova, D.Sc. (Agriculture), Professor of the Department of Ecology and Forest Protection at the Siberian State Technological University. Graduated from the Faculty of Biology at the Krasnoyarsk State University. Between 1978 and 1982, a graduate student at the V. N. Sukachev Institute of Forest and Wood SB USSR AS (Academic Adviser – A. S. Isaev). An expert in forest entomology. Th e author of over 100 published studies, including one monograph.

ix

x Authors

Anton V. Kovalev, Ph.D. (System Analysis). Senior Researcher of International Scientifi c Center for Organism Extreme States Research (Krasnoyarsk Scientifi c Center). Graduated from the Faculty of Automatization and Robototechnic at the Siberian Technological State University. Between 1999 and 2002, a graduate student at the V. N. Sukachev Institute of Forest SB RAS (Academic Adviser – V. G. Soukhovolsky). An expert in system analysis of ecological processes. Th e author of over 100 published studies, including one monograph.

IntroductionAn insect outbreak is one of the fi rst critical events in ecological sys-tems described in world literature (Exodus 10:12). Until now, however, prediction and control of insect populations damaging forest stands and agricultural crops has remained an unresolved issue. Th e current insect outbreak situation can still be described with the biblical quote: “…When it was morning, the east wind had brought the locusts…”.

Research in insect population dynamics is important for more reasons than just protecting forest communities. Insect populations are among the main ecological units included in the analysis of stability of ecological systems. Moreover, it is convenient to test new methods of analyzing population and community stability on the insect-related data, as by now ecologists and entomologists have accumulated large amounts of such data.

In this book, the authors analyze population dynamics of quite a narrow group of insects – forest defoliators. We hope, though, that the methods we propose for the analysis of population dynamics of these species may be useful and eff ective for analyzing population dynamics of other animal species.

Below is a brief description of each chapter in the book.Chapter 1 is, rather predictably, a review of the literature on modeling

forest insect population dynamics. Section 1.3 provides a brief description of the phenomenological theory of population dynamics (Isaev et al., 1984; Isaev et al., 2001).

Chapter 2 discusses the issue that is seldom addressed in the litera-ture – the choice of the way of describing insect population dynamics. In our opinion, for each defi nite task in the analysis of insect population dynamics, there is a specifi c way of data presentation: as a time series, a phase portrait, the “Lamerey stairs”, or potential function. Th erefore, we discuss diff erent ways of presenting survey data, as related to the purposes of the analysis.

We think that a necessary condition for the successful analysis of processes occurring in forest ecosystems is a certain irreverence towards the fi eld data. As fi eld ecologists, we know very well how much eff ort it takes to carry on insect population surveys on the same plot in the forest

xi

xii Introduction

for many years. On the other hand, we are aware of the inaccuracy of the fi eld data and the inevitable errors in estimates of the density of popu-lation dispersed over a vast area. Survey data should not be regarded as something incontrovertibly true but rather as a basis for research activities. Th ese activities should include repair and transformation of the fi eld data, based on the theoretical concepts developed in this research. Before using the survey data for analysis, they need to be “cleaned” as much as possi-ble, to remove the inevitable errors of surveys, without distorting the time series. Our experience shows that it is important not only to collect the data but also to treat them properly. Th erefore, Chapter 2 gives a detailed description of fi eld data repair and transformation. Th is chapter focuses on the methods used to process survey data and transform an arbitrary time series into the stationary time series, which can then be studied by using standard techniques of correlation and spectral analysis.

Chapter 3 is devoted to the analysis of weather eff ects on the devel-opment of outbreaks of taiga defoliating insects. Th is subject has been extensively discussed in the literature, especially in the last decades, as related to the possible global climate change. Here we present our under-standing of these processes.

Chapter 4 analyzes spatial coherence of population dynamics of the same insect species in diff erent habitats and the temporal coherence of population dynamics of several insect species in the same habitat. Such analysis can be used to reveal interactions between species associated with, for example, competition for food and to estimate possible responses of diff erent species to external impacts such as changes in weather and geo-physical parameters.

Chapter 5 describes parasite – host interactions for populations of for-est insects and their parasites in diff erent outbreak phases.

In Chapter 6, we present a model of food consumption by insects, which links population dynamics with food properties. We propose a quasi- economic approach to describing food consumption and introduce indicators of food consumption analogous to costs in economics. In this way, we relate the energy and population approaches to the description of the processes in the forest – insect system and approach evaluation of fecundity of individuals – very important parameters for analysis and fore-cast of insect population dynamics.

Chapter 7 is devoted to modeling time series of forest insect population dynamics by using autoregressive models. Th e chapter describes models of population dynamics of the larch bud moth and other species of the defo-liating insect community in forests of the Alps, the pine looper in Europe,

Introduction xiii

defoliating insects in the Siberian pine forests, the European oak leaf-roller in European Russia, and the gypsy moth in the South Urals. For auto regressive models, we introduce parameters of stability, stability margin, and robust stability, which are used to assess the risks of “removal” of the species from the community. Th ese models serve as a basis for developing adaptive meth-ods for short-term forecasts of forest insect population dynamics.

Chapter 8 deals with a new method of describing and modeling forest insect population dynamics, based on the presentation of critical events in the population as fi rst- and second-order phase transitions. Using the models of phase transitions, we managed to introduce conditions of the occurrence of forest insect outbreaks, describe the patterns of insect migra-tions in the forest during an outbreak, and characterize the susceptibility of populations to weather eff ects.

We consider in Chapter 9 methods of short-, medium-, and long-term forecasting of insect population dynamics based on the approaches described in the previous chapters and methods of assessing the risk of the tree stand damage and death caused by insect outbreaks. In addition to that, Chapter 9 contains a brief discussion of the problems associated with controlling the risks of insect attacks and making decisions about extermi-nation measures based on forest entomological monitoring. We may have given too little consideration to these issues, and they will need to be dis-cussed more thoroughly in a future study.

Finally, in Chapter 10, we discuss the eff ects of possible global climate change on population dynamics of defoliating forest insects. We use ADL-models and phase transition models developed in this book to assess the risks of outbreaks under various scenarios of climate change.

We hope that this book will be useful to specialists in ecology, entomol-ogy, ecological modeling, and forest protection as well as to undergraduate and graduate students of ecology and entomology.

We are grateful to our former and current Ph.D. students – S. Astapenko, Y. Bekker, O. Bulanova, P. Tsikalova, T. Iskhakov, I. Kalashnikova, P. Krasnoperova, V. Kuznetsova, M. Meteleva – for their assistance in diff erent stages of the research. We specially appreciate out deceased colleagues – Yuri P. Kondakov and Viktor M. Yanovsky, with whom we had studied forest insect population dynamics for many years.

Our studies were supported by very many grants of Russian Foundation for Basic Research No. 96-04-48340, 99-04-49450, 00-04-48990, 02-04-48769, 02-04-62038, 03-04-49723, 03-04-62037, 04-04-49821, 08-04-00217, 08-04-07052, 09-04-00412, 10-04-08236, 11-04-00173, 11-04-08064, 15-04-01192.

1

1 Population Dynamics of Forest Insects: Outbreaks in Forest Ecosystems

1.1 Approaches to modeling population dynamics of forest insects

Populations of forest insects constitute one of the components of a com-munity that includes the species we are interested in, species competing for various types of resources, parasites, predators, and host plants. All these species making up an ecosystem are influenced by weather and oth-er external factors. In accordance with the basic theses of systems analy-sis, it is not correct to study only one ecological component, which, in our case, is a forest insect population. Analysis of the processes occurring in a complex system must be based on the systems approach and investi-gation of all ecosystem components.

However, the dogmas of the systems analysis collapse once the re-searchers face their object – a population of a certain species of defoliat-ing forest insects. Long-duration measurements in the forest usually rec-ord only local population density. Sometimes, one manages to estimate the mass of individuals, their coloration, female fecundity, and the extent of parasitic infection. Other components, such as predators, usually re-main unevaluated. Moreover, in a study of a certain insect species, sur-veys cannot be performed in every developmental stage of this insect. Thus, the system as a whole “eludes” the researcher.

Forest Insect Population Dynamics, Outbreaks, and Global Warming Effects. A. S. Isaev, V. G. Soukhovolsky, O. V. Tarasova, E. N. Palnikova and A. V. Kovalev.

© 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

2 Forest Insect Population Dynamics, Outbreaks, and Global Warming Effects

In this situation, models of population dynamics serve as an integrator used to combine discrete observations and local experiments in a system, thus enabling the description of both individual characteristics of a forest insect species (the type of long-term population dynamics, intra-population relationships and interactions with parasites and predators, susceptibility to external effects on population dynamics) and general trends in the population dynamics of ecological groups of insects.

Thus, there may be two extreme approaches to constructing mathe-matical models of population dynamics of a single forest insect species. If the model of the population dynamics of a species takes into account all of its specific biological and ecological properties as well as effects of various external factors in its habitat, this approach will be too detailed and ineffective, as the modeler will need to know too many aspects of the species.

On the other hand, if the approach used for constructing the popula-tion dynamics model is too generalized, e.g., if the Lotka–Volterra model is used to describe population dynamics of absolutely all species of forest insects and if individual properties of the species can only be defined by using five coefficients of that unified model, this total approach will pre-dictably be no more fruitful than the extremely detailed one.

The phenomenological approach was previously proposed for modeling forest insect population dynamics: only the main trends in changes of pop-ulation density were studied, with various factors potentially capable of influencing the insects reduced to two types – modifying (density inde-pendent) factors and regulating (density dependent) factors (Isaev, Khlebopros, 1973; Isaev et al., 1984). Analysis of forest insect population dynamics by using phase portraits provided a basis for the qualitative model of outbreak development and classification of the types of forest insect population dynamics and outbreaks.

In this study, we use the systems approach in our search for the neces-sary generalization level in modeling population dynamics of a single insect species. The descriptions of population dynamics of all forest in-sect species will be based on the following notions:

– the notion of the existence of three stable states of the population that differ considerably in density.

The first state is degenerate; it characterizes the habitat in which popu-lation density of a given insect species is equal to zero. The other two states are characterized by nonzero values of population density. In one of these states (in the outbreak phase), population density is much higher than population density in the other state – in the stable state. Hence, there must be a system of negative feedbacks regulating the behavior

Population Dynamics of Forest Insects: Outbreaks in Forest Ecosystems 3

of the system close to each of these stable states. If a certain number of insects of one species are introduced into the habitat with the stable pop-ulation density x = 0, in some time, the reproduction coefficient of the invading population will be equal to zero, all insects will die, and the sys-tem will return to the state with the population density of this species equal to zero. Thus, it is very important to define the conditions under which the population will be in one of these stable states, in order to be able to predict population dynamics.

– the notion of the ability of the population to pass from one stable state to another and the attainability of different stable states under certain conditions.

Transitions may be regarded as the critical events in forest insect populations. The transition from the low population density – the sparse stable state – to the high population density – the outbreak phase – is of special interest for ecological theory and forest management, as insect outbreaks are among the main causes of tree damage and die-off of for-ests in the boreal ecosystem. It is also very important to understand why the population of a species passes from the zero-density state to the stable state with a nonzero density. In recent decades, invasions of new species of forest insects have become more and more frequent, and, thus, it is essential to identify the conditions under which the invading species, ini-tially represented by just a few insects, successfully colonizes the new habitat.

– the notion of two types of factors influencing forest insect popula-tion – modifying and regulating factors.

Modifying factors are density-independent factors and changes in the population density do not influence the current value of the modifying factor. The converse is certainly not true, and the sensitivity of popula-tion parameters to a change in the value of the modifying factor may be nonzero.

Values of the regulating factors depend on the current and previous states of the population. At low values of population density, sensitivity to the external field is low, modifying factors do not influence the popu-lation, and population density variations are determined by the intra-population regulation. As the density grows, sensitivity is rapidly in-creased, and then weather effects may facilitate development of an out-break. It is a difficult task to decide whether food is a modifying or a reg-ulating factor. Under certain conditions (usually at a low density of forest insect population), food (tree needles or leaves for forest defoliating in-sects) may be a modifying factor. However, as the population density is increased and insects consume greater amounts of food, host trees re-


spond by changing the quality of the food. In this case, food is also a reg-ulating factor.

– the notion of the time lag between the impact of external factors and response of the population.

While evaluating the sensitivity of population parameters to the im-pacts of external factors, one needs to take into account that the time of the impact and the time when the response is first detected may not coin-cide. Then, there is a time lag between the impact and the response. The length of the time lag may vary, influencing the behavior of the popula-tion.

This book reports studies based on the principles of the system of regulating forest insect population dynamics. On the one hand, it de-scribes properties of specific time series of forest insect population densi-ty (by using the abundant field data accumulated over the last few dec-ades). On the other hand, the authors employ the theory of phase transi-tions – a universal approach to the description of critical events, validated by physicists – to describe universal properties of the processes occurring in forest ecosystems and leading to qualitative and quantitative changes in the state of these systems.

The vast majority of forest insects do not undergo any qualitative changes, and their densities remain low for an indefinitely long time, causing no significant alterations in other components of the ecosystem. Only a small number of species may change qualitatively and quantita-tively and attain outbreak densities.

1.2 The role of insects in the forest ecosystem The role of insects in the forest is determined by their specific interac-tions with the plant community. Phytophagous insects consume part of the vegetable matter as green mass, reproductive structures, cambial tis-sues, bark, and wood. Insects transform biomass in their vital processes and quickly return it to the soil, making up for the absorbed organic mat-ter and carbon dioxide, which are necessary for successful functioning of the forest ecosystem. Thus, the ecosystem “grants” part of its biomass to insects for speedy transformation and in this way increases the overall stability of the system. By infesting weakened trees, insects improve the health of the forest. In all phases of tree stand formation, they take part in selection and self-thinning, contributing to the development of stable tree stands.


Every ecological group of insects has its own way of interacting with the plant community. Defoliating insects are external feeders and are, thus, more vulnerable to the impacts of biotic and abiotic factors. Xylophagous insects, which live in the wood and beneath the bark, are closely related to the tree as their habitat. The regulating effect of natural enemies is not a determining factor in the population dynamics of xy-lophages, as their successful development is mainly related to the availa-bility of suitable food. Insects feeding on reproductive structures (fruits, cones, seeds) have specific ecological relations with their hosts. They in-fest tree stands at the fruiting time, and their populations do not fluctuate greatly. The qualitatively dissimilar interactions of different ecological groups determine the types of their feedbacks and population dynamics.

The great majority of forest insect species have densities correspond-ing to stable populations and represent consumers, which generally pro-duce no adverse effects on growth and development of the forest com-munity. Only a small number of species are capable of dramatic density variations and, under suitable conditions, may attain outbreak densities rather quickly (in two or three years). The outbreak does not usually last long (two to three years), and then, under the impact of regulatory mech-anisms (parasites and predators, diseases, food shortage etc.), the popula-tion enters the critical and decline phases. After that, the population gradually resumes its sparse stable state, completing the outbreak cycle. Outbreaks are not, however, always harmless.

Insect outbreaks may lead to severe disturbance of the state of forest ecosystems. For example, Siberian forests are severely damaged by peri-odic outbreaks of the Siberian silk moth (Dendrolimus sibiricus superans Tschetv.), the gypsy moth (Limantria dispar L.), the saddleback looper (Boarmia bistortata Coere), the pine looper (Bupalus piniarias L.), the Asian larch bark beetle (Monochamus urussovi F.), and other insect spe-cies, which cause tree death and greatly influence the forest formation process in taiga.

Human activities also affect the resistance of forest ecosystems to pest insects. Uncontrolled tree felling, damages caused by forest fires, in-creased recreational impact, pollution due to industrial discharges, and other human-related factors weaken regulating mechanisms of ecosys-tems and facilitate insect outbreaks. Artificial forest ecosystems that re-place natural forests are, as a rule, much less stable. Particularly vulnera-ble are pure single-species stands, which can be easily killed by insects. A good illustration of this is the death of vast spruce plantations that occurred in Central Europe (Germany, Poland, Czech Republic, Slo-vakia) at the end of the 20th century due to outbreaks of the nun moth


(Limantria monacha L.) and the bark beetles (Ips. typhographus L., Pityogenes chalcographus L., and other species). Establishment of artificial ecosystems can lead to an increase in the abundances of the insect species that do not reach outbreak levels in natural forest ecosystems. Prior to establishing pine plantations in Western Siberia, the great spruce bark beetle (Dendractonus micans L.) was a rare species for that region. In the 1970s, this species infested plantations of 20-year-old pines and damaged them considerably (Kolomiyets, Tarasova, 1977). A similar situation was observed in pine plantations in the Irkutsk Oblast, which were infested by sawflies (Pristiphora erichsonii Htg., Pristiphora wesmaeli Tischb.) (Verzhutskii, 1966). In natural forests, sawflies live on young pines as sparse populations. Establishment of even-aged pine plantations with the young growth occupying vast areas created favorable conditions for these insect species and, in some cases, promoted them to the dominant posi-tion in forest ecosystems.

Strictly speaking, destruction of natural ecosystems by insect infesta-tions should not be regarded as an entirely negative event because these insects perform a special, “predatory”, function in the evolution dynam-ics of forest canopy. Transformations of the smaller ecosystems that stop being resistant to insects in a certain stage of evolution are necessary for the stability of the larger natural systems and evolution dynamics of the forest cover of a given region.

A good illustration is recurrent outbreaks of the Siberian silk moth in dark coniferous forests growing along the Yenisei River in Middle Sibe-ria. Between the mid-19th century and the mid-20th century, there were nine outbreaks of this species there, which occurred every 10–14 years (Kondakov, 1974). The outbreak zone is confined within well-defined landscape and ecological boundaries; dark coniferous forests in this zone have been eliminated by the Siberian silk moth and replaced by hard-wood stands. This is a typical example of the transformation that some of the ecosystems of a large region undergo due to certain properties of the contemporary climate, the composition and structure of tree stands, eco-logical traits of the insects, and their specific interactions with compo-nents of the forest canopy.

The natural evolution of forest ecosystems may be incompatible with human economic activity, directed towards preservation of these for-ests. If it is necessary to perform forest protection procedures aimed at increasing the stability of ecosystems and decreasing the abundances of pest insects, these procedures must be based on a thorough knowledge of regulating mechanisms and insect interactions in the forest eco-system.


Thus, insects play a dual role in the forest ecosystem. On the one hand, they consume “surplus” phytomass of healthy trees and, thus, expedite transformation of organic matter and facilitate growth of the tree stand; by infesting weakened and dead trees, they regulate the stand density and improve the health of the forest. This aspect of insect activity is beneficial for the stability of the ecosystem and its evolutionary development. On the other hand, insects can escape the control of the ecosystem and ad-versely affect its state, as the regulating mechanisms, which are mainly responsible for the control of outbreak species, act with a considerable delay.

Between the two extreme forms of the effect (the enhancement of eco-system stability and the destruction of the ecosystem) there is a scale of transitions determined by biological properties of different species, their specific interactions with regulating mechanisms, and their response to the impacts of modifying factors. Analysis of the stability of forest ecosys-tems should be aimed at determining possible scenarios of insect popula-tion dynamics and revealing the basic principles responsible for the oc-currence of one of the scenarios. This information can be used to both evaluate the state of the system and perform targeted regulation of the ecosystem in order to enhance its stability.

Modeling of insect population dynamics is a tool for achieving two aims: to identify the most significant factors of population dynamics and to predict population behavior, so as to be able to control population densities of economically important species of forest insects.

Numerous studies have been conducted on population dynamics of an-imals in general and modeling of forest insect population dynamics, in par-ticular (we are citing only a few of them: Allstadt et al., 2013; Anderson, May, 1981; Auerbach et al., 1995; Babin-Fenske, Anand, 2011; Bazikin et al., 1993; Bazikin, 1985, 2003; Bazykin, Berezovskaya, 1980; Berryman, 1981, 1982, 1988, 1991, 1992, 1994, 1995, 2002, 2002a, 2003; Berryman, Stark, 1985; Biging et al., 1980; Bigsby et al., 2011; Bjorkman et al., 2011; Brandt et al., 2013; Brown, Kulasiri, 1996; Byrne et al., 1987; Chen-Charpentier, Leite, 2014; Christie et al., 1995; Cooke et al., 2012; Coyle et al., 2014; Della Rossa et al., 2012; Dwyer, Elkinton, 1993; Elkinton, Liebhold, 1990; Elliott, Evenden, 2012; Foster et al., 2013; Fursova et al., 2003; Ginzburg, Taneyhill, 1994; Hassell, 1989; Holling, 1965, 1973; Ilyinykh et al., 2011; Isaev et al., 1982, 1984, 1997, 2001, 2009; Isaev, 1971; Isaev, Girs, 1975; Isaev, Khlebopros, 1973, 1977; Iskhakov et al., 2007; Iskhakov, Soukhovolsky, 2004; Jankovic, Petrovskii, 2014; Jeffers, 1978; Klomp, 1966; Kondakov et al., 1985; Kuznetsova, Palnikova, 2014; Liebhold et al., 1991, 1992, 1993, 2000; Liebhold, 2012; Liebhold, Elkinton,


1989; Logan, 1988; Maximov, 1989; Murray, 1977; May, 1976, 1978; Miller-Pierce, Preisser, 2012; Morse, Simmons, 1979; Murray, 2002; Myers, Cory, 2012; Nixon, Roland, 2012; Palnikova et al., 2002, 2014; Palnikova, 1998; Pleshanov, 1982; Poluektov et al., 1980; Rafes, 1978, 1980; Rasmussen et al., 2011; Reynolds et al., 2011; Riznichenko, 2003; Rogers, 1972; Ronnas et al., 2011; Rossiter, 1994; Royama, 1992; Rykiel, 1996; S Schwerdtfeger, 1952; Sawyer et al., 1993; Sharov, Colbert, 1996; Sheehan, 1988, 1989; Smith, 1968, 1974; Soukhovolsky et al., 2000, 2003, 2005, 2008, 2015; Soukhovolsky, 2004; Soukhovolsky, Iskhakov, 2004; Soukhovolsky, Palnikova, 1990; Stange et al., 2011; Suckling et al., 2012; Svirezhev, 1987; Svirezhev, Logofet, 1978; Sweaney et al., 2014; Tarasova et al., 2002, 2004; Tarasova, Soukhovolsky, 1997; Turchin, 1990; Varley, 1949; Varly et al., 1978; Viktorov, 1967; Wallner, 1987; Wilder et al., 1994; Wilder, 1999).

Analysis of the literature on population dynamics showed that there are several major strategic approaches to describing changes in the den-sity of forest insect populations. One approach is based on the analysis of population densities, which takes into account different ecological factors (climate, forest growth, landscape, etc.). Mathematically, this strategy leads to constructing simulation models, which describe the ef-fects of various ecological factors by using influence functions (Holling, 1965, 1973; Isaev et al., 2009). This approach effectively takes into ac-count the impacts of external factors and incorporates them into differ-ent modes of interaction. Its limitation is that it cannot produce analyti-cal results that would characterize fundamental properties of population dynamics.

Another approach is to develop models of intra-population interac-tions of insects and the influence of food and natural enemies on their density. These models are based on different types of differential equa-tions, and analysis of these equations reveals different types of population dynamics. The choice of functions describing these interactions, howev-er, may be quite arbitrary, as the models do not restrict this choice. Moreover, these models usually assume that model parameters are time invariant. If this assumption is invalidated, analysis of the models be-comes very difficult.

A third approach to analysis of phytophagous insect population dy-namics is based on using phase portraits and identifying characteristic parameters that reflect interactions in a forest ecosystem (Isaev, Khlebo-pros, 1973; Isaev et al., 1984; Isaev et al., 2001; Isaev et al., 2009). By em-ploying this approach, one can study not only different forms of popula-tion density regulation but also regulatory mechanisms of the entire eco-logical system, its stability and delays, positive and negative feedbacks,


the ranges of the parameter values for certain effects, dimensions of the phase portraits and their qualitative content. Furthermore, there is no need to evaluate parameters of the forest ecosystem as influenced by spe-cific environmental factors.

Specific interactions between plants and insects in the forest ecosystem are manifested as a combination of continuous and discrete processes. Insect population dynamics is characterized by synchronous generations and discrete vital processes. As the insect life cycle is “short” (one or two years) relative to the “long” process of the ecosystem change (decades and centuries), insect population dynamics can be studied in terms of continuous systems.

The simplest models used for analysis are regression models, in which population dynamics is described by some function of time F(t) of a cer-tain linear or nonlinear form, and coefficients of this function are calcu-lated by using methods of regression analysis (Biging et al., 1980; Camp-bell, Sloan, 1978a,b). If F(t) is a polynomial function of some sufficiently high degree, any, even quite complex, curve of population dynamics can definitely be approximated by using this model. However, coefficients of this model polynomial will be unstable, and the ecological meaning of these coefficients, beginning with the second-degree coefficient, will be unclear. In fact, while constructing these models, one may only identify a constant component and a linear trend of population dynamics, and even this is impossible unless the time series of the population dynamics is sufficiently long.

More effective model evaluations of the dynamics of complex systems (including forest insect populations) can be obtained by using recursive discrete multiplicative models, in which temporally adjacent values of a model variable (or variables) characterizing the system (e.g., population densities at adjacent time moments t and t + 1) are related by some func-tion or a set of such functions of the values of factors influencing the state of the system (Forrester, 1971, 2006). This approach is widely used to model gypsy moth population dynamics (Dwyer, Elkinton, 1993; Elkinton, Liebhold, 1990; Gray et al., 1991; Liebhold et al., 1993; Morse, Simmons, 1979; Sharov, Colbert, 1996; Sheehan, 1988, 1989; Wilder et al., 1994; Wilder, 1999).

In this case, modeling, however, has certain limitations. Evaluations of the density of gypsy moth population were based on a limited number of parameters: the average number of individuals (eggs or larvae) per tree, the number of eggs per egg mass, the percentage of trees with egg masses, and the average mass of an individual (egg, larva) (Benkevich, 1984; Liebhold et al., 1991; Ponomarev et al., 2012; Wilder, 1999). Evaluation


of parasite infestation of insects under study is sometimes possible, too. Weather data are provided by weather stations situated closest to the sampling sites.

The introduction of detailed descriptions of the effects produced by different factors on the population dynamics of the model insect species into the simulation model makes it too complex, the number of unknown parameters in model equations increases dramatically, and verification of this complex model becomes very inconvenient.

1.3 The phenomenological theory of forest insect population dynamics: the principle of stability of flexible ecological systems

An original approach to describing forest insect population dynamics was developed by the “Krasnoyarsk school” of forest ecologists (Isaev, Khlebopros, 1973, 1977; Isaev et al., 1982; Isaev et al., 2001; Isaev et al, 2009). It is based on the phenomenological theory, which uses regulatory mechanisms of the entire system as characteristic parameters of population dynamics.

Analysis of phytophagous insect population dynamics is based on us-ing the method of phase portraits and determining characteristic parame-ters that reveal interactions in forest ecosystems (Isaev, Khlebopros, 1973; Isaev et al., 1984; Isaev et al., 2001; Isaev et al., 2009). Using this ap-proach, researchers are not restricted to the analysis of forms of popula-tion regulation; they can examine regulating mechanisms of the entire ecological system, its stability and delays, different types of direct rela-tionships and feedbacks, ranges of parameter values for certain effects, the dimensions of phase portraits and their qualitative content. Moreo-ver, using this approach, we do not need to relate the estimates of forest ecosystem parameters to specific environmental factors. We regard this theory of forest insect population dynamics as phenomenological, i.e. describing general principles of functioning of the ecological system as a diverse natural phenomenon. This approach can be used to obtain new theoretical knowledge regarding mechanisms of forest insect population dynamics, find out the conditions under which different situations occur in nature, and develop a new classification of outbreak types. Differential characterization of population phase portraits can be used to classify the types of insect population dynamics, based on the functional impact of a particular insect species in the forest ecosystem.


The strategy of forest protection must be based on the knowledge of the laws of forest insect population dynamics. It is only natural that this issue has caused active discussions in the Russian and international lit-erature. The concept of self-regulation of insect populations seems to be the most well-founded; it has provided the basis for the synthetic theory of outbreaks. This theory, however, is unable to solve many of the cardinal problems of population dynamics because it lacks a mathematical tool that would describe the process of self-regulation in flexible ecological systems. The development of this tool based on specific properties of par-ticular groups of forest insects will make it possible to isolate effects of different factors and analyze successive phases of changes in the abun-dances of insect populations in forest ecosystems.

Changes in insect abundance can be considered as a wave process, with certain amplitudes of fluctuations that occur in forest ecosystems under the impact of environmental factors. Researchers that described biological wave processes were usually more interested in fluctuations of the system than in the mechanisms responsible for its stability. Based on the analysis of stability, though, one can both evaluate the variable and determine the conditions for reducing its fluctuations, which can be use-ful for finding the way to control insect populations in natural ecosys-tems. To formalize these processes, we should define certain general no-tions, which represent the main aspects of interactions in forest ecosys-tems. These notions, united in the principle of stability of flexible ecologi-cal systems, are based on the following theses:

1. The ecosystem, as a self-regulating biological system regarding any of its parameters (e.g., insect abundance and reproduction coeffi-cient) has the stability domain, in which it does not undergo any significant changes.

2. Every population has a stable density determined by the negative feedback and appropriately included in the general mechanism of the stability of the forest ecosystem.

3. In the phase plane of the system, there are characteristic curves, the crossing of which determines the qualitative changes in population dynamics.

4. Successions of forest ecosystems are incomparably longer than in-sect life cycles; thus, characteristic points and curves of the system can be considered static relative to the population dynamics.

Based on the principle of stability of flexible ecological systems, we can use our method to determine characteristic points and curves of the spe-cies population dynamics: 1) stable density of sparse population 1x ,


which is typical of most species in a stable ecosystem; 2) threshold curve ,ry the exceeding of which (by reproduction coefficient, at a constant density) leads to an outbreak; 3) optimal density cx in the outbreak phase, which corresponds to the maximum reproduction coef-ficient; 4) limit density Tx – the boundary of the outbreak peak; 5) metastably increasing density 2 ,x which causes long-term destructive impact of insects on forest ecosystems. Determination of characteristic points and curves of the system is useful not only for revealing principles of population dynamics of various insect species but also for assessing human-related impacts that disturb the stability of forest ecosystems. Based on these concepts, we have developed mathematical models for constructing more detailed phase portraits and predicting (as a result of their analysis) adverse effects of human impact on forest ecosystems.

When applied to forest insect population dynamics, the principle of stability can be used to explain and predict the behavior of the spatially separated “forest – phytophagous insect” system, based on the analysis of paired interactions between insect populations, including “phytopha-gous – entomophagous” local subsystems. Behavior of the “phytopha-gous – entomophagous” system is best described by using the basic por-traits that differ in the number of equilibrium modes and their charac-ter. The principal modes of population behavior are the stable state with the sparse stable population density and the oscillatory mode of the populations of increased density, including different types of out-breaks.

1.4 Classification of the factors of forest insect popu-lation dynamics

Under certain ecological conditions, changes in the population density of a forest insect species are influenced by various factors: predators and parasites, the state of host plants and the amount of the available food, weather, etc. In order to construct generalized models of population dy-namics, it is necessary to synthesize and classify all diverse factors of dy-namics by using the same approach.

The negative feedback principle, developed in the phenomenological theory, implies that reproduction coefficient is a function of density val-ues, ( ),y y x which decreases monotonically when the values of other parameters of the ecosystem remain unvaried. This function can be pre-


sented as ( ) ( ),y x Kf x where ( )f x reflects the character and intensity of the feedback, i.e. mechanisms of the system regulation, and K is a generalized modifying parameter.

To decide whether a factor is a modifying or a regulating one, it is es-sential to know the way it behaves towards population density. If a change in factor C only affects the value of parameter ,K this is a modifying factor (such as weather); if a change in factor C affects the behavior of function ( ),f x this is a regulating factor. If the correlation

max mi( , ) ( , )n 0,CC CC

y x C y x C

is valid at all permissible values of population density ,x where C is the domain (or a certain subdomain C ) of factor C values, i.e. if fluctua-tions in factor C do not cause any significant variations in the reproduc-tion coefficient, C can be considered neutral towards the population ex-amined (Isaev et al., 2001).

If the factor substantially affects the population reproduction coeffi-cient, ( )K K C and ( , ),f f x C in order to class it with modifying or regulating factors, one needs to determine the major (dominating) chan-nel of its influence on y , i.e. to find out which of the cofactors contrib-utes most to the change in the reproduction coefficient.

The classification of the factors was performed under unvaried values of population density, in other words, we examined different trans-sections of the ( , )x C half-plane by straight lines constx (Isaev et al., 2001). The resulting sections were not uniform, suggesting varying eco-logical loads of each factor. One example is a change in the significance of food for the regulation of abundance of defoliating insects. When the insect population density is sufficiently low and the food resource is un-limited, the dynamics of its biomass does not produce any noticeable ef-fect on the reproduction coefficient. In this case, food is a modifying fac-tor. During the peak of the outbreak, when the population of defoliating insects experiences food shortage, this factor functions as a regulating one.

The transformation of food from a modifying factor to a regulating one, with positive feedback, accounts for the occurrence of the metastable state of insect populations under certain ecological conditions (Berry-man, 1981, 1982; Berryman, Stark, 1985; Isaev, Khlebopros, 1973, 1977; Isaev et al., 1982; Isaev, Girs, 1975).

The functional relationship between the population density and the reproduction coefficient can be used to construct phase portraits and de-termine the characteristic points in population dynamics of forest insects


that mark qualitative changes in the behavior of the whole system. The development of the theory in this direction requires differentiation of the regulating mechanisms into delayed and direct ones.

1.5 Delayed and direct regulation mechanisms The issue of including delay in equations of population dynamics has been discussed in a number of case studies (Berryman, 1981, 1982, 1991, 1994, 2003; Berryman, Stark, 1985; Hutchinson, 1947, 1948; Murray, 2002; Poluektov et al., 1980; Smith, 1968, 1974). The delay effect is of in-terest as a manifestation of the general properties of the self-regulating ecological system.

In the analysis of the self-regulation process, the delay value can be characterized by parameter , i.e. the relaxation time of the regulating mechanism. A characteristic feature of the delay of the system is that the regulating effect of biological factors is rather determined by the density of the previous generations than by that of the current one. Assuming that the population density remains unchanged through several genera-tions, there will be no delay effect in the system. As population density in the forest ecosystem changes continuously, reproduction coefficient is not equal to 1. The difference characterizes the magnitude of delay of biotic factors relative to the population size of a given species. When the population density decreases dramatically, biotic factors first produce stronger impact on the population (negative delay) and then, if the popu-lation density rapidly grows, in some time this impact is reduced (posi-tive delay).

Delay is characteristic of all regulating mechanisms, but the value of the parameter for each of them may be significantly different. If the re-laxation time is considerably shorter than the length of the generation cycle of a given species, the regulating mechanism functions as a direct one. If the delay time is comparable to the length of the generation cycle, the regulating mechanisms operate as delayed ones.

Delayed mechanisms functionally depend on the density of the previ-ous generations and, hence, may have a long relaxation period. The func-tioning of direct (or slightly delayed) mechanisms is determined by the density of the current population. There is actually no delay, as they per-form regulating functions during only one life cycle. The presence of de-layed mechanisms with large time lags in the regulation system gives phytophagous insects a potential opportunity to “escape” from their nat-


ural enemies and attain outbreak levels. On the other hand, direct mech-anisms stabilize the rate of population dynamics during the outbreak and allow delayed mechanisms to restore the regulating effect. The predomi-nance of certain regulating mechanisms in each instance determines the qualitative pattern of population dynamics in different groups of forest insects and stability of forest ecosystems.

The differentiation of the mechanisms affecting insect populations based on the time hierarchy of the ecosystem’s components is the essence of the principle of stability of flexible ecological systems, which develops methodological aspects of the phenomenological theory of forest insect population dynamics.

To analyze insect population dynamics, reproduction coefficient can be considered as a function of factors depending on population density, quantity and quality of food, weather conditions, and the delay of regu-lating mechanisms relative to the population size of the given species.

To identify the roles of delayed and direct regulating mechanisms, we assume modifying factors to be relatively constant. We need to make this assumption or else phenomenological analysis of the population phase portrait would be impossible.

As different regulating mechanisms have dissimilar effects on insect population dynamics, one should first evaluate the magnitude of delay of each regulating factor. Obviously, the longest time lag is characteristic of biotic regulating mechanisms related to the effect produced on phytophagous insects by their natural enemies, as their generation periods are the most similar. The delay in the effect of pathogenic microorganisms functioning as “live insecticide” must be much shorter. Inter-specific regulating mechanisms can also be subdivided into delayed and direct ones. Direct mechanisms have an immediate effect, decreasing population size within the lifetime of one generation (competition, cannibalism, in-terference, fecundity variations, migrations). These mechanisms function as direct ones because the regulating effect is achieved within one genera-tion and is only very slightly dependent on the density of the previous generation. In this sense, they are practically delay-free. The second group comprises mechanisms that do not only affect the size of one generation, but also determine the possibility of a numerical change in the subsequent generation (diapause, variations in age composition, speeded-up life cycle of a portion of the generation, genetic and phenotypic polymorphism, sex ratio). The second group also includes mechanisms of the signal effect of population increase (Naumov, 1963), which change behavioral responses of the insects and their physiological state in such a way that the popula-tion density decreases. A characteristic example of such intra-population


effects, unrelated to competition, is phase variability in some Lepidoptera (Iwao, 1962), including major defoliator species (Sharov, 1953). As re-ported in numerous studies, the phase state in some insect populations is related to the population density and serves as an effective mechanism of numerical regulation.

The effect of delayed mechanisms weakens the negative feedback, i.e. makes regulation less rigorous (Isaev et al., 2001). In populations of outbreak species, delayed mechanisms make a greater contribution to the regulation system, offering the insects a potential opportunity to quickly escape from their natural enemies. It is well known, for instance, that in populations of lepidopterous pests, parasites are a major factor in numer-ical regulation of sparse populations. That was proven to be true for the most destructive pest of coniferous forests – the Siberian silk moth (Dendrolimus superans sibiricus) (Kolomiets, 1962; Rozhkov, 1963, 1965). The reason for the large delay of the regulating effect is that abundant species usually have a complex group of natural enemies, characterized by inter-specific competition and high activity of secondary parasites.

Direct mechanisms operate whatever the food availability for entomophagous organisms is, but their regulating effect is amplified as the size of the host population increases. The reason is that as the popula-tion density grows, more and more new direct mechanisms are sequen-tially involved in the regulation effect, and, thus, their relative impact increases.

***

We think that to gain insight into ecological mechanisms of regulation of population density and predict population dynamics of forest insects, we need an approach that would combine the use of universal models, de-scribing critical processes in the population, and detailed analysis of the properties of population dynamics time series for specific populations. For the modeling of forest insect population dynamics to be effective, it should be based on a compromise between the generalized approach and the individualized one. The question is not whether this compromise needs to be found. The real question is how to find it.

17

2 Ways of Presenting Data on Forest Insect Population Dynamics 2.1 Representation of population dynamics data Every researcher beginning to study population dynamics of a particular forest insect species faces a seemingly technical question about the way of presenting the data on changes in population density. It would be wrong to think that data analysis and results obtained are not influenced by the way of data presentation. The choice of the form of presentation deter-mines what aspects of the life and death processes of individuals and generations in the population will be emphasized in the study. Moreover, the way of describing the data will implicitly determine the type of ques-tions about the phenomenon under study.

The data on forest insect population dynamics can be presented by using one of the four main approaches:

time series: presentation in the { , }x t plane; the Lamerey stairs: presentation in the { ( ), ( 1)}x i x i plane; phase portrait: presentation in the { ( ), ( 1) / ( )}x i x i x i or { ( ), ( 1) ( )}x i x i x i plane; potential function: presentation in the { , ( )}x G x plane, where

1( )( )

G xp x

and p(x) is probability of attaining density х by the

population. The first three ways are well known and have been extensively used by

researchers analyzing forest insect population dynamics. The last-




mentioned way is new for describing insect population dynamics, and we have found no mention of it in the literature.

Let us examine the potential of each of these approaches to data presentation. As material for description, we have chosen the “classical” data on population dynamics of the pine looper Bupalus piniarius L. in Thuringen (Germany) between 1880 and 1940 (Schwerdtfeger, 1968) and our own data on population dynamics of the Dendrolimus pini L. popula-tion dynamics in the Krasnoturansk pine forest (in the south of Middle Siberia) between 1979 and 2014.

For the last 37 years (from 1979), the authors of this book had been studying population dynamics of defoliating insects in the Krasnoturansk pine forest (south of Middle Siberia, 54°16.315´N, 91°37.757´E). The landscape structure of the Krasnoturansk pine forest mapped by D. M. Kireev (1977) is shown in Figure 2.1.

Fig. 2.1. A diagrammatic landscape map of the Krasnoturansk pine forest;

1 – Hill Top (tops and steep vertical slopes of the hills); 2 – Narrow planes; 3 – Plakor (gentle slopes); 4 – steep slopes of gullies and ravines; 5 – Dunes

(southward and westward concave gentle slopes); 6 – Terrace (gently sloping terrace-type surfaces); 7 – Lake (flat proluvial floors)

2.2 Presenting the data on forest insect population dynamics through changes in density over time

What can one learn from a study of population density change over time? Firstly, one can obtain information on how the population density fluc-tuates. The parameters that can be used to describe density fluctuations

Ways of Presenting Data on Forest Insect Population Dynamics 19

are the highest and the lowest density values over the observation period, the average density, and standard deviation of the mean. By studying the time series, one can also identify the trend and changes in the amplitude of oscillation of the population density. Based on these data, we can characterize the range of values within which population density varies but cannot understand the reasons for the variations in this process. The data presented in this way do not explain why density fluctuations occur. The fact that the linear or nonlinear trend in the time series exists is important, but if we merely observe the trend, we will not be able to understand the reasons for population density variations.

Important parameters of population dynamics that can also be esti-mated from the time series are cyclic oscillations of population density. For this, methods of autocorrelation or spectral analysis are used (Box, Jenkins, 1974; Anderson, 1976; Kendall, Stuart, 1976). However, autocor-relation function or spectral density function can be calculated correctly only if the time series is stationary and average values and standard deviation of the mean do not change over time (Box, Jenkins, 1974). If these conditions are not satisfied, and there is a trend in population density values and (or) temporal variations in the oscillation amplitude, the time series needs to be transformed into a linear time invariant set (LTI) without losing the properties that we study. The LTI set must have the time invariant average value, i.e. time invariant variance of the aver-age value and temporally stable oscillation frequency of the variable.

Linearization of the time series of defoliating insect population density consists of the following steps:

1. Data repair; 2. The transfer to the logarithmic scale of density and data normaliza-

tion; 3. Detrending; 4. “Fixing” heteroscedasticity; 5. Regularization – filtration of the high-frequency components of the

series and noise reduction. Let us discuss the process of linearizing the time series of insect popu-

lations in greater detail. Graphs of the dynamics of the “reference” time series are shown in Figures 2.2 and 2.3.

The time series of forest insect population density may have gaps in the years when surveys were not conducted for some reasons or zero density values (which happens more often), when no insects of the study species were found during the survey. Possible reasons for the absence of the insects may be their complete absence on the sample plot or a very


Fig. 2.2. Changes in the population density of the pine looper B. piniarius

in Thuringen between 1880 and 1940 (Schwerdtfeger, 1968)

Fig. 2.3. Population dynamics of D.pini in the Krasnoturansk pine forest (sam-

ple plot “Dune”, 1979–2012) low population density – lower than the survey detection limit, which is inversely proportional to the number of sample units (trees, sample plots, etc.). The data repair is necessary because at ( ) 0,x i it is impossible to transform the data into the logarithmic scale of population density, as ln 0 . Hence, the zero density values must be replaced by the values corresponding to the survey error or the values below the minimum population density observed during the entire survey period. Repair can be done in two ways: by switching from the density variable, x, to the new variable, (1 ),x x or by replacing the zero density values with the value of density measurement errors. If the variable is replaced by

(1 ),x x in the case when 1,x (1 ) ,x xx but at 0,x


ln(1 ) 0,x x which is correct, in contrast to the case when 0x and ln0 . Then, however, the repaired series will not contain density values below 1, and variations at the low density levels will even out. If only zero density values are replaced by the values of survey errors, the amplitude of the repaired density values will remain unchanged. In our opinion, in this way we get a more accurate representation of the real situation.

No insects were found in ten surveys of the time series of the D.pini population dynamics, and this series was repaired by replacing the zero values with the minimum values of 0.01 larva per tree.

Then we should go from the linear scale of population density to the logarithmic scale in order to reduce variations in the time series and normalize the resulting data to the average value of the log of population density for n surveys:

1

1( ) ln ( ) ( ) ln ( ) ln ( ).n

ix i x i z i x i x i

n (2.1)

All time series, including time series of population dynamics can be separated into three main components: trend, cyclic components, and noise. Isolation and evaluation of each component is important for understanding ecological processes in the insect population. For example, the presence of a decreasing trend may suggest that the population is being driven out of the ecosystem. The population increase trend is associated with the species invading a new habitat, weather changes in the habitat, or a decrease in the population densities of parasites and predators – enemies of the species under study.

Furthermore, isolation of the trend is important because methods of correlation and spectral analysis can only be applied to the stationary time series (Box, Jenkins, 1974).

The trend can be isolated by using one of the two main approaches. The trend is eliminated after the basic time series { ( )}x i is replaced by the first differences of the series { ( 1) ( )}x i x i : { ( )} { ( 1) ( )}x i x i x i (2.2)

Another common technique is to find the linear regression equation ˆ( )x i A Bi and subtract the value obtained in calculation of the regres-

sion from the corresponding value of the basic series: ˆ{ ( )} ( ) ( ) ( )x i x i x i x i (2.3)

Population dynamics time series are often heteroscedastic: variance of the average value rather than the average value changes over time, sug-gesting changes in the strength of regulatory processes in the population.


In order to reduce the time trend of the population density variance, it is necessary to find the equation for the temporal dynamics of the local maxima in the time series

max ( )x i A Bi . (2.4)

If coefficient B of regression equation (2.4) is significant, the values of the time series that we are transforming at time points i should be nor-malized to the corresponding values of equation (2.4).

For our time series, the time trend is not noticeable (Fig. 2.4 and 2.5), and, thus, we do not need to perform (2.2) and (2.3). Our calculations showed the absence of the trend of maxima in our time series, therefore, we did not need to eliminate the variance trend.

After the initial time series has been transformed into the stationary time series, it would be good to perform regularization. Regularization is removal of the high-frequency (HF) component from the time series. The presence of this component may be due to certain temporal instability of the processes occurring in the population or to inaccurate evaluations of population density during surveys. Filtering of HF components is neces-sary because various mistakes, which are inevitable during surveys, could cause deviations of the values recorded from the actual values. The sim-plest method of isolation of the HF components is to filter them by using various non-recursive filters (Hamming, 1987). The choice of the filter is determined by the specific properties of the object and the type of fluctu-ations in the recorded values of population density. In our research, we used a Hann filter, which filters off HF components of the time series of frequencies above 0.25 1/year:

( ) 0.24 ( 1) 0.52 ( ) 0.25 ( 1)L s x i x i x i (2.5)

As the choice of the filter is determined by the properties of the time series analyzed, other HF filters, even the simple five- or seven-point rectangular filters, can be used (Kendall, Stuart, 1976).

After the initial time series of population dynamics of the pine looper and pine-tree lappet sequentially underwent repair, transformation of density values into the logarithmic scale, normalization, detrending by linear regression, and HF filtration with the Hann filter, we obtained the end product of transformation: LTI sets of population dynamics shown in Figures 2.4 and 2.5.

After this transformation of the time series of population dynamics, one can calculate correctly (by using the Statistica 10.0 package) func-tions of spectral density, which characterize cyclic changes in population density (Fig. 2.6 and 2.7).


Fig. 2.4. LTI set of pine looper population density

Fig. 2.5. LTI set of D.pini population density

Fig. 2.6. Spectral density of the LTI set of pine looper population dynamics

in Thuringen


Fig. 2.7. Spectral density of the LTI set of D.pini population dynamics

in the Krasnoturansk pine forest (Middle Siberia) The presence of the peak of spectral density functions ( )A f in Fig-

ures 2.6 and 2.7 suggests that the time series of population density has cyclic components. The cyclic oscillation frequency of the pine looper population density in Thuringen is 0.086 1/year (Fig. 2.6). The value inverse to the oscillation frequency characterizes the periodicity T (years) of the occurrence of pine looper outbreaks in this area during the years of observation. For the model population of the pine looper,

10.086 11.6T years. For the D.pini population, 10.0625T 16 years.

After the long-term survey data have been transformed and presented as a stationary time series, these time series can be modeled by the com-monly used methods of autoregressive analysis. These approaches to analysis and modeling of the time series of forest insect population dy-namics will be described in detail in Chapter 7.

2.3 Presenting the data on population dynamics as a phase portrait

The data on population dynamics presented in the { , }x t plane can be transformed and presented in the { , }x y phase plane. In this case, dynam-ics of the populations with non-overlapping generations is usually char-acterized by the value of population density of the ith generation ( )x i and


its reproduction coefficient ( )y i – the ratio of the population densities of two adjacent populations:

( 1)( )( )

x iy ix i

(2.6)

The aggregate of the points in the { , }x y phase plane, which corre-spond to the adjacent moments in time 1, 2,…, n, n + 1, …, form the population phase trajectory (the reproduction curve), characterizing changes in population density over time. The reproduction coefficient can be replaced by the parameter of population density growth rate

( ) ( ) ( )dx t x t t x tvdt t

. For populations with non-overlapping

populations and one survey per generation (i.e. when 1t ), ( 1) ( ).v x i x i

All points in the phase plane for which 1y or 0v characterize the stationary state of the population (Andronov et al., 1966).

Phase portraits may be narrow or broad. For populations whose dy-namics is characterized by the narrow phase portrait, every x value has one corresponding y value. For populations with broad phase portraits, there is no definite relationship between the growth rate and population density, and one density value may have several corresponding values of reproduction coefficient.

Analysis of population dynamics of forest insects in the stable sparse state, in which the insect density and growth rate are not high, should start by presenting the population dynamics in the { , }x y phase plane. Figure 2.8 shows characteristic curves and points of the phase portrait of forest insects in the stable sparse state.

Fig. 2.8. Characteristic curves and points of the phase portrait of forest insects


Values x1, x2, xc, and xr can be used to characterize the population phase portrait. Domain I characterizes the stable sparse state of the population. The point in the phase plane within the boundaries of the stability domain corresponds to some state of the population with densi-ty x and reproduction coefficient y. Curve S characterizes the adiabatic phase trajectory for the particular population. Within the boundaries of the stability domain on the phase trajectory, there is one point corre-sponding to the stable state of the population, in which its density is equal to x1, and reproduction coefficient y = 1. Point x1 attracts all popu-lation states within the stability domain. The boundaries of the stability domain in the phase portrait are characterized by curves yc and yr. As the population density drops below xc and is transferred to Domain II (left of the lower critical boundary yc), the population enters the domain of attraction of 0x and dies out at a certain rate, so that ( ) 0.x If the population density increases above the critical value xr, and it is trans-ferred to Domain III, the population enters the domain of attraction of x2. The state with density x2 and reproduction coefficient y = 1 corre-sponds to population outbreak.

Another important parameter of the population phase portrait is the population density at which its growth rate is the lowest. This value is traditionally named the Allee density xA (Allee, 1931; Liebhold, Bascompte, 2003). As shown in Figure 2.8, point {xA, yA} is not stable, and usually the population does not stay in this state for a long time, but moves to the stable state with density х1.

If the state of the population changes within the stability domain in the phase portrait, its density is maintained close to density х1, and no quali-tative changes related to the effect of the population of the forest insect species under study on different components of the ecosystem (such as trees) occur in the ecosystem. The population density within Domain I is rather low, and the amount of the phytomass consumed by the insects is not large.

In this case, it is important to identify the conditions under which the population will not leave the stability domain and its density will vary between хс and хr.

Stability of model populations is usually evaluated by using the Lyapunov stability. Population is Lyapunov stable if it resumes its initial state with some density x and reproduction coefficient y = 1 some time after the impact of some factor has caused it to leave state x1 (Andronov et al., 1981).

In the static phase portrait of the population, there may be five sta-tionary states with y = 1, three of which (x0 = 0, x = x1, and x = x2) are


Lyapunov stable (Fig. 2.8). Point x0 = 0 characterizes the situation with no insects of the study species in the region. Point xc (the lower critical density) is unstable. If at some moment, population density is below xc, the population inhabiting the study region will die out. If at some moment, the population density is above xc, the population inhabiting the study region will tend to x1, which characterizes the stable sparse state. Stable (or metastable) point х2 with reproduction coefficient also equal to 1 characterizes population outbreak. Unstable upper critical point xr characterizes the boundary between the x1 and x2 attraction domains.

The Lyapunov stability of theoretical models of population dynamics is evaluated by using the stability criteria proposed by Gurvitz, Raus, Mikhailov and others (Andronov et al., 1981; Bautin, Leontovich, 1976; Kim, 2007). However, a stumbling block to the use of theoretical criteria for evaluating the stability of actual insect populations is that for forest insect populations in certain habitats, critical values хс, хr, and х1 of the phase portraits are not known and cannot be found based on rather short available time series of population dynamics.

As reproduction coefficient and population density cannot change in-definitely and must be limited by certain values, first, we need to deter-mine the boundaries of the phase portrait – limit curves y and y (Fig. 2.9). The biological meaning of these limit curves is that under small fluctuations in modifying factors, these curves limit the phase space within which the phytophagous insect – ecosystem interactions can occur.

The limit curves have another important property. The aggregate of the points that make up the boundaries of the phase portrait characterize the maximum time lag of delayed regulating mechanisms. The 0y y static curve, with the time lag equal to zero, divides the phase portrait into two parts: with the positive time lag and with the negative one. The intersection of 0y y with the 1y curve determines the stability point of the system, 1.x

The boundaries of the phase portrait under small fluctuations of modi-fying factors are characteristic of a given ecosystem; they change over the time comparable with the succession time periods in the forest ecosys-tem. The boundaries of the phase portrait can be changed significantly if a different ecosystem is examined.

If the effects of natural enemies, as delayed regulating mechanisms, are gradually excluded from the ( , )x y system, i.e. if their relative contri-bution is decreased, the lower curve will approach the upper one and


the phase space will tend to zero. Then the stability point will move along the 1y straight line, and in the extreme case, 1 Tx x . This trend in phytophagous insect stable density variation characterizes ecosystems in which human impact reduces populations of natural enemies and dimin-ishes their regulating effect. In such tree stands, certain insect species become more abundant and form stable high-density populations.

Fig. 2.9. The phase portrait of the phytophagous insect population dynamics. I – the region of slightly delayed regulating mechanisms (limited by curves y and ry ); II – the region of maximally delayed regulating mechanisms (limited

by curves ry and cy ); III – the region of direct regulating mechanisms (limited by curves cy and y ). Curves: y – the lower boundary of the phase portrait,

y – the upper boundary, ry – threshold, cy – buffer, 0y – static. The outbreak phase trajectory: bc – the population buildup phase, cd – peak, de – decline, eh – low density, ha – recovery of stable density. Characteristic points: 1x – stable population density, rx – threshold density, cx – optimal

density, rTx – minimum density during the peak phase, Tx – maximum density If natural enemies served as direct regulating mechanisms, the upper

and the lower boundary curves would tend to static curve 0y , and the stable density would retain its value 1x . Then the phase space would tend to zero. Thus, if we assume that there are no delayed regulating mecha-nisms in the ecological system, its phase portrait will be generally repre-sented by one line, 0y y .


Phase portrait structure. If the boundaries of the phase portrait are de-termined by the presence of delayed regulating mechanisms, its structure depends on the character of interactions between the insects and their natural enemies. As already noted, a qualitative change in these interac-tions occurs at the threshold points of food availability 1 2( , ) and is characterized by the /y derivative. If the set of threshold values 1 and 2 is transferred to the ( , )x y plane, three characteristic domains are formed within the boundaries of the phase portrait (Fig. 2.9). Depending on whether 0y occurs in Domain I, II or III, the interaction of the species with its natural enemies can be qualitatively different.

Within the phase portrait, curves 3 and 4, corresponding to the aggre-gates of points of the first threshold value 1( ) and the second one 2( ), delineate the domain of the maximum delay of regulating mecha-nisms (II). In this domain, it is easiest for the insects to escape from their natural enemies because this is the only domain where there can be positive feedback leading to an increase in the number of pests.

In Domain I, regulating mechanisms of the ecosystem have a short time lag ( ),T and, thus, these mechanisms function as slightly de-layed ones.

Domain III is the region of direct intra-specific mechanisms ( ).T As there is no delay in this domain, fluctuations in population reproduc-tion coefficient follow the fluctuations in external factors.

Depending upon the ecological features of the species and its interac-tions with its natural enemies, the portrait can correspond to the follow-ing situations:

1. If delayed mechanisms control the number of phytophagous insects in Domain II rigorously enough, all the three domains of the phase portrait represent a stability region, and the intersection of 0y with the y = 1 line is the point of the stable size of population 1x . In the stabil-ity region, the system drifts around the stability point, and the highest velocity must be observed in Domain II.

This structure of the phase portrait is typical of the majority of phytophagous forest insects. These include both indifferent species, whose populations are rather stable, and species with widely varying population densities. However, these species (such as most of wood-eating insects) do not actually reach outbreak levels because variations in their population densities are determined by fluctuations of modifying factors. As soon as the fluctuation stops, the regulating mechanisms rapidly return the population into the stable state. Such systems are characterized by the absence of positive feedback in the case of popula-


tion density increase. This does not allow the population to completely escape from its natural enemies and reach a qualitatively different level – outbreak.

2. If the delayed regulating mechanisms cannot wholly control popula-tions of phytophagous insects, Domains II and III should be considered as the outbreak region. In this case, curve 3 will correspond to threshold curve ry , curve 4 – to the buffer curve, limiting the region of outbreak development, and curve 2 – to the upper boundary of the phase portrait (Fig. 2.10).

Fig. 2.10. The structure of the insect outbreak region. 1 – the safe section,

2 – the critical section, 3 – ecosystem destruction; ,d d – intersection points of the boundaries of the sections and the 1y line.

Then, the stability region can only be in Domain I, limited by the low-

er boundary curve (1) and threshold curve ry . This phase portrait is typical of the species that can escape from their natural enemies and dramatically increase their population size. These species are the most hazardous forest pests because during outbreaks they can damage the forest ecosystem considerably and even ruin it. Outbreaks of these species are qualitatively different from those of the species belonging to the previous group. This can be accounted for by specific interactions of the population with the forest ecosystem as it moves along the phase trajecto-ry of the outbreak.

The outbreak trajectory in the phase portrait. As found previously (Isaev, Khlebopros, 1973, 1974), in the undisturbed (stable) ecosystem, insects occur in sparse, stable populations. The reproduction coefficient of a sparse population is close to unity and its density is close to 1x . In


the absence of external fluctuations, the system can stay at point 1x over indefinitely long periods of time, as the rate of numerical change 1y , and, hence, acceleration will be equal to zero. If random external fluctua-tions unbalance the population, sooner or later the negative feedback will make the system return to stable state 1x . The larger the feedback, the sooner the system comes back.

Forest insects have different infestation capacities, and most of them do not achieve outbreak levels. Their population densities and reproduc-tion coefficients fluctuate little, mainly because of the large feedback factor.

For quite a number of forest insects that have one point of numerical stabilization, reproduction coefficient y can be described using the fol-lowing function: ,xy APe where parameter А corresponds to the effect of the food factor, its quality and quantity, on the population; Р is the effect of modifying weather conditions; characterizes the feedback value.

Point 1x – stable population density – corresponds to 1y and 0 and is located in the domain of the effect of direct mechanisms; then 1 ,xAPe

Solving this equation for x , we obtain

11 ln( ).x AP

As evident from the equation, the rate at which the density of the sparse population 1x increases with growing А or Р values is inversely related to the value of regulating factor .

In undisturbed ecosystems, the value of , characterizing indifferent species, is sufficiently stable and rather high. Human activities (treatment of vast areas with pesticides, removal of nectariferous plants, burning of forest litter, etc.) often cause a “sudden” numerical increase in the popu-lations of normally innocuous species.

Species capable of wider density variations have special relationships with the ecosystem. In the healthy forest, these species also occur in sparse populations, i.e. are in stable state 1x . In more favorable habitats, the 1x value is higher while in less favorable ones, it is lower. Yet, sparse populations of abundant species have a high potential for numerical growth, which is ultimately determined by the specific habitat. The quality of the habitat determines the magnitude of the impact produced


by natural enemies. For sparse populations, the regulating effect of natu-ral enemies is most pronounced in the optimal habitats, and this signifi-cantly restricts the counteraction of other environmental factors.

The stable state of sparse populations is “preset” by the ecological sys-tem itself. Variations in 1x within the limits of stable functioning of the ecosystem are determined by variations in modifying and regulating factors, depending on the character of the ecosystem. Thus, density 1,x which corresponds to the stable state of the population, is not the same in different types of ecosystems. In optimal biotopes, feedback factor grows somewhat, thus compensating for the change in 1x . This process is most evident in populations of the insect species that have a rather small value of the negative feedback factor. In biotopes with the most unfavorable weather conditions (high altitudes, habitat borders, etc.) the effects of regulating factors are weakened, and modifying factors (mostly weather conditions) play a dominant role.

Thus, every insect species has a definite population density, 1x , which corresponds to a definite type of forest ecosystem. Stabilization of the population size within these limits is a basis for dynamic stability of the ecological system and fitness of the species.

If the amplitude of fluctuations in modifying factors is large enough to “eject” the population from the stability region, outside the threshold curve ry , an outbreak begins. The outbreak consists of five qualitatively different phases: population buildup, peak, decline, low density, and recovery of stable density. Let us examine population performance in different phases of the outbreak, taking into account the effects of de-layed and direct regulating mechanisms.

The population buildup phase (segment bc) takes place in the domain of maximally delayed regulating mechanisms (II). When the population enters this domain, the negative feedback is replaced by the positive one and the delay time of regulating mechanisms grows to its maximum. Thus, the population escapes from its natural enemies and rapidly grows numerically, with the reproduction coefficient also increasing. Should the factors restricting the system dynamics be excluded, the population size would grow endlessly because the difference between the population density and the regulating effect of the species’ natural enemies would increase continuously. However, the numerical increase is limited by Domain III – the region of direct regulating mechanisms. The buildup phase ends in the characteristic point c, which corresponds to the maxi-mum reproduction coefficient. This is the outbreak optimum, associated with the qualitative change in the state of the population.


The positive feedback observed during the population buildup phase operates not only through the escape from natural enemies but also through the functioning of intra-specific mechanisms that are responsi-ble for the increase in pest population density.

After the population passes through point c, the outbreak peak phase begins (segment cd in Fig. 2.10). It occurs in Domain III, in which regu-lation is mainly performed by direct mechanisms. Their participation in the regulating system inverts the sign of the feedback, causes the repro-duction coefficient to decrease, and gradually shortens the time lag. Thus, natural enemies “catch up” with their prey, whose numerical growth is hindered by the operation of direct regulating mechanisms. During the peak phase, competition becomes more severe, and, thus, survival and fecundity are reduced, leading to a decrease in the pest population densi-ty. Large numbers of insects migrate, leaving the outbreak site, and the sex ratio changes in favor of males. Specialized parasites acquire greater significance in density regulation.

If we assume that in Domain III natural enemies and diseases are not capable of controlling phytophagous insect population density, the outbreak trajectory in the peak phase will end in point Tx , where popula-tion density is stabilized at the highest possible level determined by the regulating effect of direct mechanisms only. The quickest possible impact of natural enemies on the pest population will cause the phase trajectory to go through threshold point rTx . In every particular case, the peak phase ends between these limit values, after crossing the buffer curve (point d in Fig. 2.10).

At the end of the peak phase and during the transition to the decline phase (segment de) delayed mechanisms acquire special significance. They actually take over from direct regulating mechanisms, whose effect cannot return the system to its initial state. The presence of natural enemies, whose abundance increases much faster than that of their host, causes the outbreak trajectory to intersect the line 1y , where the rate of pest population increase is equal to zero and the negative feedback has the largest value.

During the decline phase, the system returns to Domain II, where the effect of delayed mechanisms increases manifold due to the dramatic increase in the abundance of natural enemies and, hence, the decrease in the amount of food available for them. Thus, during the decline phase, phytophagous insect population density subsides at an increasingly high rate. The intra-population mechanisms also act to decrease the popula-tion density.


The movements of the system to Domain II during both the buildup phase and the decline phase have one common feature, determined by the presence of the positive feedback. During the buildup phase, the positive feedback enables the host population to escape from its natural enemies while during the decline phase it works in favor of natural ene-mies, allowing them to quickly “catch up” with their prey. It is interesting that the density of a given population increases and decreases at similar rates. This fact was experimentally proven in the study that analyzed population dynamics of the Siberian silk moth Dendrolimus superans sibiricus Tschetv. (Kondakov, 1974).

Once the population has crossed threshold curve ry , it is back in the stability domain (I). However, prior to population stabilization, there is a phase of extremely low density (segment eh), with the lowest reproduc-tion coefficients and the strongest impact of natural enemies. The latter condition is accounted for by the persistent positive feedback and a rapid decrease in the delay time of regulating mechanisms. The low-density phase is followed by the gradual recovery of the stable state, with the reproduction coefficient increasing first and population density gradually stabilized later.

During the final phase of the outbreak, the population recovers its stable density (segment ha). When the outbreak is over, the population reaches the stable state and retains stable density for a certain period of time.

In nature, however, outbreaks can persist. In the phase portrait of in-sect species capable of sustained outbreaks, static curve 0y is transferred from the stability domain (I) to the outbreak domain (II). If 1 rx x , the intersection point of static curve 0y and line 1y stops being the nu-merical stabilization point, and Domain I cannot be considered as the stability region.

In this specific situation, the phytophagous insect population density can remain close to outbreak levels almost continuously. When an out-break cycle is over, the pest population density does not stabilize in point 1x , so even under small fluctuations of modifying factors, the population again reaches outbreak levels and goes through the five phas-es of the phase trajectory.

A necessary condition for the functioning of such systems is a relative-ly weak impact of the insects on forest ecosystems. Even during out-breaks, the insects must not unbalance forest ecosystems. An example of sustained outbreak is population dynamics of the larch bud moth (Zeiraphera diniana Gn.) in forests of the Swiss Alps (Baltensweiler, 1958, 1964, 1970, 1978).


Pest outbreaks frequently cause serious destruction of the tree stand, which naturally leads to the death of the major portion of their popula-tion. Potential impact of insects on forest ecosystems can be predicted from the behavior of the reproduction curve crossing the characteristic structural parts of the outbreak region. The outbreak region is limited by threshold curve ry and limit curve y ; it is located in the 1y part of the phase plane. When any of the phase trajectories of outbreak devel-opment crosses the 1y line, i.e. when the peak phase is succeeded by the decline phase, the trajectory has abscissa d at which l Tx d x (Fig. 2.10).

Based on the intensity of the effect of insects on the forest ecosystem, the outbreak region can be divided into three different sections. The section safe for the ecosystem is limited by the characteristic outbreak phase trajectory crossing point ,d where d is the largest population density at which the ecosystem does not suffer any significant impact of the pest. If outbreak phase trajectories stay within the boundaries of this section, the forest ecosystem retains its structure and composition. Eco-nomically speaking, such outbreaks do not have to be controlled; moni-toring of the pest population dynamics usually suffices.

The section of potential hazard in Fig. 2.10 is situated between two phase trajectories crossing points d and ,d which lie on the 1y line, where d is the lowest population density at which the ecosystem is completely destroyed. Under the impact of modifying factors on the population dynamics, the system can be “ejected” from the second sec-tion into either the safe section (1) or the ecosystem destruction sec-tion (3), and in the latter case, consequences will be detrimental. Forest ecosystems suffering the impact of crowded pest populations should be protected by taking special measures.

The use of the phase plane to describe dynamic processes in the popu-lation, instead of the traditional presentation in the time – population density plane, can be very effective for describing fundamental properties of the population.

Which of the parameters indicating changes in population density should be used to describe forest insect population dynamics? In terms of elementary algebra, for example, it does not matter whether reproduction coefficient or relative rate of population growth is used.

Indeed, ( 1) ( ) ( 1)( ) 1 ( ) 1( ) ( )

x i x i x iy i y ix i x i

. If ( ) 1y i or 0v ,

during the survey, population density did not change, and the population was in the steady state.


However, if we construct the phase portrait based on reproduction co-efficient, zero population density values in one of the survey years may be a problem. If in year i, ( ) 0,x i and in year ( 1),i density ( 1) 0,x i calculations of the reproduction coefficient or relative growth rate of the population will give an ecologically meaningless value: ( ) .y i

Another issue that should be taken into account while choosing the parameter in the phase plane is the accuracy of measurements. Let the population be in a stable stationary state and its density be low. Theoreti-cally, at any absolute value of the density of a stable population, its repro-duction coefficient must be equal to 1. During actual surveys, however, if the population density is low, any small accidental mistake may result in ( 1) 0.02,x i instead of the theoretically expected density of a popu-lation that has remained stable for two years of surveys: x(i) =

( 1) 0.01.x i Then, however, instead of the theoretically expected reproduction coefficient ( ) 1,y i we obtain ( ) 2y i (!). Results of sur-veys of forest insect populations with low densities show that reproduc-tion coefficient values can deviate significantly from 1.y

Let z be indirectly estimated from the recorded values x1 and x2. If, for

instance, 1 2 ,z x x then the relative error 2 2

1 2

1 2

( ) ( )x xzz x x

(Zaidel, 1985). The lower the absolute value of the difference 1 2( ),x x

the larger the relative error of evaluating this difference. If 1

2

xzx

,

'( )z ( )

f xz Xf x

. Then

2 2

1 21 2

1 1 .( ) ( )

f fz x xz f x x f x x

Thus, for indirect values determined in different ways, relative error will vary depending on how this indirect value has been determined. The relative error of the indirect value determined from the difference be-tween parameters measured with the same error will be smaller than the relative error of the indirect value determined from the ratio of the meas-ured values. Thus, if, for the description of the phase plane, we introduce the parameter of the growth rate of the insect population with one-year generation ( ) ( 1) ( )v i x i x i , the relative error of the growth rate will be smaller than the relative error of the indirect value of reproduction

coefficient ( 1)( )( )

x iy ix i

.


Figures 2.11 and 2.12 are phase portraits of B.piniarius population in Thuringen and D.pini population in the Krasnoturansk pine forest (Mid-dle Siberia).

Figures 2.11A and 2.12A give only a vague idea of population dynam-ics of these species. At a low population density, reproduction coeffi-cient could be much higher than 1 and even reach 20. Reproduction coefficient ( ) 20y i corresponded to population densities ( ) 0.01y i

A

B Fig. 2.11. Phase portraits of B.piniarius in Thuringen. A – in coordinates

{ln x(i), ln [x(i + 1)/x(i)]}; B – in coordinates {L(i), L(i + 1)}


A

B Fig. 2.12. A phase portrait of D.pini population in the population density –

reproduction coefficient plane in habitat “Dune” (1979–2012). (A – in coordinates {ln x(i), ln [x(i + 1)/x(i)]}; B – in coordinates {L(i), L(i + 1)}

and ( 1) 0.2,y i i.e. pine-tree lappet population densities in those years were very low, but reproduction coefficient was very high, considerably higher than 1. Of course, in this case, reproduction coefficient cannot be regarded as an indicator of the state of the population.

Thus, the theoretically accurate presentation of insect population dy-namics in the population density – reproduction coefficient plane is not quite suitable for the analysis of the population survey data. The large error in calculations of reproduction coefficient for low population densities leads to considerable variations in the reproduction coefficient values at low population density, and results of calculating this parameter based on the survey data may be substantially different from the theoreti-cal value, which is equal to 1 (Fig. 2.13).


Fig. 2.13. Distribution of the reproduction coefficient values of D.pini popula-

tion in habitat “Dune” (1979–2012) The data can be presented in the phase plane in the form of the so-

called Lamerey stairs in the {x(j), x(j + 1)} plane. In this case, stable states of the population are characterized by points of the bisector x(j) = x(j + 1). Figure 2.14 presents the model series of pine looper population dynamics as the Lamerey stairs.

Fig. 2.14. Presentation of the model series of pine looper population dynamics

as the Lamerey stairs. However, the capabilities of these two ways of data presentation are

basically the same. We think that the choice of the phase portrait or the Lamerey stairs for presenting the data is a matter of taste and personal preference of the researcher or (rather) his/her teachers.


2.4 The probability of the population leaving the sta-bility zone and reaching an outbreak density: A model of a one-dimensional potential well

Forest insect outbreak is characterized by a rapid (over two to three years) and significant (by several orders of magnitude) increase in insect population density on vast areas (Isaev et al., 2001). This is a very inter-esting event both because it is important to determine ecological mecha-nisms of such dramatic changes in the population and because forest insect outbreaks damage and kill forests, while an accurate forecast may initiate timely protective measures.

The risk of an outbreak R may be characterized as potential damage to the forest or its death in the next few years. The current outbreak risk at time t can be defined in the same way as it is defined in the problems of the risk theory (Korolev et al., 2007): e.g., as the product of the prob-ability (( ))p x t that the population will reach some density x(t) higher than the critical value Хr multiplied by the damage W(t) caused by insects. Damage W(t) may be defined as the product of the expected population density x(t) multiplied by the phytomass M(x) of food (leaves or needles) consumed by an individual in the population of density x(t): ( ) ( ( )) ( )R t p x t x t M (2.7), where ( ) ( ) .rx t x t x

Based on (2.7), the average risk R can be determined by knowing the density function ( )f x of the expected value x:

0

( ) ( ) ( ).R M x x f x xd x (2.8)

Outbreak risk assessment may be used as a basic parameter when mak-ing a decision about taking protective measures.

In the simplest case, we may assume that M = const, which is reasona-ble enough but not quite correct, as during the outbreak, the mass of larvae and, hence, the mass of the food consumed by one larva, vary (Isaev et al., 2001). In this simplified case, outbreak risk will be deter-mined by the probability of the population density exceeding a certain critical value, xr. It follows from (2.8) that in the case when р( x) = 0, the risk of an outbreak is also equal to zero.

Thus, two questions arise: How many and what sort of data are neces-sary in year t to assess the probability of outbreak occurrence in year


(t + k)? and how can one estimate a possible level of damage to tree stands caused by insects? Clearly, k cannot be large – no more than 2–3 years. Short-term forecasts of forest insect population dynamics must be based on the current parameters of the population, also taking into account the sensitivity of the population to the effects of regulating factors (parasites and predators) and modifying factors (primarily, weather), and the quality and amount of available food. In the simplest case, the description of the state of the population may be only based on the data on population density. In this case, however, the forecast will be unable to predict the directions of changes in population dynamics. Populations of the same density may be in different phases of the out-break cycle: either in the population buildup phase, which starts the outbreak, or in the critical phase, which is usually followed by a dramatic decrease in population density (Isaev et al., 2001).

Hence, parameters describing the current state of the population must include not only its current density but also the direction of change in the density. Three approaches to data presentation may be used to describe population dynamics in this way: presentation in the (time t – population density x) plane, presentation in the plane of temporally adjacent popula-tion densities x(t) and x(t + 1) (the “Lamerey stairs”), and presentation in the {x, y} phase plane, where y = dx/dt are changes in the population density. For a forest insect population with non-overlapping generations, rate y can be determined from the data on population densities of two adjacent years:

( ) ( ) .dx x t t x tydt t

The forecast of the occurrence of insect outbreak is necessary if the population is in the stable sparse state, and its density is quite low (Isaev, Khlebopros, 1973; Isaev et al., 2001). To define the limits of possible changes in the state of the population within the stability zone, the lower and upper boundaries of this zone, described by curves ( )y x and ( ),y x are plotted on the phase plane (Fig. 2.15).

It is assumed that the boundaries of the stability zone do not change over time and characterize the population of a given species under defi-nite natural conditions (Isaev et al., 2001). As the population passes from the stable sparse state to the outbreak, both population density and its growth rate increase dramatically, and the point on the phase plane is ejected beyond the boundary of the stability zone. However, the closeness of the current population density x1 to the critical value xr (point 1 in Fig. 2.15) is not a definitive indicator of the outbreak risk. At a lower


population density, x2, its state may be “closer” to the boundary curve (point 2 in Fig. 2.15). Thus, current population density alone is not a sufficient parameter for outbreak risk assessment, as the low value of x is not the only parameter indicative of the risk of outbreak devel-opment. The y1 to y2 ratio should also be taken into account.

Fig. 2.15. Phase portrait boundaries and the zone of stability

of forest insect population If in year t, the population is within the stability zone, one can deter-

mine the conditional probability that in year (t + k), the population will be “ejected” beyond the stability zone and reach some metastable state with population density xm, and the insects will begin to consume great amounts of plant phytomass. If k = 1, conditional probability

1( ( 1) / ( ))p A t A t will determine the necessary but not sufficient condi-tion of outbreak occurrence in year (t + 1) and, to a first approximation, give a short-term forecast of outbreak occurrence.

Let us assume, as a working hypothesis, that the risk for the point characterizing the state of the population and located within the stability zone to be ejected beyond its right-hand boundary is determined by the distance between this point and the boundary, ( ).y x The probability that the point will be ejected beyond the stability zone is determined by either the strength of the effects of external modifying factors or (if these external factors produce no effect) the level of population density fluctua-tions. Unfortunately, the values of the critical density of population xr and parameters of boundary curves ( )y x and ( )y x in phase portraits are usually free parameters of population dynamics models, and the distance between point (x(t), y(t)), characterizing the population state at time t, and curve ( )y x can only be estimated from the long-term data


on the species population dynamics. Furthermore, the use of the data on population density of defoliating insects for predicting population dy-namics is limited because the structure of the phase portrait is deter-mined by unknown parameters of the influence of regulating factors (parasites and predators) and modifying ones (such as weather).

Thus, it is necessary to develop methods that could be used to evaluate critical parameters in the phase portraits of forest insect population dynamics and to make a short-term forecast and risk assessment of outbreaks of these populations. In this book, we do not try to find the solution to the entire problem of predicting risks of outbreaks, but rather discuss one aspect of this problem, which is related to predicting critical changes in the population density of forest defoliating insects. For the description, we used well-known time series of forest insect population dynamics – the time series of population dynamics of the pine looper B. piniarius in Germany (Schwerdtferer, 1968) and the time series of population dynamics of the larch bud moth Zeiraphera griceana Hubner in Switzerland (Baltensweiler, 1964).

Based on the hypothesis stated above and suggesting that the risk of ejection of the population from the stability zone, beyond its right-hand boundary, is determined by the distance between this point and the boundary ( ),y x let us estimate how close the current state of the popu-lation is to the upper threshold curve ( )y x by transforming the coordi-nates (shifting and rotating the axes) in the phase plane {x, y} and moving to the coordinate system related to the lower boundary curve y of the population phase portrait:

ln ; ln ,

cos sin ,sin cos .

x x y yz x yh x y

(2.9)

Transformation (2.9) is performed as follows: the axes of the {x, y} phase plane are turned in such a way that the line describing the lower boundary of the stability zone in the phase portrait becomes the new x-axis. In this case, all points characterizing the stable sparse state will be located in the new plane, below the line corresponding to the critical curve ( ).y x Then, the new variable will describe the probability of exceeding the critical density along the y-axis, and instead of two varia-bles, we will use only one, new, variable in the analysis and the model.

Figure 2.16 shows the data in double logarithmic coordinates on the density and reproduction coefficient of the larch bud moth in the Engandine Valley (Switzerland).


Fig. 2.16. The larch bud moth: a phase portrait based on observations over

120 years in the Engandine Valley (Switzerland). 1 – stable state; 2 – ourbreak As can be seen in Figure 2.16, slope ratios of the equations of the upper

and lower boundaries in the phase portrait do not differ significantly, and, thus, in the double logarithmic coordinates, boundary lines can be regarded as parallel to each other. Transformation (2.9) is the transition to new coordinates, z and h. Figure 2.17 shows a transformed phase portrait of the larch bud moth.

Fig. 2.17. A phase portrait of the larch bud moth after transformation

in the coordinates {z, h}. 1 – stable state; 2 – outbreak


The same transformation can be performed for the data on the density and growth rate of the pine looper in Thuringen forests (Fig. 2.18).

Fig. 2.18. A phase portrait of the pine looper in Thuringen (Germany) after

transformation in the coordinates {z, h}. 1 – stable state; 2 – outbreak As shown in Figures 2.17 and 2.18, in the new coordinates, the value of

variable h characterizes the distance between an arbitrary point in the plane and the upper critical line .h Thus, by transforming (using 2.9) the phase portrait coordinates, we obtain one variable, h, which can be used to characterize the probability of the “ejection” of the population beyond the boundary of the stability zone. Transition from the phase plane to the one-dimensional scale can be treated as transformation of the two-dimensional phase portrait into the one-dimensional potential well (Fig. 2.19).

Fig. 2.19. A model of the one-dimensional potential well for estimating

the probability that a population of one insect species will be ejected beyond the boundaries of the stability zone (h0 – well height, z0 – well width)


Let us imagine that the population, whose state is now characterized by only one variable, h, is in a so-called potential well. The height of the well is equal to the critical value h0, and the outbreak begins when the popula-tion is ejected from the potential well, i.e. when h > h0. The closer h(t) to the critical value h0, the more likely the ejection at time (t + 1) is.

The additional parameter – the width of the potential well, z – will ac-tually characterize the width of the population phase portrait. The type of the potential well can be used as a basis for classifying population dynam-ics of a species (Table 2.1).

Table 2.1. Classification of the types of population dynamics based

on the type of the potential well Height of the potential well

Width of the potential wellNarrow Wide

High Stable dynamics with a narrow phase portrait

Prodromal dynamics with population buildup events

Low Eruptive dynamics, fixed outbreaks

Eruptive dynamics: out-breaks proper

How close to the limiting value should the population be to be “eject-

ed” into the outbreak zone under the impacts of endogenous and (or) exogenous factors? The available time series of population dynamics can be used to estimate population density fluctuations in the potential well. The density function ( )f h of value h will be used as the indicator, if h < h0. Figure 2.20 shows the density function ( )f h of the state of the pine looper in the potential well and the conditional probability of the population being ejected beyond the boundaries of the stability zone as related to the current value of h.

Thus, the necessary condition for the occurrence of the outbreak of a forest insect species is the low height of the potential well. This parameter is, in its turn, determined by the small value of the slope angle between the x-axis and the upper threshold curve y (x), together with the small values of xr. However, to evaluate the critical density xr and the slope angle of the upper threshold curve ( ),y x one needs sufficiently long time series of population surveys in a given habitat. Unfortunately, these data are seldom available to entomologists, and, thus, the question arises if one can assess the probability of population ejection from the stability zone by using a limited dataset on this population. To find an answer to this question, in Chapter 8 we will describe models in which outbreaks of defoliating insects are treated as analogous to phase transitions in physi-cal systems.


Fig. 2.20. The population density function f(h) (1) of the state of the pine looper

in the potential well and the conditional probability (2) of the population being ejected beyond the boundaries of the stability zone as related

to the current value of h

2.5 Presenting the data on forest insect population dynamics as a potential function

One of the most powerful methods of analysis of physical systems is the use of the so-called potential functions, whose minimal values character-ize steady states of physical systems (Landau, Lifshits, 1964). Potential functions, however, can be used to describe nonphysical systems, too (Abaimov, 2013). The advantage of this approach is that it offers an opportunity to introduce the extreme principle – attaining the minimal value of the potential function in stable or metastable states of the system under study. In this case, instead of using kinetic equations to describe system dynamics, one can consider minimizing some functional, which is often a simpler task.

Let us introduce potential function G(x) as a parameter of probability р(х) of population survival at density х:

1( )( )

G xp x

(2.10)

It follows from (2.10) that at ( ) 0p x , ( )G x , and at ( ) 1p x , ( ) 1G x . Thus, potential function characterizes population survival


at different densities х ( 0 x ), and the greater G(x) for a particular population, the higher the probability that this population will die out.

Suppose that the stable states of the population are characterized by the minimum (either local or global) of potential function G(x). If poten-tial function G(x) has a concave shape (Fig. 2.21), there is a global mini-mum of this function, and there is only one value of density x1 at which population survival has maximum probability.

Fig. 2.21. Potential function G(x) with one global minimum

The greater the deviation of the absolute value of the current popula-

tion density x(t) from x1, the lower the probability of reaching this value and the higher the probability that even if this value is attained, popula-tion will not stay in this state for long, and its density will change, tending to approach x1.

The convex shape of the potential function, under the condition of population stability in the point of minimum of the potential function, indicates that the population with the potential function of this type is globally unstable and cannot be present in ecosystems for long time periods.

Finally, potential function may have several local minima rather than one global minimum. In this case, stable states of the population are assumed to correspond to local minima of potential function G(x). For instance, potential function G(x) for a bistable system with two stable (or metastable) values of population density, х1 and х2, is characterized by the presence of two local maxima of G(x) (potential wells) (Fig. 2.22).

Function G(x) in Figure 2.22 is described by using the following pa-rameters:

local minima of potential function G(x1) and G(x2) and the respec-tive values of х1 and х2 population densities (at x1 < x2);


Fig. 2.22. Potential function G(x) for a bistable system

the local maximum of function G(xr) – the height of the potential barrier G(xr) and value of population density xr at which potential function G(x) reaches local maximum.

Through these “basic values”, one can determine additional parame-ters:

the difference 2 1x x x between densities х1 and х2 – the range of population densities in stable states; the differences 2 1( ) ( )G G x G x between values of potential functions G(x2) – G(x1); the depth of potential wells: 1 1( ) ( )rG G x G x and

2 2( ) ( )rG G x G x ; half-width of the potential barrier ud u dx x x , where xu and xd are values of population density at which

1 2( ) max( ( ), ( ))( )2

rG x G x G xG x , and 1 2( ) max( ( ), ( ))2

rG x G x G x

is half-height of the potential barrier; absolute values of derivatives dG

dx of the potential function left

and right of point х = xr. At sufficiently low dGdx values, the po-

tential will be described as “soft”; at high dGdx values, the poten-

tial function will be characterized as “hard”. The derivative values

can be roughly replaced by 1

1

( ) ( )r

r

G x G xx x

and 2

2

( ) ( )r

r

G x G xx x

values.


Furthermore, to describe population dynamics, we introduce charac-teristic times over which population stays in stable states, or characteris-tic frequencies ω1 and ω2 of population oscillation in each of the potential wells and characteristic frequencies ω12 and ω21 of population transitions from the state with density x1 to the state with density x2 and back.

When presenting the data as a potential function, we certainly lose in-formation of the population temporal dynamics. Nevertheless, the com-bination of the main and additional parameters, which minutely charac-terize the shape of potential function G(x), gives an idea of population transitions between states with different densities and provides a basis for classifying forest insects by the type of population dynamics.

Figure 2.23 shows potential function G(x) for the model pine looper population in Thuringen.

Fig. 2.23. Potential function G(x) for the pine looper population in Thuringen

From the data in Figure 2.23, we can determine two stable states of the

population: the lower level density, characterized by a broad minimum, with pupal densities ranging between 1min exp( 4) 0.02x pupa per m2 and 1max exp( 1) 0.37x pupa per m2, and the pupal density at the upper level of 2 exp(2) 7.4x pupae/m2. The value 1( ) 6.7 7.5G x

2( ) 12G x suggests that the probability of the population being in the state with density х1 is higher than the probability of the population being in the state with density х2. The difference 2 1( ) ( ) 5G G x G x , i.e. it is more “advantageous” for the population to have low density. The potential barrier of relative height (1, ) 13G r is too high for the buildups of the pine looper population to occur frequently.


Figure 2.24 shows potential function of the D.pini population in habi-tat “Dune” in the Krasnoturansk pine forest.

Fig. 2.24. Potential function of the D.pini population in habitat “Dune”

in the Krasnoturansk pine forest The shape of the potential function of the D.pini is noticeably different

from the shape of the potential function for the pine looper. The deep minimum G(L) = 1.7 indicates that most of the time, the D.pini popula-tion stays in the state with the density of the LTI set of L = –0.2. The high potential barrier, (1, ) 30G r , shows that the transition to the upper metastable state with L = 0.6 occurs rather seldom.

Important parameters characterizing the probability of an outbreak are values (xr – x1) and G(xr) – G(x1). The lower these values, the more prob-able it is that the system will be transferred from the left-hand potential well to the right-hand one, i.e. that the outbreak will occur.

In the previously proposed classification of population dynamics and types of outbreaks, every insect species was assigned to one type of dynam-ics only (Isaev et al., 2001). Based on the type of the population phase portrait, one can assign the outbreak to one of the four types: the outbreak proper, the fixed outbreak, the sustained outbreak, and the reverse out-break. If the types of population dynamics and outbreaks are classified by using potential functions, all differences in population dynamics can be characterized through gradual changes in potential function parameters. Moreover, the gradual classification method can be used to characterize intermediate types of population dynamics and outbreaks and describe the situations in which a population in some habitat changes the type of out-break. This approach will be described in greater detail in Chapter 8.


*** As the data on population dynamics can be presented in different ways (as a time series, a phase portrait or a potential function), the question arises as to which presentation should be used in the analysis of dynamic processes in forest insect populations. We think that these presentations complement each other, and the choice of one of them will be deter-mined by the problem the researcher needs to solve. Time series seem to be suitable for short-term forecasts of population dynamics; potential functions are convenient for describing long-term behavior of popula-tions; and population stability can be effectively estimated by analyzing phase portraits of this population.

53

3 The Effects of Weather Factors on Population Dynamics of Forest Defoliating Insects

3.1 The necessary and sufficient weather conditions for the development of outbreaks of defoliating insects in Siberia

Many researchers have tried to explain changes in the insect population density by the influence of the weather (Barthod, 1994; Candau, Fleming, 2011; Flower et al., 2014; Hodar et al., 2012; Hunter et al., 2014; Ilyinsky, 1952; Jepsen et al., 2011; Jones, Wiman, 2012; Klapwijk et al., 2013; Kondakov, 1974; Larsson, Tenov, 1984; Mattson, Haack, 1987; Pimentel et al., 2011; Rafes, 1978; Regniere et al., 2012; Speight, 1986; Steinbauer et al., 2012; Tamburini et al., 2013; Vorontsov, 1963; Wallner, 1987; Young et al., 2014; Zhang et al., 2014).

First, however, we should define the necessary and sufficient weather conditions favoring outbreaks of defoliating insects.

The condition is necessary if the outbreak is always preceded by specif-ic changes in the weather (e.g., summer drought when Siberian silk moth larvae are active). The condition is sufficient if every specific change in the weather is followed by an outbreak. The difference between a necessary and a sufficient condition is that the necessary condition may be insufficient and there will be no outbreak in spite of the favorable weather.

To check whether the effects of weather factors are necessary and suf-ficient conditions for the emergence of outbreaks, in this study we use the classical Bayes’ formula in probability theory (Ventsel, Ovcharov, 2010).




The necessity of the effect of the weather on outbreak development was determined from the conditional probability 0( / )P OUT W W of the event OUT – outbreak – occurring within the time period analyzed, if the weather parameter W exceeds a certain critical value, W0, in the years preceding the outbreak:

00

( , )( / )( )

P OUT W WP OUT W WP OUT

, (3.1)

where 0( , )P OUT W W is the probability of the value of the modifying factor exceeding the threshold value W0 within a certain period of time preceding the outbreak; ( )P OUT is the probability of the emergence of the outbreak within the given time.

The sufficient condition for the occurrence of the outbreak under spe-cific weather changes is also calculated from Bayes’ formula as the condi-tional probability 0( / )P W W OUT :

00

0

( , )( / )( )

P OUT W WP W W OUTP W W

, (3.2)

where 0( )P W W is the probability of the value of the modifying factor exceeding the threshold value within the given time period.

The conditional probabilities 0( / )P OUT W W and 0( / )P W W OUT are interrelated. For example:

0 00

( / ) ( )( / ) .( )

P OUT W W P W WP W W OUTP OUT

(3.3)

If the sufficiency condition of specific weather changes for the occur-rence of the outbreak is satisfied and 0( / ) 1P W W OUT , outbreak forecasts are very simple to make: a weather change will necessarily be followed by an outbreak. If the necessity condition of specific weather changes for the occurrence of the outbreak is satisfied and

0( / ) 1P OUT W W , every outbreak will be preceded by specific weather changes, but not every specific weather change will be followed by an outbreak.

If the value of the conditional probability 0( / )P OUT W W is close to the value of the unconditional probability ( )P OUT of outbreak occur-rence and the value of the conditional probability 0( / )P W W OUT is close to the value of 0( )P W W , the corresponding weather factor does not influence the development of the outbreak. The significance of the

The Effects of Weather Factors on Population Dynamics 55

difference between the values of the conditional probabilities and the values of the corresponding unconditional probabilities and, hence, the significance of the relationship between the change in the weather factor and the development of an outbreak can be determined by using a conventional statistical procedure.

However, calculations based on Bayes’ formula do not indicate directly what weather factors should be used to evaluate conditional probabilities. Therefore, based on the specific seasonal development of an insect spe-cies, one has to find the weather parameters for which conditional prob-abilities 0( / )P OUT W W and 0( / )P W W OUT are close to 1.

3.2 Weather influence on the development of the pine looper Bupalus piniarius L. outbreaks

To study the relationship between weather changes and development of outbreaks, it is necessary to have long-term survey data on the density of the population in the stable sparse state. For the outbreak species, howev-er, the only available data are usually the date when the outbreak is de-tected and the area it covers. When an outbreak is detected in Siberian forest stands, as a rule, the insect population density is much higher than the critical value, suggesting that the outbreak starts between one and three years before it is detected. Taking into account this inaccuracy of the available data on the past outbreaks, we estimated the necessity and sufficiency of weather conditions for the development of pine looper outbreaks.

The most considerable fluctuations in the pine looper density have been observed in the ribbon pine forests in the Minusinsk Depression and in Altai. In Minusinsk ribbon pine forests, six outbreaks have oc-curred since the 1930s. The outbreaks were detected in 1939, 1944, 1954, 1962, 1972–1974, and 1988. In Altai, there were five pine looper out-breaks between 1928 and 1996.

Weather parameters were determined by using the weather data on the monthly average temperature and precipitation obtained at the nearest weather stations and the hydrothermal coefficient (HTC) proposed by Selyaninov. We used the weather data for the periods between May and September for the years between 1928 and 1997.

For comparative analysis of the coherence between weather and out-breaks, we have introduced the notion of pre-outbreak years. This term will be used to denote the years preceding the year in which the outbreak


was detected. Changes in the weather conditions in the pre-outbreak years are decisive for the development of population.

For every month of the pre-outbreak years and every weather parame-ter, we found long-term annual average values. In our analysis, they function as threshold values of weather parameters W0.

The value P(W > W0) characterizes the percentage of years of the study period in which weather parameters exceeded the threshold value, W > W0. The value P(OUT, W > W0) characterizes the percentage of the pre-outbreak years in which weather parameters exceeded the threshold value, W > W0. P(W > W0 / OUT) is the probability of outbreak develop-ment after the weather parameters have exceeded the threshold value, i.e. the probability of the fulfilment of the sufficiency condition. P(OUT / W > W0) is the probability of weather parameters exceeding the threshold value in the specified pre-outbreak years

The data obtained by using this procedure can give answers to the fol-lowing questions:

1. How much difference is there between the probability of fulfill-ment of the sufficiency condition and the probability of fulfilment of the necessity condition?

2. Which weather conditions have the greatest effect on population development?

3. What seasons can be defined as critical for the population devel-opment?

4. Weather parameters of what years are the most significant for the development of an outbreak?

In evaluations of the conditional probability 0( / )P W W OUT

0

0

( , )( )

P OUT W WP W W

, the denominator characterizes the percentage of years

in which the values of the specified weather factors exceed the threshold value. Calculations, however, must not include the years when the values of the weather parameters exceed the threshold value but the reproduc-tion coefficient is ( , ) 1Ry x y . This may happen when the post-outbreak population is in the low-density state because the insects have been killed by parasites, predators, and diseases, but weather conditions are favorable for the occurrence of an outbreak. If we ignore this condition, we allow the denominator 0( )P W W to increase, which will lead to underesti-mating the value 0( / )P W W OUT .

The conditional probability of the sufficiency of the weather effect on the population development 0( / )P W W OUT has different values


depending on whether we include the years of the post-outbreak low-density state in the analysis or exclude them (Table 3.1). Thus, the accu-racy of the evaluation is increased. It is also evident that the level of significance of the difference between the conditional probability of the sufficiency of weather effects on population development and the uncon-ditional probability of the outbreak development for the specified period is somewhat increased depending on the accuracy of the analysis used (including or excluding the post-outbreak low-density years).

Table 3.1. Conditional probabilities of the occurrence of the pine looper out-

break, including and excluding the low-density years

Statistical parame-ters*

SeptemberIncluding low-density

yearsExcluding low-density

years

Т, С Precipita-tion, mm HTC Т, С Precipita-

tion, mm HTC

<Х> 10.01 40.41 1.04 10.02 36.60 0.94 ( )P OUT 0.18 0.18 0.18 0.24 0.24 0.24

0( )P W W 0.52 0.36 0.36 0.53 0.32 0.35 0( , )P OUT W W 0.09 0.02 0.02 0.12 0.03 0.03

0( / )P OUT W W 0.50 0.13 0.13 0.50 0.13 0.13 0( / )P W W OUT 0.17 0.06 0.06 0.22 0.09 0.08

t-test 0.05 1.10 1.10 0.08 1.10 1.19 * <Х> is the long-term annual average of the weather parameter; ( )P OUT is the outbreak probability; 0( )P W W is the probability of exceeding the critical value for three pre-outbreak years; 0( / )P OUT W W is conditional probability of the necessity of certain weather conditions for outbreak development;

0( / )P W W OUT is conditional probability of the sufficiency of certain weather conditions for outbreak development; t – test.

The test of significance t was determined from the formula

0( ) ( / ) ,P OUT P W W OUTtSd

0 0( ) (1 ( )) ( / ) (1 ( / )),

1P OUT P OUT P W W OUT P W W OUT

SdN n

where N is the number of years in the period studied and n is the number of specified pre-outbreak years.

Unfortunately, there is no definitive answer to the question: Weather conditions of which years had the strongest effect on the population?


Therefore, we calculated the probabilities for several combinations of pre-outbreak years. As the basic combination, we used the combination of years with the most significant difference between the conditional proba-bility of the necessity of weather effects on population development and the unconditional probability of the outbreak development. The following distribution was found for the Minusinsk Depression (Table 3.2).

Thus, in this case, the combination of years in which weather effects are the strongest are two sequential years one year before the outbreak. This, however, is not always right. Analysis of the July HTC has shifted the years we are searching for closer to the outbreak year (Table 3.3).

In the Altai Territory, the most significant effects are always produced by the weather in the two years immediately before the outbreak (Ta-ble 3.4).

Table 3.2. Determination of the combinations of pre-outbreak years

in which the May weather had the strongest effect on the development of the pine looper outbreak in the Minusinsk district (based on HTC)

Statistical parame-ters

Pre-outbreak years * 0, 1, 2 1, 2 1, 2, 3 2, 3

( )P OUT 0.27 0.18 0.27 0.180( )P W W 0.45 0.45 0.45 0.45

0( , )P OUT W W 0.11 0.07 0.16 0.140( / )P OUT W W 0.42 0.38 0.58 0.75

0( / )P W W OUT 0.25 0.15 0.35 0.30t-test 0.23 0.40 0.77 1.64

* Different combinations of the pre-outbreak years. Numbers denote years counted backwards; 0 is the year when the outbreak supposedly began.

Table 3.3. Determination of the combinations of pre-outbreak years in which the July weather had the strongest effect on the development of the pine looper

outbreak in the Minusinsk district (based on HTC)

Statistical parame-ters

Pre-outbreak years0, 1, 2 1, 2 1, 2, 3 2, 3

( )P OUT 0.27 0.18 0.27 0.180( )P W W 0.61 0.61 0.61 0.61

0( , )P OUT W W 0.14 0.05 0.11 0.090( / )P OUT W W 0.50 0.25 0.42 0.50

0( / )P W W OUT 0.22 0.07 0.19 0.15t-test 0.68 2.03 1.19 0.56


Table 3.4. The significance of the difference between the conditional and unconditional probabilities of the necessity of weather effects

(based on HTC) in different combinations of pre-outbreak years in Altai

Month Statistical parameters Pre-outbreak years

0, 1, 2 1, 2 1, 2, 3 2, 3

May 0( )P W W 0.45 0.45 0.45 0.45

0( / )P OUT W W 0.67 0.80 0.73 0.70t-test 1.56 2.40 2.14 1.53

June 0( )P W W 0.48 0.48 0.48 0.48

0( / )P OUT W W 0.67 0.60 0.47 0.50t-test 1.35 0.70 0.08 0.12

July 0( )P W W 0.46 0.46 0.46 0.46

0( / )P OUT W W 0.33 0.40 0.47 0.50t-test 0.93 0.37 0.02 0.20

August 0( )P W W 0.43 0.43 0.43 0.43

0( / )P OUT W W 0.60 0.70 0.53 0.50t-test 1.15 1.62 0.67 0.37

September 0( )P W W 0.38 0.38 0.38 0.38

0( / )P OUT W W 0.33 0.40 0.47 0.50t-test 0.31 0.13 0.62 0.70

The differences between the conditional probability (necessity of

weather effects) and the unconditional probability in the Altai Territory are very large (Table 3.4). Their significance is very high for almost any combination of years. This suggests the nearly definitive conclusion that for the pine looper outbreak to develop in the Altai pine forests, for several years, the hydrothermal coefficient must be higher than its critical value. In other words, the temperature must be rather low and the humidity must be high. This is consistent with the field data, indi-cating that the pine looper is a moisture-loving and cold-resistant species. Moreover, the cold and humid spring may encourage the in-sects to move to the tree crowns early and escape from their natural enemies.

Evaluations of the necessary conditions for the development of pine looper outbreaks in the ribbon pine forests of the Minusinsk Depression are listed in Table 3.5. The table includes the data for the pre-outbreak years for which the t-test had the highest values.

The unconditional probability of the weather parameter exceeding the threshold value fluctuates about 0.5 for the entire period (Table 3.5),


indicating that favorable and unfavorable weather conditions are approx-imately equally probable. Comparison of conditional and unconditional probabilities of the necessity of weather effects may show what weather could have the strongest effect on the development of outbreaks in this region in each of the seasons.

Similar analysis can be done for the Altai Territory (Table 3.6). Weather conditions favorable for the development of the pine looper

clearly differ between habitats. The probability of the development of a pine looper outbreak in the pine forests in the Minusinsk Depression is largely determined by the amount of rainfall between May and Sep-tember. In Altai, however, this relationship is only valid for May and August, while in midsummer, temperature becomes a more significant factor. In addition to this, in pre-outbreak years, the amount of precipi-tation in May is higher than the long-term annual average value.

Table 3.5. Necessary conditions of weather effects on the development of pine

looper outbreaks in the ribbon pine forests of the Minusinsk Depression

Month Weather parameter*

Probabilities Signifi- cance**

0( , )P OUT W W 0( )P W W 0( / )P OUT W W t-test Minusinsk weather station

May

T 0.05 0.43 0.25 1.01 Precipita-tion 0.11 0.41 0.63 1.09

HTC 0.14 0.45 0.75 1.64

June

Т 0.09 0.55 0.50 0.22 Precipita-tion 0.14 0.48 0.75 1.51

HTC 0.14 0.48 0.75 1.51

July

Т 0.07 0.50 0.38 0.63 Precipita-tion 0.14 0.39 0.75 2.03

HTC 0.14 0.39 0.75 2.03

August

Т 0.05 0.41 0.25 0.89 Precipita-tion 0.11 0.45 0.63 0.86

HTC 0.11 0.43 0.63 0.98

Septem- ber

Т 0.09 0.52 0.50 0.11 Precipita-tion 0.02 0.36 0.13 1.65

HTC 0.02 0.36 0.13 1.65


Table 3.6. Necessary conditions of weather effects on the development of pine looper outbreaks in the ribbon pine forests of the Altai Territory



0( , )P OUT W W 0( )P W W 0( / )P OUT W W t-test Barnaul weather station (the Altai Territory)

May

Т 0.01 0.48 0.10 3.24 Precipita-tion 0.10 0.48 0.70 1.35

HTC 0.12 0.45 0.80 2.40

June

Т 0.09 0.55 0.60 0.28 Precipita-tion 0.07 0.52 0.50 0.45

HTC 0.07 0.48 0.50 0.70

July

Т 0.04 0.52 0.30 1.35 Precipita-tion 0.06 0.49 0.40 0.53

HTC 0.06 0.46 0.40 0.37

August

Т 0.04 0.43 0.30 0.82 Precipita-tion 0.10 0.45 0.70 1.53

HTC 0.10 0.43 0.70 1.62

Septem- ber

Т 0.07 0.57 0.50 0.37 Precipita-tion 0.07 0.41 0.50 0.53

HTC 0.07 0.38 0.50 0.70

* Т – temperature; HTC – hydrothermal coefficient of the month.

3.3 Siberian silk moth Dendrolimus sibiricus Tschetv. population dynamics as related to weather conditions

Siberian silk moth Dendrolimus superans sibiricus Tschetv. is the phytophagous insect that causes the greatest damage to trees in Siberian taiga forests. During outbreaks, the number of larvae of this species reach several thousand per tree, and they completely defoliate and kill fir, spruce, and Siberian pine trees.

Outbreaks of the Siberian silk moth cover vast areas. An outbreak that occurred in the forests of the Krasnoyarsk Territory in the 1990s dam-


aged tree stands on 479.9 thousand hectares: 125.5 thousand ha of the stands were weakly damaged (25% defoliated), 67.4 thousand ha were moderately damaged (25–50%), 49.8 thousand ha were strongly damaged (50–75%), and 237.3 thousand ha were completely defoliated (Baranchikov et al., 2002).

Pan-regional outbreaks of forest insects severely damage forests in all regions, and, thus, it is imperative to be able to predict their occurrence and development. Such forecasts are usually limited to monitoring the synchronizing factor and predicting the development of a pan-regional outbreak when the value of the synchronizing factor exceeds certain threshold values. To reduce the range of consequences of insect out-breaks and minimize the damage caused by the pests to the forests, it is necessary to devise methods for detecting woodlands highly susceptible to outbreaks, effectively predict forest insect outbreaks, develop optimal strategy of forest protection, diagnose outbreak types, and predict spatial distribution of pest populations.

The relationship of the Siberian silk moth population dynamics in the Yeniseiskii and Boguchanskii Districts of the Krasnoyarsk Territory to weather conditions was analyzed by using the approach similar to the one described above, which was used to evaluate the influence of the weather on pine looper population dynamics. Five outbreaks of the Siberian silk moth occurred in this region between 1931 and 1995: in 1935, 1950, 1962, 1978, and 1989 (Kondakov, 1974; Isaev et al., 2001). We studied the effect of the weather on population buildup in summer months (between May and September).

The sufficiency condition of the weather effect on population devel-opment was studied with and without taking into account the time of the post-outbreak low-density phase.

Table 3.7 gives the data on HTC in July at Boguchany. The data in Table 3.7 suggest that the life and development of the pop-

ulation are mainly influenced by the weather conditions of the two years before the outbreak, and if these conditions are favorable to the popula-tion, the probability of an outbreak is very high.

A similar approach was used to analyze weather data in the Yeniseiskii district (Table 3.8).

Analysis of weather conditions in the Yeniseiskii district shows that the weather conditions of two years preceding the outbreak were favora-ble for the development of the population (similarly to the Boguchanskii district). In the Yeniseiskii district, however, the August weather influ-ences the development of the outbreak to a greater extent.

For more detailed analysis, see Tables 3.9 and 3.10.


Table 3.7. Determination of the combinations of pre-outbreak years in which weather had the strongest effect on the development

of the Siberian silk moth outbreak in the Boguchanskii district (based on HTC)

Month Statistical parameters Pre-outbreak years0, 1, 2 1, 2 1, 2, 3 2, 3

May 0( )P W W 0.40 0.40 0.40 0.40

0( / )P OUT W W 0.40 0.50 0.53 0.50t-test 0.00 0.56 0.91 0.56

June 0( )P W W 0.48 0.48 0.48 0.48

0( / )P OUT W W 0.60 0.70 0.53 0.40t-test 0.85 1.35 0.38 0.44

July 0( )P W W 0.42 0.42 0.42 0.42

0( / )P OUT W W 0.40 0.60 0.60 0.60t-test 0.15 1.02 1.23 1.02

August 0( )P W W 0.42 0.42 0.42 0.42

0( / )P OUT W W 0.47 0.50 0.53 0.50t-test 0.35 0.48 0.80 0.48

September 0( )P W W 0.41 0.41 0.41 0.41

0( / )P OUT W W 0.53 0.60 0.43 0.56t-test 0.87 1.11 0.15 0.85

Table 3.8. Determination of the combinations of pre-outbreak years

in which weather had the strongest effect on the development of the Siberian silk moth outbreak in the Yeniseiskii district (based on HTC)

Month Statistical parameters Pre-outbreak years0, 1, 2 1, 2 1, 2, 3 2, 3

May 0( )P W W 0.43 0.43 0.43 0.43

0( / )P OUT W W 0.40 0.30 0.47 0.50t-test 0.21 0.79 0.24 0.39

June 0( )P W W 0.45 0.45 0.45 0.45

0( / )P OUT W W 0.47 0.70 0.60 0.50t-test 0.14 1.54 1.06 0.30

July 0( )P W W 0.37 0.37 0.37 0.37

0( / )P OUT W W 0.33 0.40 0.40 0.40t-test 0.26 0.18 0.21 0.18

August 0( )P W W 0.48 0.48 0.48 0.48

0( / )P OUT W W 0.60 0.70 0.67 0.70t-test 0.85 1.35 1.35 1.35

September 0( )P W W 0.40 0.40 0.40 0.40

0( / )P OUT W W 0.47 0.50 0.40 0.30t-test 0.45 0.56 0.00 0.61


Table 3.9. Necessary conditions of weather effects on the development of Siberian silk moth outbreaks in the fir forests of the Boguchanskii district



0( , )P OUT W W 0( )P W W 0( / )P OUT W W t-test Boguchany weather station

May

Т 0.09 0.46 0.40 0.42 Precipita-tion 0.11 0.49 0.47 0.17

HTC 0.12 0.40 0.53 0.91

June

Т 0.08 0.54 0.33 1.46 Precipita-tion 0.11 0.46 0.70 1.45

HTC 0.11 0.48 0.70 1.35

July

Т 0.03 0.47 0.20 1.99 Precipita-tion 0.16 0.44 0.67 1.63

HTC 0.14 0.42 0.60 1.23

August

Т 0.05 0.52 0.30 1.35 Precipita-tion 0.12 0.45 0.53 0.59

HTC 0.12 0.42 0.53 0.80

Septem- ber

Т 0.09 0.52 0.67 0.89 Precipita-tion 0.09 0.41 0.60 1.11

HTC 0.09 0.41 0.60 1.11

3.4 Synchronization of weather conditions

on vast areas as a factor of the occurrence of pan-regional outbreaks

Pan-regional outbreaks are characteristic of eruptive species with high migratory activity (the Siberian silk moth, the larch bud moth, the en-grailed moth, etc.). They usually occur on extensive areas, which are many times greater than the habitats of insect populations responsible for such outbreaks (Isaev, 2001). The ecological parameters, total area, and the dynamics of regional and pan-regional outbreaks are considerably different from those of local outbreaks. Pan-regional outbreaks occur when densities of the local populations of the pest species dramatically increase almost simultaneously on an area of several hundred kilometers.


Table 3.10. Necessary conditions of weather effects on the development of Siberian silk moth outbreaks in the fir forests of the Yeniseiskii district

in the Krasnoyarsk Territory



0( , )P OUT W W 0( )P W W 0( / )P OUT W W t-test Yeniseisk weather station

May

Т 0.08 0.43 0.33 0.70 Precipita-tion 0.05 0.48 0.30 1.07

HTC 0.05 0.43 0.30 0.79

June

Т 0.03 0.57 0.20 2.52 Precipita-tion 0.11 0.48 0.70 1.35

HTC 0.11 0.45 0.70 1.54

July

Т 0.05 0.51 0.30 1.26 Precipita-tion 0.09 0.43 0.60 0.97

HTC 0.09 0.37 0.40 0.21

August

Т 0.06 0.49 0.27 1.69 Precipita-tion 0.11 0.45 0.70 1.54

HTC 0.11 0.48 0.70 1.35

Septem- ber

Т 0.11 0.54 0.70 0.98 Precipita-tion 0.11 0.46 0.70 1.45

HTC 0.05 0.40 0.30 0.61 This case is different from outbreaks of the gypsy moth in the U.S., where the invasion front extends over the forested area (Sharov et al., 1995). Siberian silk moth populations do not migrate from one or several pri-mary outbreak sites, but population density increases almost simultane-ously on the entire territory (Kondakov, 2002).

To answer the question whether synchronous changes in weather conditions on vast areas favor the development of outbreaks, it is neces-sary to analyze weather data in the points of the regions with a develop-ing pan-regional outbreak that are located sufficiently far away from each other. The coherence of the weather conditions in this area may have a synchronizing effect on the development of large-scale out-breaks.

A thorough study of the coherence between weather conditions and development of pan-regional outbreaks was conducted for the regions


situated along the Angara River, based on the data of the Yeniseisk and Boguchany weather stations. The distance between these weather stations is about 350 km. To compare temperature conditions, we calculated cross-correlation functions of temperature changes. For analysis of the precipitation data, we used the method of constructing contingency tables (Upton, 1978).

A measure of relationship between two random processes in the time domain is their cross-correlation function ( )xyr (Kendall, Stuart, 1971):

1( ) lim ( ) ( ) ,

2

T

xy TT

r x t y t dtT

(3.4)

where x(t) – y(t) are signals compared (weather time series); T is the survey period; τ is the time shift (days) between weather series.

Cross-correlation function varies between –1 and +1. The closer its absolute value ( )xyr is to 1, the higher the degree of synchrony of the processes under study. Variations in parameter τ cause changes in the “shift” of one signal (the temperature time series in this case) relative to the other. The value of τ at which r is the highest is usually interpreted as a time lag shown by one signal relative to the other. If in this case τ = 0, the weather is completely synchronized on the area between Boguchany and Yeniseisk. If τ is not equal to 0, but lies within the range between –10 and 10 days, the weather conditions in this area exhibit coherence.

To analyze temperature coherence of the weather conditions, we cal-culated the cross-correlation function of long-term time series of daily temperatures based on the data of the Boguchany and Yeniseisk weather stations. Calculations were performed by using the Statistica 6.0 software package. To eliminate the seasonal trend, we used the “first difference” method or the “modified second-order correlation coefficient” (Lukashin, 2003). The series of parameter values was transformed into the series of differences between two sequential parameter values, and, then, we studied the coherence between the series of the first differences of temperature series.

After this transformation, we calculated cross-correlation functions for the spring (May), summer (June–August), and autumn (September) seasons for each year between 1931 and 1995. Note that the pan-regional Siberian silk moth outbreak that occurred in this area was the largest in the recent years. The outbreak was stopped by joint efforts of specialists from different countries and thanks to the substantial financial support of the Russian government and the World Bank.


Figure 3.1 shows a typical shape of the cross-correlation function of the daily temperature time series based on the data obtained at the Yeniseisk and Boguchany weather stations (June–August 1991).

Fig. 3.1. Cross-correlation function of the daily temperature time series

based on the data obtained at the Yeniseisk and Boguchany weather stations (June–August 1991)

The cross-correlation function peaks at the shift 0, suggesting spa-

tial synchrony in weather changes in the Yeniseiskii and Boguchanskii districts. The closer coefficient ( 0)xyr to 1, the higher the degree of synchrony.

Statistical significance r was calculated by using Student’s t-test, ac-cording to the following formula:

21

2

r

nrt , (3.5)

where n is the number of degrees of freedom for the time series. For the significance level р = 0.95 for the period between June and Au-

gust, the table value of the t-test was 2.33критt . Hence, from (3.5), we calculated the critical value of the correlation coefficient .critr

1

22

21 .critcrit

nrt

(3.6)

The values of correlation coefficient below 0.239критr are non-significant at the significance level р = 0.95.


Our calculations show that the degree of synchrony between tempe-rature variations over the Angara region was high in all years (Fig. 3.2).

Fig. 3.2. Synchronization of temperature variations of May weather

in the Angara region between 1931 and 1995. Lines at the time scale point to the years when outbreaks began. 1 – correlation coefficient; 2 – critical value

Figures 3.2–3.4 show that temperature variations in the study area are

synchronized to a high degree. The highest degree of synchrony is ob-served for the air temperatures in May and in the period between June and August. Only once does the significance test of temperature syn-chrony in spring and summer drop below the critical value, suggesting that temperature variations in both regions are almost always synchro-nized with each other. Temperature variations in September are syn-chronized to a lower degree, as in a number of years, the temperature synchrony is non-significant.

Fig. 3.3. Synchronization of temperature variations of June-August weather

in the Angara region between 1931 and 1995. 1 – correlation coefficient; 2 – critical value


Fig. 3.4. Synchronization of temperature variations of September weather in the Angara region between 1931 and 1995. 1 – correlation coefficient;

2 – critical value The highest correlation coefficients for the temperature in the study

regions were obtained for spring and summer (between May and Au-gust). These values are much higher than the critical value of correlation coefficient. September temperature (Fig. 3.4) is less synchronized, but correlation coefficients for temperature in September of pre-outbreak years are also high.

Analysis of the cross-correlation function shows that most of the tem-perature variations in the study regions are not shifted relative to each other. Only a one-day delay is observed from time to time in weather change in Boguchany relative to Yeniseisk.

Table 3.11 lists the data on correlations between temperature varia-tions in the study region in the pre-outbreak years.

As suggested by the statistical analysis presented in Table 3.11, tem-perature variations in the Angara regions in the pre-outbreak years are always synchronized, at least in one season. Hence, for outbreaks to develop over a vast area, correlation between temperature variations must remain high for one of the seasons.

The method of calculation of cross-correlation function can also be used to analyze the synchrony of rainfall at the site of the Siberian silk moth pan-regional outbreak. However, there may be no significant correlation between the rainfalls in the regions situated rather far away from each other, as this parameter is determined by the total amount of the rain formed and its random distribution over the territory. In other words, the rain that has formed and fallen on the territory of the Yeniseiskii district may not reach the Boguchanskii district, and, vice versa, if the clouds have reached the Yeniseiskii district but the rain has not formed yet, it must mainly fall in the Boguchanskii district.


Table 3.11. Correlation coefficients of temperature variations in the Angara region in the pre-outbreak years

Year Correlation coefficient of temperature

in the Angara regionsMay June–August September

1939 0.75 0.47 0.661940 0.86 0.47 0.601941 0.59 0.62 0.661942* 0.32 0.58 0.501951 0.50 0.55 0.311952 0.57 0.57 0.491953 0.61 0.54 0.541954* 0.38 0.39 0.441964 0.79 0.59 0.201965 0.50 0.54 0.431966 0.81 0.53 0.741967* 0.68 0.52 0.651979 0.68 0.39 0.051980 0.74 0.48 0.581981 0.73 0.59 0.721982* 0.70 0.53 0.741991 0.48 0.58 0.141992 0.70 0.67 0.431993 0.66 0.56 0.531994* 0.73 0.70 0.25Average for the entire period 0.62 0.52 0.51

* – years when outbreaks began; the critical value of correlation coefficient is 0.239. Bold type denotes the values of correlation coefficient higher than the average.

Calculation of the cross-correlation function for estimating the quanti-

tative coherence between precipitations does not show any correlation between the amounts of precipitation in the Yeniseiskii and Boguchanskii districts (Fig. 3.5).

It would be interesting to evaluate qualitative coherence between rain-falls in the study regions as synchronization of the events. In this study, we used contingency tables – sample estimates of probability distribu-tions of multidimensional random variables.

Let us assume that two properties, X and Y, are being analyzed. The first takes r values, 1, 2, …, r, and the second takes s values, 1, 2, …, s.


Fig. 3.5. The characteristic shift of precipitation time series relative to each

other. The delay less than 0 characterizes the precipitation delay in Boguchany relative to Yeniseisk, the delay more than 0 characterizes

the precipitation delay in Yeniseisk relative to Boguchany. A two-dimensional contingency table (frequency table) is a matrix, in

which, at the intersection of the ith row and the jth column, there is num-ber nij, denoting the number of objects possessing the ith value of the first property and the jth value of the second one (i = 1, … , r; j = 1, … , s).

In other words, the contingency table has the following form:

11 12 1

21 22 2

1 2

.

s

sij

r r rs

n n nn n n

n

n n n

It is usually presented as a table with explicitly denoted names of prop-erties and their values and the marginal totals:

X Y Row marginal totals 1 2 … j … s

1 2

… i

… R

n11 n12 … n1j … n1s n1. n2. … ni. … nr.

n21 n22 … n2j … n2s

… … … … … …ni1 ni2 … nij … nis

… … … … … …nr1 nr2 … nrj … nrs

Column marginal totals n.1 n.2 … n.j … n.s n


The rightmost column is composed of the row marginal totals. Val-ue ni. is equal to the sum of the ith elements (i.e. the number of the objects for which the first property takes value i). The lowermost row is com-posed of the column marginal totals. Value n.j is equal to the sum of the elements in the jth column (i.e. the number of the objects for which the second property takes value j). n is the sample size; it is equal to the column marginal total (or the row marginal total).

The simplest contingency table has dimensions 2 × 2 (the table con-sists of two rows and two columns, corresponding to two coherent varia-bles, each of which takes two values).

Table 3.12. The structure of the contingency table for two weather parameters

Precipitation in the specified period

of the season

Temperature in the specified period of the season Marginal

frequencies Below the norm (T)

Above the norm (T )

Below the norm (P) n11 n12 n1. = n11 + n12

Above the norm ( P ) n21 n22 n2. = n21 + n22

Marginal frequencies n.1 = n11 + n21 n.2 = n12 + n22 n.. = n1. + n2. =

n.1 + n.2 If there is no relationship between the parameters, the following equa-

tion holds:

11 21 11 21

11 12 21 22

.n n n nn n n n n

This equality is taken as the definition of independence of properties. If the equality is broken in the case of n11/(n11 + n12) > (n11 + n21)/n, there is a positive correlation between the properties; if the inequality has the opposite sign, the relationship is negative.

The relationship between alternative properties is measured by param-eter 2ˆ (Gruza, Rankova, 1983):

2 22 2

2

, 1 , 1

( . . / )ˆ 1

. . . .ij i j ij

i j i ji j i j

n n n n nn n

n n n n.

The properties must be recognized as dependent if 2 0. However, this statement will be true only if the data comprising the contingency table belong to the entire population. If these data are obtained by ran-dom sampling, this statement may be wrong (because of possible repre-sentativeness errors). Thus, the main aim of using criterion 2


in sampling studies is to set such critical value 2cr that the probability of

obtaining values of 2 above the critical value, because of the sample randomness, is very low. Thus, if 2 is greater than 2

cr , the hypothesis about the absence of relationship between the properties must be rejected for the level of significance chosen in the study.

The significance level α means the probability of error in rejection of the statistical independence hypothesis.

The convenience of using criterion 2 is based on the presence of the tables containing critical values of this criterion for different significance levels and dimensions of the task. In the analysis of the contingency table of nominal properties, the number of degrees of freedom is determined from the formula ( 1)( 1),k r s where r and s, again, denote the number of gradations of the specified properties. The significance level α is usually taken equal to 0.01, 0.05 or 0.10.

The critical value 2cr is determined by the intersection of the row cor-

responding to the given value k and the column corresponding to the specified significance level α.

Using the contingency table, we estimated the degree of synchrony of rainfalls in June and July in the Angara region. We calculated the coher-ence of precipitation for summer months of each year between 1931 and 1991 (Table 3.13). The property values were daily indicators of the pres-ence (value 1) or absence (0) of precipitation.

Table 3.13. Coherence of precipitation in the Angara region (June–August 1993), based on the data of the Boguchany and Yeniseisk weather stations

Boguchany Yeniseisk Marginal frequen-cies Yes (1) No (0)

Yes (1) 22 8 30No (0) 11 51 62Marginal frequencies 33 59 92

The data in Table 3.13 suggest that rainfalls in the summer months

of 1993 showed a high degree of synchrony in the two districts, as the fraction of the years with the equal distribution of precipitation is 0.79.

2 = 27.16 (at 2cr = 5.99).

In the same way, we obtained a set of 2 values for all years of the study period (Fig. 3.6).


Fig. 3.6. Coherence of precipitation in the Angara region in June–August between 1931 and 1995. 1 – coherence of precipitation, 2 – critical value

The degree of coherence of precipitation at Boguchany and at

Yeniseisk varies widely; however, in the pre-outbreak years (except the years preceding the 1942–1945 outbreak), the degree of synchrony is rather high.

The data on coherence of precipitation in the pre-outbreak years are given in Table 3.14.

The highest coherence of precipitation events is observed in summer, and for every outbreak, there is a pre-outbreak year (or years) in which the significance level of precipitation coherence is higher than the critical value (Table 3.14).

The degree of coherence of autumn precipitation events is much lower than that of the summer ones, but it also exceeds the critical value 2 in many of the pre-outbreak years. The lowest level of coherence of precipi-tation events in the Angara region is observed in May: the significance level never rises above the critical value in the pre-outbreak years. Hence, spring precipitation cannot be a factor synchronizing the onset of out-breaks in the Angara region.

A more reliable parameter in this case may be hydrothermal coeffi-cient (HTC). Analysis of HTC coherence in these regions suggests the synchrony of the combination of the temperature and precipitation parameters (Fig. 3.7–3.9).

Table 3.15 lists the data on HTC coherence in the Angara region in pre-outbreak years.

The highest coherence of HTC values is observed in spring and au-tumn (Table 3.15). Moreover, in the three years preceding the most extensive pan-regional outbreaks of 1954 and 1994, HTC coherence was very high, with a time shift of no more than 10 days. On the other hand, the 1982 outbreak was quite moderate and short, and in the years preceding that outbreak, correlation coefficients of HTC were relatively low.


Table 3.14. The coherence of precipitation events in the Angara region in pre-outbreak years

Year Significance level based on 2 May June–August September

1939 0.03 4.60 4.461940 2.64 4.37 5.001941 0.14 6.27 1.151942* 0.76 19.05 4.001951 0.02 6.62 9.541952 4.29 7.07 6.271953 4.19 20.18 6.721954* 0.86 9.67 2.331964 0.00 5.48 3.201965 0.02 15.69 2.331966 0.33 11.68 4.041967* 0.41 3.12 5.491979 0.01 19.33 2.921980 0.58 13.13 6.651981 2.70 12.47 5.131982* 1.55 10.37 8.441991 1.77 4.03 12.131992 1.29 2.98 6.981993 0.02 27.16 2.041994* 0.52 9.16 6.45

* – years when outbreaks began; bold type shows values above 2 5.99.cr

Fig. 3.7. Synchrony of the May HTC values in the Angara region between 1931

and 1995. 1 – correlation coefficient; 2 – critical value


Fig. 3.8. Synchrony of the June–August HTC values in the Angara region

between 1931 and 1995. 1 – correlation coefficient; 2 – critical value

Fig. 3.9. Synchrony of the September HTC values in the Angara region between

1931 and 1995. 1 – correlation coefficient; 2 – critical value

*** Weather is one of the main modifying factors that influence changes in the densities of forest insect populations and the occurrence of outbreaks. Simplified approaches, which do not differentiate between the necessary and sufficient conditions of the influence of modifying factors on forest insect population dynamics and which use the simplest parameters of the relationship between modifying factors and changes in population densi-ty, such as correlation coefficients, often lead to wrong conclusions. In our opinion, identifying the relationship between weather changes and changes in population density may be difficult because of the delayed response of the population to the impact of modifying factors and the integrated effect of the weather factors on insects. Another difficulty is the possible temporally nonstationary sensitivity of insects to various weather parameters. Thus, identification of weather effects on population density needs to be based on detailed analysis employing mathematical and statistical methods.


Table 3.15. Correlation coefficients of HTC in the Angara region in pre-outbreak years

Year Correlation coefficient of HTC

May June–August September1939 0.54 0.23 0.351940 0.39 0.25 0.491941 0.64 0.22 0.641942* 0.65 0.32 0.601951 0.68 0.26 0.651952 0.70 0.32 0.341953 0.88 0.18 0.351954* 0.25 0.20 0.461964 1.00 0.26 0.681965 0.71 0.23 0.341966 1.00 0.18 0.601967* 0.71 0.23 0.801979 0.60 0.28 0.651980 0.44 0.30 0.291981 0.35 0.49 0.331982* 0.48 0.30 0.961991 0.47 0.41 0.571992 0.37 0.19 0.731993 0.37 0.52 0.821994* 0.99 0.23 0.82

Average value for the entire period 0.57 0.31 0.53

* – years when outbreaks began; 0.239 is the critical value of correlation coeffi-cient. Bold type shows values of correlation coefficient higher than the corre-sponding average values.

79

4 Spatial and Temporal Coherence of Forest Insect Population Dynamics

4.1 Coherence and synchronicity of population dynamics

Coherence of population dynamics of various insect species is an im-portant component that should be taken into account in the assessment of the effects of different factors on insect population dynamics. Coher-ence of population dynamics is indicative of the presence of an ecological mechanism responsible for the concordance between the time series of population dynamics of different species in the same habitat or the same species in different habitats. Thus, indicators of the coherence of insect population dynamics can also be used to indirectly assess the effects of different factors influencing these populations.

For the last few decades, coherence and spatiotemporal synchroniza-tion of population dynamics have been studied by a number of authors (Bascompte, Sole, 1998; Bjornstad et al., 1999; Bjornstad, 2000; Bone et al., 2013; Buonaccorsi et al., 2001; Choi et al., 2011; Curran, Webb, 2000; Foster et al., 2013; Hanski, Woiwood, 1993; Haydon, Steen, 1997; Haynes et al., 2012; Henttonen et al., 1985; Herrero et al., 2012; Kapeller et al., 2011; Liebhold, Kamata, 2000; Miller, Epstein, 1986; Myers, 1998; Peltonen et al., 2002; Raimondo et al., 2004; Ranta et al., 1998; Schowalter, 2012; Sutcliffe et al., 1997; Van Rossum, Triest, 2012).

They showed that the degree of coherence between the population dy-namics of the same species in different habitats monotonically decreases with the distance between these habitats. If the degree of coherence be-tween population dynamics does not decrease with the distance between




habitats, and this distance is considerably greater than the dispersal radius of the study species, this is indicative of the global spatial coherence of this species, which is determined by the population response to the impact of a powerful modifying factor (Liebhold et al., 2004).

Mechanisms responsible for the synchronous increase in the sizes of insect populations on a vast area have not been studied in detail yet. It has been assumed that synchronization of local insect outbreaks may be caused by an endogenous factor. Among these synchronizing factors may be changes in solar activity (Chizhevsky, 1973), summer droughts on vast areas (Kondakov 1974, 2002), etc. As the solar activity rhythm determines changes in the effects of solar radiation simultaneously on the entire planet, but outbreaks of different insect species are neither temporally nor spatially synchronized, it has been assumed that the synchronizing factor is the combination of the solar activity rhythm and local planetary rhythms (Maximov, 1989). A number of authors think that the reason for the coherence of the population dynamics of the same species in different habitats may be the Moran effect, which is related to the uniformity in weather patterns on an extensive area and similarity in responses of populations to weather changes in different habitats (Moran, 1953; Baars, Van Dijk, 1984; Bjornstad, Bascompte, 2001; Bjornstad et al., 1999; Liebhold et al., 2000; Maron, Harrison, 1997; Pollard, 1991; Williams, Liebhold, 1995; Volney, Fleming, 2000; Chizhevsky, 1973).

To quantify coherence of population dynamics of a particular species in different habitats or several species in the same habitat, one needs to have long-term surveys (lasting for a few decades) of these insect species on permanent sample plots. Unfortunately, such surveys are seldom done, and estimates of coherence based on short time series are not reliable (Anderson, 1971; Jenkins, Watts, 1969; Marple, 1990).

Forest insects can show two main types of coherence: spatial coherence of population dynamics of the same insect species in different habitats and temporal coherence of two species in the same habitat.

To quantify the coherence of two time series of population dynamics, we used a cross-correlation function. Calculation of the cross-correlation function ( )xy k of two stationary time series {x} and {y} with the mean values μх and μy and standard deviations σx and σy was based on the following formula (Box, Jenkins, 1970):

( ( ) ) ( ( ) )

( ) .x yxy

x y

E x t k y tk (4.1)

where Е is the expectation operator, 0, 1, 2,...k is the time shift.

Spatial and Temporal Coherence of Forest Insect Population Dynamics 81

Based on the shape of the cross-correlation function, one can estimate the coherence of the dynamics of two time series. Figure 4.1 shows exam-ples of stationary time series and the shapes of cross-correlation func-tions for different degrees of coherence of these time series.

А1. Time series of synchronous populations

А2. Cross-correlation function of these series

В1. Time series of coherent populations

Fig. 4.1. The types of coherence of two stationary time series


В2. Cross-correlation function of these series

С1. Time series of temporally non-coherent populations

С2. Cross-correlation function of these populations Fig. 4.1 (cont.). The types of coherence of two stationary time series

For the synchronous time series, maxima and minima coincide tem-

porally; the maximum of the cross-correlation function is observed at the time shift between the series k = 0 and at xy(0) 1 (Fig. 4.1, А1 and А2). For the coherent series with the k phase shift, the maximum of the cross-correlation function and value xy(k) 1 (Fig. 4.1, В1 and В2). For the non-coherent series, xy(k) 0 at any k (Fig. 4.1, С1 and С2).


Thus, to estimate the coherence of the population dynamics time series of the same species in different habitats or two species in the same habi-tat, one needs, first, to transform the time series into the stationary form, and then, to calculate the cross-correlation function for the transformed time series.

Cross-correlation functions were calculated by using the Statistica 6.0 package.

4.2 Spatiotemporal coherence of the population dynamics of defoliating insects in pine forests of Middle Siberia

The defoliating insects inhabiting the Krasnoturansk pine forest are repre-sented by the species usually occurring in pine forests in Siberia (Isaev et al., 1997; Palnikova, Kondakov, 1982; Palnikova, 1987; Palnikova, 1998; Tarasova, 1982; Epova, 1999; Yanovsky, 2003). These are the pine looper Bupalus piniarius L., the tawny-barred angle Semiothisa liturata Сl. (Geometridae), and the Dendrolimus pini L. (Lasiocampidae).

From 1979, population densities of these species were determined eve-ry year between August 10 and 20, by beating the trees to knock off the insects into a sheet spread on the ground. Annual surveys of defoliating insect populations were conducted in five types of habitats: Hill Top, Narrow Plane, Dunes, Terrace, Lake (Figure 2.1). The data of population surveys in different habitats are given in Table 4.1.

Table 4.2 gives the long-term annual average densities (insects per tree) of the study species populations and standard deviations of the mean in different habitats.

The numerator denotes long-term annual average population densi-ties; the denominator shows standard deviations of the mean.

As can be seen from the data in Table 4.2, population densities of the defoliating species in all habitats were low during the entire study period, and in most cases, they were no more than one individual per tree. Fig-ure 4.2 shows typical curves of the pine looper population dynamics in habitats “Dune” and “Narrow Plane”.

Based on the long series of field surveys, we studied the spatial and population coherence of population dynamics of the community of forest insect species in different habitats. The purpose of the study was to estimate spatial coherence of the population dynamics of the same insect species in different habitats and temporal coherence of the population dynamics of different insect species in the same habitat.


Tabl

e 4.1

. Den

sitie

s of p

opul

atio

n (in

divi

dual

s per

tree

) in

Kras

notu

rans

k pi

ne fo

rest

Year

Bu

palu

s pin

iariu

s L.

Sem

ioth

isa li

tura

taСl

.D

endr

olim

us p

iniL

.

Terr

ace

Dun

es

Nar

row

Pl

ane

Lake

H

ill

Top

Terr

ace

Dun

esN

arro

w

Plan

eLa

keH

ill

Top

Terr

ace

Dun

es N

arro

w

Plan

eLa

keH

ill

Top

1979

0

0.04

0 0.

2 0

00

00.

1 0.

06

0.06

0.04

1980

0

0 0

0 0

0.04

00

00.

080.

16

0.06

0.

120.

060.

1219

81

0 0.

06

0 0.

02

0 0

0.04

00

0.04

0.38

0.

38

0.14

0.12

0.28

1982

0

0 0

0 0

00

00

00.

02

0.02

0.

020.

060.

0419

83

0 0

0 0

0.04

0

00

00

0.02

0.

04

00

019

84

0 0.

06

0 0

0 0.

020

00

00

0 0

00

1985

0

0.02

0

0 0.

04

00.

020

00

0 0

00

019

86

0.04

0.

5 0.

18

0.14

0.

16

0.1

0.08

0.02

0.12

00

0 0

0.02

0.04

1987

0.

1 2.

66

0.7

0.08

0.

2 0.

020.

160.

030.

460.

080.

1 0.

1 0.

130.

060.

219

88

0.2

9.44

3.

38

0.6

3 0.

040.

360.

150.

330.

080.

04

0.08

0.

10.

050.

1619

89

0.28

13

.51

5.6

1.56

10

.88

0.02

0.06

00.

160.

080

0.12

0.

050.

060.

1219

90

0.76

13

.16

4.04

0.

24

5.4

00.

040

0.16

0.08

0.08

0.

08

0.24

0.16

0.12

1991

0.

52

6.88

2.

64

0.2

1.6

0.12

0.08

0.04

0.16

0.08

0.44

0.

84

0.72

0.12

0.28

1992

0.

28

0.96

0.

88

0.16

0.

52

0.28

0.32

0.08

0.12

0.04

1.72

2.

2 1.

20.

040.

6819

93

0.64

0.

48

0.32

0.

04

0.84

1.

561.

320.

240.

120.

43.

08

1.08

0.

40.

120.

2419

94

0.04

0.

44

0.04

0

0.2

0.28

0.4

00.

080

2.96

2.

96

0.96

0.08

0.68

1995

0.

05

0.25

0

0 0

0.2

0.2

0.05

0.1

0.05

0.2

0.2

0.05

00

1996

0.

05

0.1

0 0.

05

0 0.

150.

150.

050.

10.

10

0.2

00

0.05

1997

0.

05

0.05

0.

05

0 0

0.1

0.05

0.05

0.05

0.05

0 0

00

019

98

0.05

0.

05

0.15

0

0 0.

10

00

00

0 0

00


Ta

ble 4

.1 (c

ont.)

. Den

sitie

s of p

opul

atio

n (in

divi

dual

s per

tree

) in

Kras

notu

rans

k pi

ne fo

rest

Year

Bu

palu

s pin

iariu

s L.

Sem

ioth

isa li

tura

taСl

.D

endr

olim

us p

iniL

.

Terr

ace

Dun

es

Nar

row

Pl

ane

Lake

H

ill

Top

Terr

ace

Dun

esN

arro

w

Plan

eLa

keH

ill

Top

Terr

ace

Dun

es N

arro

w

Plan

eLa

keH

ill

Top

1999

0

0.2

0.1

0 0

0.05

00

00

0.05

0.

05

00

020

00

0 0.

1 0.

15

0 0

00

00

00

0 0

00

2001

0

0.1

0.2

0.05

0

00

00

00

0 0

0.05

020

02

0 0.

2 0.

65

0 0.

2 0.

050.

10

0.05

0.05

0 0

00

020

03

0.15

0.

5 1.

35

0.25

0.

6 0.

50.

30.

050.

150.

150.

15

0 0.

050.

050.

0520

04

0.2

0.55

2.

65

1.05

0.

5 0.

550.

350.

20.

950.

550

0.2

00.

20.

0520

05

0 0.

35

0.55

0.

35

0.85

0.

70.

650.

22.

050.

450.

2 0.

1 0.

050.

20.

1520

06

0.05

0.

45

0.2

0.05

0.

15

0.2

0.3

0.2

0.4

0.4

0.8

0.85

0.

150.

650.

520

07

0.15

0.

35

0.25

0.

15

0.1

0.15

0.5

0.15

0.15

0.2

1.1

0.95

0.

40.

40.

8520

08

0.05

0.

25

0 0.

05

0.1

00.

050.

050

02

1.25

0.

40.

21.

320

09

0.1

0.15

0.

05

0 0.

05

0.05

00

00

0.15

0.

1 0.

050

0.05

2010

0.

1 0.

55

0.7

0.1

0.25

0

00

0.05

00.

05

0.05

0

00

2011

0.

05

0.55

0.

1 0.

25

0.35

0

00

00

0.05

0

00

020

12

0.2

1.2

0.05

0.

35

0.65

0.

050

00

00

0.07

0.

050.

050

2013

0

0.3

0.15

0.

5 0.

3 0.

10.

10.

050.

10.

050.

1 0

0.05

00.

0520

14

0.25

0.

5 0.

35

0.2

0.1

0.6

0.2

0.15

0.3

0.15

0.3

0.2

0.15

0.2

020

15

0 0.

35

0.2

0.4

0.1

0.35

0.05

0.05

0.65

0.08

0.1

0.1

0.25

0.25

0.05

2016

0

0.05

0.

1 0.

1 0

0.05

0.1

0.1

0.6

0.17

0.1

0.15

0.

050.

050.

1


Table 4.2. Long-term annual average densities (insects per tree) of the study species populations and standard deviations of the mean in different habitats

Species Habitat*

Terrace Dune Narrow Plane Floors Hill Top

Semiothisa liturata Сl. 0.16/0.30 0.16/0.27 0.05/0.07 0.17/0.38 0.09/0.14

Bupalus piniarius L. 0.12/0.19 1.59/3.56 0.76/1.36 0.17/0.33 0.79/2.07

Dendrolimus pini L. 0.42/0.84 0.35/0.67 0.16/0,29 0.08/0.13 0.18/0.30

Fig. 4.2. Density (individuals tree–1) of the pine looper B. рiniarius larvae

in habitats “Dune” (1) and “Narrow Plane” (2) between 1979 and 2012 Analysis of the data in Tables 4.1 and 4.2 showed that no temporal

trend was observed in the population densities of these species, and that their population dynamics in the given habitats can be regarded as a stationary process, with cyclic and random components. To study the periodicity of the population dynamics, we transformed the time series by using the approach described in Chapter 2 and calculated the func-tions of power spectra of the insect species in all habitats for the trans-formed series (LTI series).

Figure 4.3 shows the curves of the power spectra of typical LTI series of pine looper population dynamics.

The power spectrum curves in Figure 4.3 have statistically significant peaks at f = 0.067 Hz, i.e. for these series, there are periodic population

buildups, every 1 1 150.067f

years on average.


Fig. 4.3. Spectral density of the transformed time series of the pine looper

(B.piniarius) populations in Krasnoturansk pine forest habitats (1 – Terrace; 2 – Dune; 3 – Narrow Plane; 4 – Lake; 5 – Hill Top)

However, cyclic fluctuations in the population density of defoliating

insects are not observed in every habitat. For example, for the D.pini population, cyclic fluctuations are observed in two habitats: “Dune” and “Terrace”. In the habitats with cyclic population dynamics, the peaks of the function of power spectrum for both the B.piniarius and the D.pini occur with the same frequency, f = 0.067 1/year, i.e. the period T between two adjacent peaks is 15 years for the population densities of both the B.piniarius and the D.pini .

Figure 4.4 shows LTI series of the population dynamics of the B.piniarius and the D.pini in habitat “Dune”, and Figure 4.5 presents the cross-correlation function for the time series of these two species in the same habitat.

The statistically significant shift in the cross-correlation function in Figure 4.5 indicates that there is a three-year lag between the maxima of the time series of the B.piniarius population dynamics and the maxima of the time series of the D.pini population dynamics in this habitat.

Cross-correlation functions for evaluation of the temporal coherence of different defoliating species in the same habitat were calculated for every species in every habitat (Table 4.3).

The data in Table 4.3 show that for all habitats in the study area, there is a 1–4 year delay of the population dynamics of the tawny-barred angle and the pine-tree lappet relative to the population dynamics of the pine


Fig. 4.4. LTI series of the population dynamics of the B.piniarius (1)

and the D.pini (2) in habitat “Dune”

Fig. 4.5. Cross-correlation function between the LTI series of the B.piniarius

and the D.pini population dynamics in habitat “Dune”

looper. The higher the average density of pine looper population com-pared to the average densities of the other defoliators in this habitat, the greater the lag of the dynamics of these species relative to the pine looper dynamics.

Table 4.4 characterizes spatial coherence of the pine looper and pine-tree lappet populations in different habitats.

The data above the main diagonal in Table 4.4 show that although the time series of pine looper populations in different habitats differ in their absolute values (see Table 4.2), their temporal change is synchronous, except for the series in habitat “Terrace”, which shows a time lag of one – two years.


Table 4.3. Characterization of cross-correlation functions for evaluation of the temporal coherence of different species in the same habitat*

Pairs of species Landscape features (habitats)

Terrace Dune Lake Narrow Plane Hill Top

pine looper – tawny-barred angle –3/0.56 –4/0.61 –3/0.39 –1/0.52 –4/0.25

pine looper – pine-tree lappet –3/0.66 –4/0.76 –3/0.88 –2/0.56 –3/0.40

tawny-barred angle –pine-tree lappet –1/0.59 –1/0.72 –1/.38 –1/0.84 –2/0.58

* The numerator denotes the delays (years) of the peaks in dynamics of the second species relative to the first one; the denominator shows the maximum value of the cross-correlation function. Table 4.4. Characterization of cross-correlation functions of a particular species

in different habitats (the data for the B.piniarius are above the main diagonal, and the data for the D.pini are below the main diagonal)

Habitat Habitat

Terrace Dune Narrow Plane Floors Hill Top

Terrace Х –2/0.77 –1/0.76 –1/0.57 –1/0.78 Dune 0/0.79 Х 0/0.93 0/0.63 0/0.90Narrow Plane 0/0.95 0/0.89 Х 0/0.82 0/0.90Floors 0/0.80 0/0.99 1/0.89 Х 0/0.78Hill Top 0/0.80 0/0.99 1/0.89 1/0.99 Х

For D.pini populations, the degree of spatial coherence of population

dynamics is considerably higher than for the pine looper population (Table 4.4, the data below the main diagonal). The population dynamics of this species are synchronous in different habitats.

Analysis shows that synchronous population dynamics of different species of defoliating insects can occur in habitats located several kilome-ters apart. At the same time, our calculations show that the time series of the population dynamics of the same species are not necessarily coherent and synchronous, even in habitats situated rather close to each other. The non-coherence of the time series of the same insect species in different habitats suggests that neither the migration-related effect nor the Moran effect occurs in this case, although the habitats are situated at distances of only a few kilometers away from each other.


4.3 Spatiotemporal coherence of population dynamics of defoliating insects in the Alps

Regular surveys of insects defoliating larch – Zeiraphera griseana Hub., Oporinia autumnata, Exapate duratella, Pristiphora laricis, and Spilonota laricana – were conducted in four habitats in the Alps (Baltensweiler, 1991):

Goms in the Upper Rhone River Valley (Kanton Wallis, Switzer-land) – 1300–1600 meters above sea level; Oberengadin (Kanton Graubunden, Switzerland) – the valley is situated at 1600–1800 m above sea level; Valle Aurina (Provincia Autonoma di Bolzano-Alto Adige, Italy); Lungau (Land Salzburg, Austria).

Figure 4.6 is a map showing where the habitats are located in the Alps.

Fig. 4.6. Study areas in the Alps: A – Goms in the Upper Rhone River Valley;

B – Oberengadin; C – Valle Aurina; D – Lungau Of the species mentioned above, the larch bud moth Zeiraphera

griseana Hub. is a dangerous pest of the European larch in the Alps (Baltensweiler, Fischlin, 1988). Larch bud moth outbreaks occur at alti-tudes of 1600–2100 m every 8.47 ± 0.27 years and last 2.93 ± 0.21 years.

Surveys of insects defoliating the European larch in the Alps were con-ducted in the same manner as the surveys of the insects defoliating the Scots pine in Siberia described above, and the only difference was the greater distances between the study habitats. Table 4.5 gives these dis-tances.

We used the data of surveys of larch defoliators in four habitats (Baltensweiler, 1991) for further analysis. The data were transformed, and then cross-correlation functions characterizing coherence of the


dynamics of the same species in different habitats and different species in the same habitat were calculated for the LTI series.

Figure 4.7 shows population dynamics of LTI series of the larch bud moth and Pristiphora laricis, and Figure 4.8 shows population dynamics of LTI series of Pristiphora laricis and Spilonota laricana in Oberen-gadin. Table 4.5. Distances (km) between habitats in the Alps where surveys of popula-

tions of insects defoliating the European larch were carried out

Habitat HabitatGoms Oberengadin Valle Aurina Lungau

Goms 0 119 294 467Oberengadin – 0 181 357Valle Aurina – – 0 176

Fig. 4.7. LTI series of Zeiraphera griseana (1) and Pristiphora laricis (2) popula-

tion dynamics in the Swiss Alps (data found in (Baltensweiler, 1991))

Fig. 4.8. LTI series of Pristiphora laricis (1) and Spilonota laricana (2)

population dynamics in the Swiss Alps (data found in (Baltensweiler, 1991))


Based on the transformed LTI series of the defoliating insect popula-tion dynamics, we calculated cross-correlation functions of the time series of the defoliating insect population dynamics. Figure 4.9 shows the cross-correlation function of Zeiraphera griseana and Spilonota laricana in Oberengadin.

Fig. 4.9. The cross-correlation function of the time series of Zeiraphera griseana

and Spilonota laricana population dynamics in Oberengadin The time series of the population dynamics of these species in the

same habitat are practically coherent, and the series of Spilonota laricana dynamics shows a lag of only one year relative to the series of Zeiraphera griseana dynamics.

Coherence is also observed between the time series of populations of two other species – Oporinia autumnata and Exapate duratella – in Oberengadin (Fig. 4.10).

Table 4.7. Lists parameters of the lag k and the maximum value

of the cross-correlation functions in the same habitat (Oberengadin)

species number

species number2 3 4 5 6 7

1 4/0.62 2/0.60 0/0.65 1/0.72 1/0.69 1/0.812 –1/0.68 –4/0.55 –3/0.74 –2/0.81 –3/0.82 3 –2/0.50 –1/0.73 0/0.71 –1/0.74 4 1/0.78 2/0.63 1/0.765 1/0.86 0/0.916 1/0.94


Fig. 4.10. The cross-correlation function of the time series of Oporinia

autumnata and Exapate duratella population dynamics in Oberengadin Time series of population dynamics of different species in the same

habitat subjected to the identical impacts of external modifying factors are evidently coherent but not synchronous, and the phase lags between the dynamics of different species are not equal to each other. Thus, Moran’s hypothesis does not hold, and responses of different defoliating insect to the same external effects may differ in phase. The population density of the leading species – Zeiraphera griseana – consistently increases before the population densities of other species in the community do.

It would be very interesting to estimate the coherence of the same spe-cies in different habitats situated far away from each other. In this case, weather conditions may differ, and they may not be a factor of synchro-nization or coherence of the species population dynamics in different habitats.

Figure 4.11 shows the cross-correlation function of the larch bud moth in habitats Oberengadin and Goms, which are situated 119 km apart in a straight line.

Figure 4.11 demonstrates that larch bud moth populations in these habitats are coherent, but the species population dynamics in Goms shows a lag of three years relative to the species population dynamics in Oberengadin. Thus, the data on the population dynamics of the larch bud moth in Oberengadin can be used to predict the population dynam-ics of this species in Goms.

Table 4.8 presents cross-correlation functions for larch bud moth pop-ulations in different habitats in the Alps.

The data in Table 4.8 suggest that an outbreak first starts in the “lead-ing” habitats – Oberengadin and Valle Aurina – and then, in two or three


years, in the two other, “lagging”, habitats. Thus, analysis of cross-correlation functions offers an approach to the forecast of population dynamics in lagging habitats based on the data on the species population dynamics in other, leading, habitats.

Fig. 4.11. The cross-correlation function of the larch bud moth

in habitats Oberengadin and Goms

Table 4.8. Cross-correlation functions for larch bud moth populations in different habitats in the Alps

Habitat Habitat

A Oberengadin

B Goms

C Valle Aurina

D Lungau

A Oberengadin 1 –3/0.81 0/0/84 –1/0.84

B Goms – 1 2/0.80 1/0.80

C Valle Aurina – – 1 –1/0.85

4.4 Global coherence of pine looper population dynamics in Eurasia

It is very interesting to study the global coherence of the population dynamics of a single species, whose populations are several hundred or


even thousand kilometers apart, which eliminates migratory processes and local weather effects as factors of the coherence of the species popu-lation dynamics in different habitats. In this case, possible coherence may be only associated with the response of these populations to the effects of global geophysical and climate factors, such as solar activity (Chizhevsky, 1973). However, reliable statistical analysis of the coherence of time series must be based on continuous surveys of the same species in habitats located far away from each other, at the same time, for 30 years or longer. Such data exist, and Figure 4.12 shows the cross-correlation function of the time series of pine looper population dynamics obtained in two habitats in Scotland, which are 150 km apart, during 1953–87 (Broekhuizen et al., 1993, 1994).

Fig. 4.12. The cross-correlation function of the time series of pine looper popu-

lation dynamics obtained in two habitats in Scotland (4´15 W, 57´32 N and 2´50 W, 56´25 N) between 1953 and 1987 (as reported by Broekhuizen et al., 1993, 1994)

The values of the cross-correlation function of these time series are

non-significant for all phase delay, suggesting the absence of coherence of the pine looper population dynamics in these habitats. Thus, at a distance of a few kilometers, the time series of the same species (pine looper) could be synchronous or coherent, but at a distance of a few hundred kilometers, no synchronization or coherence occurred, and external modifying factors that could synchronize pine looper population dynam-ics in habitats several hundred kilometers apart in Scotland were not effective.


4.5 Synchronization of the time series of gypsy moth population dynamics in the South Urals

Tree stands of the Chelyabinsk Oblast are inhabited by a West-Siberian geographic subspecies of the gypsy moth (Gninenko, 1998). This subspe-cies occurs in the north-west of Asia – steppe, forest-steppe, and south-taiga stands of the Trans-Ural region, West Siberia, and North Kazakh-stan. Surveys of gypsy moth population were carried out in three land-scape-geographical latitudinal zones: mountain forest, forest-steppe, and forest (Vyaznikov, Sokolov, 2006).

The mountain forest zone (MFZ) includes the north-west mountain-ous part of the Chelyabinsk Oblast, stretching as far as the boundary of Bashkortostan; its area is 21,100 km² (or 23.9% of the Oblast area). Mountain ridges occupy 90% of the area. The average annual precipita-tion is 600–800 mm, the wooded lands occupy 18400 km², the forested area is 15,800 km², the percentage of forests is 74.9%, and birch stands occupy about 640,000 hectares.

The forest-steppe zone (FSZ) is located in the north-east of the Oblast. The total area of the zone is 32,200 km² (36.4% of the Oblast area); the average annual precipitation is 400–450 mm, the wooded lands occupy 8300 km², the forested area is 7400 km², the percentage of forests is 23.0%, and birch stands occupy about 400,000 hectares.

The steppe zone (SZ) occupies the south of the Oblast. The total area of the zone is 35,200 km² (39.7% of the Oblast area); the average annual precipitation is 250–300 mm, the wooded lands occupy 3000 km², the forested area is 2000 km², the percentage of forests is about 6%, and birch stands occupy about 142,800 hectares.

The spatial coherence of gypsy moth populations in different natural zones of the South Urals may be associated with both migratory process-es and similar weather conditions in all zones.

To evaluate the degree of coherence of population dynamics, we calcu-lated cross-correlation functions of the time series {Lj(s)} of gypsy moth population dynamics in different natural zones. Our calculations showed that the degree of synchrony between the gypsy moth population dynam-ics in the FSZ and SZ is rather high (Fig. 4.13).

With the time delay k = 0, the cross-correlation coefficient for the time series of gypsy moth population dynamics in the FSZ and SZ was the highest – 0.65. The explanation of the synchrony of population dynamics in different natural zones could be based on the assumption that the synchrony of population processes follows from the synchrony of weath-er processes in the study areas (Moran, 1950, 1953). To check this as-


sumption, we calculated autocorrelation functions of the time series of the total hydrothermal coefficient (HTC) for May–July in these natural zones and cross-correlation functions characterizing the relationships between weather parameters and the current population densities (Fig. 4.14 and 4.15).

Autocorrelation coefficients of the HTC time series for the FSZ do not differ significantly from zero (Fig. 4.13), suggesting the absence of perio-dicity in weather changes. Coefficients of the cross-correlation function of weather conditions and population density for the FSZ are small, too (although they are significant for k =1 and 2) (Fig. 4.14). Similar results were obtained for the SZ.

Fig. 4.13. The cross-correlation function between the time series {Lj(s)}

of gypsy moth population dynamics in the FSZ and SZ

Fig. 4.14. Autocorrelation function of the total HTC for spring

and summer (May–June–July) in the FSZ


Fig. 4.15. The cross-correlation function of the time series of the total HTC for spring and summer and logarithmically transformed density {L(s)} in the FSZ Thus, there is no evidence suggesting considerable synchronization of

gypsy moth population dynamics in different natural zones caused by the effect of the weather. Migration of adults cannot be a synchronizing factor, either, as the habitats where surveys of the gypsy moth were conducted are separated by several dozen and even hundred kilometers. Variations in the global geophysical field, which are characterized by the Wolf numbers (ftp://ftp.ngdc.noaa.gov; http://www.gao.spb.ru), might be a synchronizing factor of population dynamics. However, our calcula-tions of the cross-correlation function between the time series of the Wolf numbers and the time series of the gypsy moth population dynam-ics in different habitats in the South Urals showed no significant relation-ship between the dynamics of these time series, and, thus, geophysical factors do not synchronize insect population dynamics at characteristic distances of several hundred kilometers.

*** Methods of correlation and spectral analysis are useful for studying coherence of population dynamics of defoliating insects in their local habitats. Coherence may be caused by different mechanisms, depending on the distance between the habitats and their landscape properties. It is very important to identify ecological mechanisms responsible for the coherence between population density oscillations to be able to under-stand coherent spatial and temporal changes in population dynamics. The constancy of the phase shift between pest population density series in different habitats may be used as an effective indicator of the future


changes in the population state. If the habitats are situated relatively close to each other (no more than several kilometers apart), coherence may be caused by migration of insects and weather factors. Migration of adults cannot be a synchronizing factor when the habitats where surveys of the gypsy moth are conducted are separated by several dozen and even hundred kilometers. In this case, variations in the global geophysical field, which are characterized by the Wolf numbers (ftp://ftp.ngdc.noaa.gov; http://www.gao.spb.ru), might be a synchro-nizing factor of population dynamics. However, our calculations of the cross-correlation function between the time series of the Wolf numbers and the time series of the gypsy moth population dynamics in different habitats in the South Urals showed no significant relationship between the dynamics of these time series, and, thus, geophysical factors do not synchronize insect population dynamics at characteristic distances of several hundred kilometers. The factors of the global and local coherence of single species population dynamics remain unclear so far.

101

5 Interactions Between Phytophagous Insects and Their Natural Enemies and Population Dynamics of Phytophagous Insects During Outbreaks

5.1 Entomophagous organisms as a regulating factor in forest insect population dynamics

Entomophagous organisms constitute an integral component of trophic chains in forest insect communities and play a notable role in population dynamics of phytophagous insects. Therefore, many researchers have addressed various aspects of the parasite – host interactions in their stud-ies (Baltensweiler, Fischlin, 1988; Barbosa, Schultz, 1987; Berryman, 1988, 1992, 2002; Gould et al., 1990; Klemola et al., 2014; Liebhold et al., 2000; Rafes, 1978; Reilly et al., 2014; Viktorov, 1976; Viktorov, Guryanova, 1974; Vorontsov, 1984; Znamensky, 1977). The most abundant literature data can be found on the species composition of parasites infesting phytophagous species that have economic importance. Nearly all studies investigating parasites of the most harmful insects in Siberian forest also identified the species composition of the parasites and examined their biology (Boldaruev, 1969; Kolomiets, 1962, 1987, 1989, 1990; Kolomiyets, Artamonov, 1994; Markova, Manzhela, 2013; Prozorov et al., 1963).




Analysis of results of field studies suggests that for many phyto-phagous species, entomophagous organisms are powerful regulators of population dynamics. For instance, the study performed during three outbreaks of the gypsy moth Lymantria dispar L. in oak forests of the Saratov Oblast revealed about 40 species of entomophagous insects. The most effective of them were an egg parasitoid Anastatus japonicus Achm., parasites of larvae Apanteles melanoscelus Ratz., Apanteles liparidis Bouche., and Parasetigena silvestris R.-D., and a parasite of pupae Blepharipoda scutellata R.-D. The most effective parasites of the gypsy moth in the phase of the high-density population are dipterous parasites B. scutellata and P. silvestris (Lyamtsev, 2003).

During an outbreak of the gypsy moth on the Privolzhskaya Vozvyshennost, where the gypsy moth outbreak was accompanied by buildups of populations of the brown-tail moth, the common lackey, and several species of cutworms, loopers, and leaf-rollers, in the first year of the outbreak peak, parasite-caused mortality of gypsy moth 1–3 instar larvae was 44.9%, 4–6 instar females – 74.9%, and female pupae – 80.2%. That caused a dramatic decline in the gypsy moth density (Panina, Belov, 2012).

The Siberian variety of the gypsy moth shows high migratory activity in the stages of early-instar larvae and adults (Kondakov, 1963). Hence, at the outbreak peak, outbreak sites are very mobile, and the gypsy moth escapes from parasitism. However, the East-Asian variety of the gypsy moth, inhabiting the Russian Far East, has reduced migratory activity, and, thus, outbreak sites are rather stationary. That is why, the same tree stands are damaged over 2–3 successive years. As the outbreak declines, activity of entomophagous insects in the outbreak sites increases dramat-ically (Epova, Pleshanov, 1988).

Egg parasitoids make a great contribution to the decrease in the popu-lations of the Siberian silk moth (Dendrolimus superans sibiricus Tschetv.) and the white-lined silk moth (D. superans albolineatus Mats.) in the Russian Far East (Yurchenko, Turova, 2002). In 1975 and 1976, between 51% and 96% of the eggs of the Siberian silk moth died in cedar-broadleaved forests. In 1987, in the region of abundant population of the Siberian silk moth, parasitized eggs constituted 64% in cedar-oak forests and 47% in valley forests. In the larch forests of the Amur Oblast, in 1982, 1983, 1989, and 1990, parasitized eggs constituted 50% in the years of Siberian silk moth population buildup, while in the years of the out-break decline, their percentage increased to 80–99%.

In Central Yakutia, in the years when the Siberian silk moth caused severe damage to larch forests (1998–2001), the Siberian silk moth popu-

Interactions Between Phytophagous Insects and Their Natural Enemies 103

lation buildup was accompanied by a gradual buildup of egg parasitoid populations, and in 2000, 70–100% of the eggs of the Siberian silk moth were infected by Telenomus tetraeneus Thorns, and the pest population declined considerably (Chikidov, 2009).

Different forest insect species may be more susceptible to parasitism in different life stages. For some forest defoliators, the main density regula-tors are egg parasitoids, other species are more frequently attacked in the larval stage, usually by the parasites of later instars, and still others – by larval-pupal or pupal parasites (Gninenko, 1998; Kolomiyets, 1962; Kondakov, Sorokopud, 1982; Martynova, 1967; Raspopov, 1973). Studies conducted in outbreak sites of web-spinning sawflies in North Kazakh-stan (Gninenko, Simonova, 2001) showed that egg parasitoids of Trichogramma sp. might be very important regulators of population densities of these phytophagous insects. In the outbreak site of the pine false webworm Acantholyda erythrocephala L. located in the Urumkaiskiy forestry in the Kokchetav Oblast, this parasite killed 88.7% eggs in 1983, 59.8% eggs in 1984, and 60.4% eggs in 1987. As a result of that, the outbreak that had lasted for 10 years subsided in 1988.

In European forests, 49%, and in some cases 60–70%, of the eggs of the pine looper (Bupalus piniarius L.) were parasitized by Trichogramma (Schwerdtfeger, 1939; Subklew, 1939). The high rate of parasitism of pine looper eggs by Trichogramma (reaching 100% in some sites) was one of the major reasons why pine looper outbreaks subsided in forests of Poland in 1961–1962 (Sliva, 1969). N.N. Egorov (1959) reported that parasites were the reason of pine looper outbreak decline in ribbon pine forests in 1931–1936. P. M. Raspopov (1973) reported the rates of para-sitism of pine looper pupae in the Chelyabinsk Oblast between 85% and 100% in different years.

Almost all researchers note that the rate of parasitism increases rapidly when the density of the host population reaches its maximum and the amount of food available to the host drops dramatically (Kolomiyets, 1989; Kondakov, 2002; Logoida, 1992; Raspopov, 1973; Yurchenko, Turova, 2002).

Populations of some defoliating insect species, however, are not affect-ed by parasites. It is well known that parasites do not influence consider-ably the long-term population dynamics of the birch sawfly Croesus (Nematus) septentrionalis L. (Sokolov, 1997), the winter moth Operophthera brumata L. (Dubrovin, 1986; Embree, 1965; Mrkva, 1968; Rubtsov, Utkina, 2011), and the sawfly Acantholyda stellata Christ. (Kolomiyets, 1967). For pine looper populations in Siberian forests, in contrast to pine looper populations in Europe, the rate of parasitism by


egg parasitoids is not high, and these parasites are practically unim-portant (Kolomiyets, 1977; Prozorov, 1956).

Parasites are not the main regulators of gypsy moth (L. dispar L.) and nun moth (Lymantria monacha L.) populations in tree stands of the Urals (Koltunov, 2006; Koltunov et al., 2014). There are 69 known entomophagous species that can attack the nun moth in Siberia (Kolomiyets, 1990), but, as a rule, the main factor causing outbreaks of these species to decline in the Urals and Siberia is viral epizootic (Bakhvalov et al.,1998; Golosova, 2003; Sokolov, Sokolova, 2010).

Generally speaking, the list of examples of different parasite – host in-teractions cannot prove or disprove anything. Valid arguments are need-ed to correctly explain differences in the parasite – host interactions for different insect species under different conditions.

A possible approach is to see how the behavior of parasite and host populations is described by different mathematical models. For example, the classical Lotka-Volterra model describes interactions between the model host (phytophagous insect) population of density x and the para-site (entomophagous insect) population of density y (Bazykin, 2003; Berryman, 1992).

( ) ,

,

dx kx A x bxydtdy cY mbxydt

(5.1)

where k, A, b, c, m are some constants. As a parameter of susceptibility of the host population to the impact of

parasites, we can use variable dxdy

. Then, from (5.1) we obtain

( ) .kx A x bxydxdy cy mbxy

(5.2)

The type of the parasite – host interaction will be determined by the

sign of value dxdy

. The case when 0dxdy

(cooperative effect) is ecologi-

cally unrealistic, but there may be cases when 0dxdy

(the effect of the

parasites decreases host population density) and 0dxdy

(there is no

effect of parasites on the host population).


If the host population density is low, A x and ( ) 0( )

x Ak bydxdy y mbx c

provided that .Ak by If the host population density is high and close

to A, ( )

bxydx bdy y mx c mx c

. If the host population density is suffi-

ciently high, b is sufficiently large and ,mx c then 0dxdy

, i.e. the

effect of the parasites causes host population density to decrease. These conditions, however, may not hold, and then at a high host population

density, 0dxdy

, i.e. the parasite has no significant effect on the host

species. Thus, the Lotka-Volterra model (5.1) suggests the possibility of vari-

ous kinds of interactions between parasite and host populations. In order to determine the type of interactions between particular spe-cies of parasites and hosts under definite conditions, one needs to find values of the coefficients of the systems (5.1). This, however, will in-volve a lot of effort, and, for example, it will be extremely difficult to estimate the density of the host population and its rate of parasitism in the stable sparse state, when the phytophagous insect population densi-ty is very low.

In this book, we present a model of phytophagous insect – entomophagous insect interactions that enables semi-quantitative esti-mation of the type of these interactions and determines the conditions under which entomophagous organisms can regulate the density of phytophagous insect populations.

For analysis of the phytophagous – entomophagous insect interactions in different phases of the outbreak cycle of defoliating insects, we used the data on population dynamics and rates of parasitism of pine looper (Bupalus piniarius L.) and engrailed moth (Ectropis (=Boarmia) bistortata Gz.) pupae.

Surveys of the pine looper and the rate of parasitism of its pupae were conducted by the authors during 1978–91 in pine forests of the Krasno-yarsk Territory, Altai, and North Kazakhstan. A detailed description of pine looper habitats, population dynamics, and outbreak phases was given elsewhere (Palnikova et al., 2002).

The pine looper is a Palearctic species occurring in the south of Siberia and the Russian Far East (Epova, Pleshanov, 1995). The pine looper was first found in Siberia in the Tubinskiy forest range of the Krasnoyarsk


Territory (Kuraginskii timber company in the West Siberian Territory) in 1930. Pine looper outbreaks spread over the entire Tubinskiy forest range, situated between 53 and 55°N and between 93 and 96°E. The insects damaged about 1 million hectares of fir stands (Prozorov, Zakrevsky, 1939). S.S.Prozorov reported that the peak of the pine looper outbreak was in 1931–1932, and in 1933, it naturally subsided, after more than 50% of the fir trees damaged by pine looper larvae had died (Prozorov, 1955).

5.2 A “phytophagous – entomophagous insect” model

For evaluation of the effects of parasites on the forest insect population, we used a model of the phytophagous – entomophagous insect system characterized by a considerable delay in the response of the entomophagous insect to the change in the density of the phytophagous insect population (Isaev et al., 2001). The phytophagous – entomophagous insect system dynamics, in which the characteristic time 1 of the host population is significantly shorter than the characteris-tic time 2 of the parasite population, will be described with the following system of equations:

( )( ) ,

,

dx kx A x x B mzdt

dz ax bzdt

(5.3)

where 1

2

1 ; ywx

is the rate of parasitism of individuals in the

host population; 0 w 1; 1

wzw

is the normalized rate of parasitism

by entomophagous insects, 0 z . At the zero parasitism rate w, the normalized parasitism rate z = 0.

At a very high parasitism rate, when w 1, z . System (5.3) is used to describe the behavior of various biological sys-

tems in which regulation of one component by another is considerably delayed. Model (5.3) (the so-called FitzHugh – Nagumo model describes excitation of neuron membranes (Anishchenko et al., 2003).


The steady states of the system (5.3), when 0dx dzdt dt

, can be char-

acterized with zero isoclines 0, 0dx dzdt dt

:

( )( ) 5.4.1

5.4.2

kz x A x x Bma cz xb b

(5.4)

The stable state of the parasite – host system with the host population

density х1 and normalized parasitism rate 11

1

yzx

is represented by the

point of intersection of zero isoclines (Fig. 5.1).

Fig. 5.1. Zero isoclines of model (5.4);

1 – zero isocline (5.4.1); 2 – zero isocline (5.4.2) When the population density increases to values higher than density

х1, but lower than хr, the system returns to state (х1, z1). If the population density exceeds value хr, the population quickly passes to the state with


density х2, with z = z1. Point х2, however, is not stable, and the system will drift, gradually decreasing population density, but considerably increas-ing the rate of parasitism. Then, as soon as population density reaches хm, it drops dramatically, and the population returns to the state (х1, z1).

For the parasite – host system (5.3), we propose the following classifi-cation of the stages of changes in population density x and the rate of parasitism of host insects w:

1. Stable state: х = х1, w = w1; in this state, the values of population density and parasitism rate oscillate around values x1 and w1, with a constant phase shift between changes in density and parasitism rate; the larger the phase shift, the larger the amplitude of popula-tion density oscillations; in fact, this case can be described with system (5.1);

2. Outbreak start: x > xr, w < w1; the population density increases rap-idly, the rate of parasitism decreases dramatically, allowing the host population to escape from the parasite;

3. Outbreak peak: the host population density x x2; parasitism rate w slowly grows; the host population reaches its highest density, and an outbreak occurs;

4. Population decline and crisis: х decreases, while w continues to grow;

5. Extremely low density: х drops dramatically, and the host popula-tion density decreases below х1; the rate of parasitism, w, begins to decrease;

6. Recovery of the stable sparse state: х х1, w w1; the host popu-lation density increases, tending to density х1 in the stable sparse state; the rate of parasitism also tends to the stable value w1.

Table 5.1 gives the data on the engrailed moth population density and the rate of parasitism.

Table 5.1. The states of the engrailed moth population in the phases

of outbreak peak, decline and crisis Date Average density of pupae m-2 Parasitism rate, % September 1932 145.2 24.5 51.6 1.3 % May 1933 74.6 13.4 45.5 0.9 % September 1933 49.4 9.3 76.6 2.6 %

As can be seen from the data in Table 5.1, the rate of parasitism of the

engrailed moth pupae increases with a decrease in host population densi-ty. This is in good agreement with models (5.3) and (5.4).


More detailed analysis of the data on engrailed moth population densi-ty on different sample plots in the outbreak site reveals the regions within which different phases of the outbreak cycle were observed: the start, the peak, the decline, and the crisis of the outbreak (Table 5.2).

Table 5.2. The rate of parasitism of engrailed moth pupae

in different population outbreak phases

Year Outbreak phase Population

density, pupae m2

Logarithm of the average

density

Parasitism rate, %

1932 1 – Population buildup 8 2.00 0.19 0.49 0.02

1932 2 – Development towards the peak 30 3.4 0.08 0.57 0.03

1932 3 – The peak 175 5.2 0.13 0.51 0.02 1933 4 – Decline 100 4.6 0.18 0.80 0.03 1933 5 – Beginning of crisis 6 1.9 0.34 0.82 0.08

The data in Table 5.2 suggest that changes in the host density and par-

asitism rate in the outbreak phases observed in 1932–1933 are in good agreement with model (5.3) (Fig. 5.2).

Indeed, in accordance with this model, in the phases between the out-break start and peak, the rate of parasitism remains unchanged; it begins to increase only when the population enters the decline and crisis phases.

Fig. 5.2. A segment of the trajectory of the parasite – host system

for the engrailed moth in different outbreak phases (1–5 are outbreak phases, see Table 5.2)


Unfortunately, there are no data for the engrailed moth population in the stable sparse state and in the phase of extremely low density, as no surveys had been conducted in the nearly unreachable (at that time) taiga forests of Central Siberia before the outbreak; and after the outbreak, when the fir stands were killed, there was no practical need for them (Prozorov, 1955).

Surveys of pine looper populations in the pupal phase provided more complete data on the host – parasite interactions. The plot in Figure 5.3 shows the relationship between the pine looper population density, outbreak phase, and the rate of parasitism of the pupae.

Fig. 5.3. A segment of the trajectory of the parasite – host system for the pine looper in different outbreak phases (1–3 are outbreak phases, see Table 5.3) As shown in Figure 5.3, in the phases of population buildup and peak

(1 and 2), the normalized rate of parasitism z is not influenced by the host population density and has the lowest values. This result is con-sistent with model (5.4). The increase in the rate of parasitism of the pupae observed in the crisis phase (3) is also in good agreement with model (5.4). Finally, the parasitism rates are the lowest when the pine looper population resumes its stable sparse state, with the population density decreasing to 0.03 pupa m–2. Unfortunately, as the densities of pine looper populations in the stable sparse state in Siberia and North Kazakhstan are very low, it is very difficult to accurately evaluate the rates of parasitism in them. For example, during surveys of the pine looper populations in the Krasnoturansk pine forest in 1980, no pupae (!) were found on 243 1-m2 sample plots. The surveys of the pine looper popula-tions in the Minusinsk forests in 1983 revealed 6 pupae on 563 sample


plots. Clearly, no reliable evaluations of parasitism of pupae are possible at such low population densities.

In our opinion, model (5.3) can explain the differences in phyto-phagous – entomophagous insect interactions for different forest insect species under different ecological conditions. Let us assume that preda-tor – prey interactions and effects of epizootics on the phytophagous insect population density can be described by models (5.3) and (5.4), similarly to the effects of entomophagous insects. Then, if the phytophagous insect population is simultaneously affected by parasites, predators, and diseases, the leading factor of the phytophagous insect population dynamics will be the factor with the shortest characteristic time. Thus, if the response of the entomophagous insect population to the change in phytophagous insect population density is faster than the responses of bacterial or viral populations, entomophagous insects will be the leading factor in the dynamics. Otherwise, the outbreak will subside due to epizootics or the impact of predators.

Another factor that can influence phytophagous insect population density is availability and quality of food. In this case, as an analog of variable z in model (5.3) for the phytophagous – entomophagous insect model, we can introduce the value of “food potential” G, defined as follows:

1 ,(1 )

GQ M

(5.5)

where M is the amount of the phytomass of leaves or needles in the tree stand, Q is the food quality index ( 0 1Q ), characterizing availability of the phytomass to the individuals of the consumer species.

At 1Q , all phytomass present in the tree stand is available to the phytophagous species; at 0Q , none of the present phytomass is avail-able to the phytophagous species. In the early phase of the outbreak, M is sufficiently large, while G is small, and population density is increasing. As the amount of phytomass is reduced and (or) its quality is impaired due to development of antibiosis resistance of the host tree in response to its loss of phytomass, the value of G increases (in the same way as the rate of parasitism z increases in the phytophagous – entomophagous insect model), and the phytophagous insect population density begins to de-crease. Naturally, to be able to use the “food” version of model (5.3), we need to evaluate the food quality index Q (availability of the phytomass to phytophagous insects) (Soukhovolsky et al., 2008; Tarasova et al., 2015).

The results of the analysis of the parasite – host system for forest in-sects suggest that for constructing a model of population dynamics of the


parasite – host system, we need the data on both the population density of the forest insect species monitored and the parasitism rate in the host population. Simple and accurate evaluation of the parasitism rate can be done in the phases of outbreak peak and crisis. However, in the phases of extremely low density and the start of an outbreak, it is rather difficult to obtain reliable values of parasitism. It is also difficult to estimate the strength of the predators’ effect on the phytophagous insect population.

***

Our analysis shows that the rate of parasitism can be estimated in a rather simple and reliable way in the phases of outbreak peak and population decline. It will be rather difficult to make a reliable estimate of the rate of parasitism of the phytophagous insect population in the very low-density phase, stable state, and at the beginning of an outbreak. It is also rather difficult to estimate the effect of predators on the phytophagous insect population.

The results of the analysis of the phytophagous – entomophagous in-sect system for forest insects suggest that for constructing a model of population dynamics of this system, we need the data on both the popu-lation density of the forest insect species monitored and the parasitism rate in the host population.

113

6 Food Consumption by Forest Insects

6.1 Energy balance of food consumption by insects: an optimization model

One of the major factors influencing population dynamics of forest insects is fecundity of individuals. Fecundity, in its turn, is determined by the mass of adults and the closely related mass of pupae or prepupal (late) instar larvae. For females of gypsy moth (Lymantria dispar L.) and Sibe-rian silk moth (Dendrolimus sibiricus Tschetv.), the number of eggs is linearly related to the mass of the late instar larvae (Isaev et al., 2001). At the end of this causative chain, a relationship arises between the mass of individuals, the amount of the available food, and its biochemical properties.

The existence of these relationships has been proven in numerous la-boratory experiments on rearing larvae (Ansari et al., 2012; Baranchikov, 1987; Bauerfeind, Fischer, 2013; Kirichenko, Baranchikov, 2007; Kula et al., 2014; Kumbasli et al., 2011; Low et al., 2014; Ponomarev et al., 2008). These experiments evaluate the balance ratios between the amount of the food consumed and the mass of the larvae consuming this food.

Energy consumed by a larva with food is expended for the increase in the mass of the larva, compensates for metabolic expenditure, and is excreted as feces (Baranchikov, 1987).

E M R H , (6.1)

where E is energy consumed by the larva; M is the increase in the biomass of the larva; R is metabolic expenditures; H is energy lost with the feces.




Energy balance of food consumption is described by using such pa-

rameters as utilization coefficient C FUCC

, assimilated food efficiency

PAFEC F

, and consumed food efficiency PCFEC

, which character-

izes ecological efficiency of feeding of the phytophagous insect (Slansky, Scriber, 1985). Note that the three parameters of food consumption by insects are independent of each other, and .CFE UC AFE

These ratios alone, however, cannot describe the feeding behavior of insect larvae. For the specified value C of the consumed food, the balance equation (6.1) is characterized by the presence of two independent varia-bles (any pair of variables P, R and F). Hence, there is no unambiguous solution to equation (6.1), and values of food consumption parameters may be arbitrary.

Thus, let us examine a modified approach to describing food con-sumption by defoliating insects, which can provide accurate solutions to the balance equation of food consumption energy.

A model of optimal food consumption by insects was proposed to de-scribe this behavior (Iskhakov et al., 2007; Soukhovolsky et al., 2000; Soukhovolsky et al., 2008). The schematic in Figure 6.1 shows the distri-bution of energy, E, consumed by the larva with food, in accordance with this model.

Fig. 6.1. The distribution of energy consumed with food during the course

of development of the larva. H = (1 – р1)Е is energy lost with the feces, S = р1Е is energy consumed by the larva, M = р2S = р2р1Е is energy

transformed into the biomass of the larva (including exuviae), Rp is metabolic expenditures, Rc is expenditures for food assimilation; f(E) is energy expended by an individual traveling in search of food

Food Consumption by Forest Insects 115

Value 11

p E pE

is utilization coefficient, UC. Value 2 1 21 2

E p p E p pE E

is parameter showing consumed food efficiency, CFE; value 1 22

1

p p E pp E

is assimilated food efficiency, AFE. In the model of optimal food consumption, the balance equation for

the distribution of energy E, consumed by the larva with food, can be written as follows:

21 1 1 2 1 2(1 ) ( , ),E p E ap E p p E bp p f m E (6.2)

where 11 p E is energy lost with the feces; 21ap E is expenditures for

food preparation and transformation to make it suitable for consump-tion; 1 2p p E is energy transformed into the biomass of the individual;

1 2bp p E is metabolic expenditure for the growth of the individual; ( , )f m E is energy expended by the larva traveling in search of food. In contrast to the conventionally used equation (6.1), the balance

equation (6.2) divides metabolic expenditures into two independent components: expenditures for food preparation and expenditures for metabolism.

In rearing experiments, the larvae are kept in small netted enclosures and fed regularly. Under these conditions, larvae do not need to travel in search of food, and we assume that ( , ) 0f m E . Then, after simple transformations of (6.2), we obtain

1 21 ( 1) .ap b p (6.3)

Equation (6.3) characterizes the energy balance of the larva in relative units. In the {p1, p2} plane, equation (6.3) describes a straight line crossing the X-axis at the point 1/a and the Y-axis at the point 1/(1 + b). The straight line (6.3) may be considered as the locus of all values of р1 and р2 that do not contradict energy considerations. The straight line (6.3) places restrictions on the values of р1 and р2, but it is not sufficient for calculating their values, as the values of two unknown quantities cannot be found if there is only one known equation relating these quantities. One more equation is needed to determine these values.

We assume that food consumption by larvae is an optimal process, i.e. that this process occurs with the highest efficiency, q (all other things being equal):

1 21 2 max .p p Eq p p

E (6.4)


The р2р1 product is a fraction of the total energy Е expended for the synthesis of the larval biomass. In the {p1, p2} plane, the locus of all eco-

logically feasible values of р1 and р2 is a hyperbola, 21

qpp

. Having

united the energy and population considerations, we may conclude that р1 and р2 must have such values as to attain the maximum р1р2 product, taking into consideration the budget constraint (6.3) on the values of р1 and р2.

Quantity q has meaning if the following conditions are fulfilled:

1 20; 0.p p (6.5) Evaluate р2 from (6.2)

2 11 1 .

1p ap

b (6.6)

As р2 >0, the following condition must be fulfilled:

11 0.ap (6.7)

Express (7.4) by taking into account (6.6) if condition (6.7) is fulfilled:

1 2 1 11 1 max .

1q p p p ap

b (6.8)

The optimal values of 1p̂ and 2p̂ , giving the highest q, can be easily

found from 1

0dqdp

and 2

21

0d qdp

:

11ˆ

2p

a; 2

1ˆ .2( 1)

pb

(6.9)

If results of the experiments on rearing larvae can provide the values

of p1 and p2, one can determine ecological cost 1

1ˆ2

ap

of food prepara-

tion and ecological cost (b + 1) of the synthesis of larval biomass

2

11 ˆ2b

p (Fig. 6.2).

If the ecological cost of food preparation is high, condition (6.7) may stop being fulfilled. This actually means that the expenditures for food preparation are higher than the energy consumed with food. In econom-ics, in this case, the consumer becomes unable to buy the commodity. In ecology (as will be shown later), if condition (6.7) is not fulfilled, the individual consumer of food will die.


Fig. 6.2. Relationships between variables in model (6.8)

Thus, the optimal consumption model (6.2)–(6.4) can evaluate ecolog-

ical costs of food consumption, determine consumption efficiency, and assess the risk of death of an individual.

Note that the model of food consumption by insects is formally the same as consumption models in the economic theory. In economics terms, the straight line (6.3) is the straight line of the budget constraint for the system consuming two products, and coefficients a and (b + 1) are equated to the costs of these products; equation (6.8) is analogous to the so-called indifference curve, and q is the efficiency of choosing certain values of costs (Lancaster, 1972; Hicks, 1993).

Model (6.2)–(6.4) was verified in laboratory experiments on rearing larvae of the black-veined white Aporia crataegi L. (Lepidoptera, Pieridae). The larva reaches a length of 4.5 cm; it is ash-grey or bluish-grey with a dark head, with two wide orange or brownish transverse bands; the body is sparsely coated with short light hairs (Korshunov, 2002). Early-instar larvae are gregarious, and fifth-instar larvae are soli-tary feeders (Zolotarev, 1950).

In spring, when buds open, the larvae leave their shelters and, first, feed on buds and, then, on leaves and flowers. Later, they skeletonize leaves, leaving the veins intact. The larvae of the black-veined white feed on leaves of hawthorn (Crataegus sanguinea Pall.J), bird cherry (Prunus padus L.), mountain ash (Sorbus sibirica Kryl.), and various fruit trees


(Korchagin, 1971; Nekrutenko, 1974; Vasilyev, Livshits, 1984). In years of outbreaks, the larvae leave the trees completely defoliated.

In suburban forests at Krasnoyarsk, the black-veined white damages bird cherry trees growing along the rivers and rivulets and defoliates plantations of apple trees, shadberry trees, and other fruit trees.

We conducted a laboratory experiment to estimate the energy balance of the black-veined white larvae and their food consumption rate. The larvae were collected from the stand of bird cherry trees growing in the Yemelyanovskii District, west of Krasnoyarsk, in May 2013. Fifty larvae were reared in netted enclosures. One hundred older-instar larvae were collected, and 50 of them were placed in netted enclosures. Food con-sumption and assimilation by all instars was determined by the “gravi-metric” balance method (Schroder, Malmer, 1980; Vshivkova, 1989; Waldbauer, 1968).

Usually, in balance experiments, calculations are based on the dry weight of the larvae, food, and feces (Baranchikov, 1987). In our experi-ments, however, calculations of food consumption efficiency were based on the weights of the larvae, food, and feces that included water con-tained in them, as consumption efficiency calculations based on the data on the dry and wet weights of the food and larvae differ only slight-ly. Indeed, let (1 – ) be the proportion of water per unit mass of the larva and (1 – ) be the proportion of water per unit mass of the food (0 < , < 1). Then, the difference between parameters of food con-sumption efficiency q and qd, when calculated for the wet and dry weight,

respectively, will be 1dM M Mq qE E E

. There is no differ-

ence between q and qd if 1 0 or 1 . Let – = . Then,

the ratio may be presented as 1 . The proportions of

water in the leaves of coniferous and broadleaved species and in the larvae (including the black-veined white larvae) are similar – 65–70% of the total weight (Kramer, Kozlovsky, 1979; Li, Zachariassen, 2006).

Hence, , and is about 0.1, i.e. the difference between the values

of food consumption efficiency determined for the dry and wet weights is no more than 10%. This is quite acceptable for field experiments, espe-cially if we remember that in the dynamic experiment, measurements of the dry weight of the larvae are based on the water content values


of the control larvae. Then, the determination of the dry weight of the larvae in the dynamic experiment contains a calculation error of un-known value, which occurs because the dry weight of the treatment group larvae is determined from the average values of water content in the control group larvae, which certainly is not the same as the weight of a separate larva in the treatment group.

Therefore, we did not determine the oven-dry weight in the experi-ment. As we did not determine the dry weight of the larvae, there was no point in determining the dry weight of the food, and, thus, the model of the optimal food consumption by insects was verified by using the data on the weights of the larvae, food, and feces that included the weight of the water contained in them.

The results were used to verify the model of optimal food consump-tion by insects.

Based on the data obtained in rearing experiment, we determined con-sumption balances for every individual, calculated the values of р1 and р2, and, based on them, calculated ecological costs of consumption, a and (b + 1), and consumption efficiency, q. Figures 6.3 and 6.4, in the {p1, p2} plane, show the respective values for black-veined white males and females; Table 6.1 gives consumption balances for males and females.

Fig. 6.3. Food consumption balance for females of black-veined white

(1 – data calculated from experimental results; 2 – budget constraint equation; 3 – efficiency equation)


Fig. 6.4. Food consumption balance for males of black-veined white

(1 – data calculated from experimental results; 2 – budget constraint equation; 3 – efficiency equation)

Table 6.1

Parameters Males Females

mean error of the mean mean error

of the mean p1 0.908 0.004 0.934 0.003p2 0.084 0.005 0.079 0.006a 0.551 0.003 0.535 0.002b + 1 6.750 0.504 6.713 0.517q 0.076 0.004 0.074 0.006

As can be seen from Figures 6.3 and 6.4, the levels of energy efficiency

of food consumption, a, and energy efficiency of biomass synthesis, (1 + b), differ somewhat between larvae. The product, q = p1 p2, is, how-ever, practically constant for all individuals of the same sex, suggesting that the relationship between coefficients р1 and р2 is described by the equation of hyperbola (6.4).

Evaluation of the effectiveness of food consumption is important for the assessment of the risk of tree phytomass loss caused by insects and determination of females’ fecundity. The effectiveness of food consump-


tion by the larva is determined by two “costs”: “cost” a of food assimila-tion and “cost” (b + 1) of the synthesis of an individual’s biomass. Hence, there are two possible opposite types of strategies of food con-sumption by insects. One strategy is to use the food that has a high assimilation “cost” but a low “cost” of the synthesis of larval biomass. One of the indications of the insects following this strategy is the large amount of insects’ feces in the tree stand. This effect is observed during outbreaks of the Siberian silk moth (Baranchikov et al., 2002; Kondakov, 1974). Calculations based on experimental data on rearing larvae of the Siberian silk moth fed on Siberian larch needles (Kirichenko, Baranchikov, 2007) give the following values of the food assimilation cost and biomass synthesis cost: а = 0.90 and (b + 1) = 3.18 (Soukho-volsky, Tarasova, 2010).

The other, opposite, consumption strategy is to use the food that has a low assimilation “cost” but a high “cost” of the synthesis of larval bio-mass. This is the typical strategy for black-veined white larvae, for which a = 0.54 – 0.55 and (b + 1) = 6.71 – 6.75 (Table 6.1). Thus, although food consumption efficiencies of the Siberian silk moth and the black-veined white are similar – q 0.087 and q 0.075, respectively – these species differ considerably in the ecological costs of food consumption.

Based on the two types of food consumption strategy, there are two types of insect outbreaks. The type one outbreaks are related to a decrease in the mass of individuals in the population during the development of the outbreak; the type two outbreaks are characterized by an increase in the mass of individuals during the course of the outbreak (Isaev et al., 2001).

By using the notion of food consumption “costs”, one can estimate the insects’ response to changes in the properties of their host plants. For instance, development of antibiosis reactions in plant leaves as a response to insect-caused damage, leads to an increase in the food assimilation “cost” with the “cost” of the synthesis of larval biomass remaining almost unchanged (Tarasova et al., 2004). This will result in a reduction in food consumption efficiency, a decrease in the relative biomass of individuals, and a decline in fecundity of females.

The effectiveness of food consumption is related to females’ fecundity. The fecundity of females of forest insects is known to be linearly related to the mass of pupae or adults (Isaev et al., 2001). The mass of individu-als, in its turn, is related to the effectiveness of food consumption, as shown in Table 6.2.

Female fecundity is known to be linearly related to the pupal mass (Isaev et al., 2001), as confirmed by the data in Table 6.2 (Fig. 6.5).


Table 6.2. Relationship between the fecundity of black-veined white females and food consumption efficiency

Larva No.

Mass ΔE of food consumed in the

experiment, gq

Mass of the larva at the end of

experiment, g

Mass of the

pupa, g

Fecundity F, (eggs)

5 2.96 0.05 0.18 0.16 506 3.44 0.06 0.25 0.22 83

12 2.09 0.09 0.29 0.27 8613 3.05 0.10 0.30 0.32 9721 3.19 0.07 0.25 0.24 8626 2.37 0.07 0.31 0.29 8227 2.31 0.07 0.26 0.22 8428 2.26 0.10 0.30 0.29 9429 2.64 0.07 0.26 0.25 8646 2.7 0.05 0.20 0.18 8048 2.6 0.06 0.30 0.25 89

Fig. 6.5. Relationship between the mass of a black-veined white pupa

and fecundity F of the adult Figures 6.6 and 6.7 show the relationship of a female’s fecundity, F, to

the mass of food consumed by the individual, Е, and to food consump-tion efficiency, q.

As can be seen from Figure 6.6, fecundity F of an individual is not re-lated to the mass E of the food consumed by it. Figure 6.6 shows, howev-er, that the efficiency of food consumption is linearly related to the fe-cundity of an individual, and the higher the efficiency of the individual, q, the greater its fecundity.


Fig. 6.6. Relationship between the mass of the food consumed by black-veined

white larvae, E, and adults’ fecundity, F

Fig. 6.7. Relationship between fecundity F of adults and efficiency

of food consumption q by black-veined white larvae The ecological costs of food consumption and the efficiency of food

consumption by black-veined white larvae can be compared with the experimental data and results of calculating the ecological costs of con-sumption and efficiency for gypsy moth (Lymantria dispar L.) larvae (Vshivkova, 1989; Iskhakov et al., 2007; Soukhovolsky et al., 2008).

Figure 6.8 gives values of р1 and р2 for gypsy moth females. As shown in Figure 6.8, in spite of differences in individual masses, the

efficiencies, q, are very similar for all individuals.


Fig. 6.8. The distribution of food consumed by gypsy moth females

(dots are data for individuals)

Figure 6.9 shows the costs of food assimilation and the costs of the

synthesis of larval biomass for gypsy moth larvae feeding on leaves of different tree species (relative to the costs of assimilation and synthesis for larch needles), calculated from the data reported by T.Vshivkova (1989).

Fig. 6.9. Energy efficiency (relative to larch needles) of the leaves of various tree

and shrub species fed to gypsy moth larvae (B – birch, W – willow, L – larch, A – alder, As – aspen, MA – mountain ash, BC – bird cherry, DR – dog rose)


Surprisingly, although the costs of food assimilation and the costs of biomass synthesis differ between tree species, the efficiencies of food consumption, q, are similar for larch, birch, dog rose, and willow foliage. The efficiency of feeding on alder, aspen, bird cherry, and mountain ash leaves is considerably lower than on larch needles. These calculations show that gypsy moth can be described as a broad oligophage, which feeds on different tree species with similar efficiency.

In early stages of development, when the mass of an individual is low, condition 11 0ap may be fulfilled, and the larva will get the energy necessary for it to grow. As the mass of the larva is increased, a may grow rather quickly, and, then, 11 ap will become negative. In this case, energy expenditures for food preparation will be higher than the energy that the individual could use later. Then, the energy balance of food consump-tion will become negative, and the larva may die in spite of food availability.

An illustration to this can be found in a study by Kirichenko and Baranchikov (2007). Siberian moth larvae of two races – larch (LR) and fir (FR) races – were fed with the needles from larch or fir. They passed through all developmental stages and pupated (Table 6.3). When, howev-er, fir race larvae were fed with Scots pine (Pinus sylvestris L.) needles, they died before they reached the third instar. Based on the data reported by Kirichenko and Baranchikov (2007), we calculated parameters of efficiency and “costs” of food consumption by Siberian moth larvae of the larch and fir races reared on different types of food (Table 6.3).

Table 6.3. Energy balance of Siberian silk moth larvae feeding

on different tree species

Instar Food type/ race of larvae

Larch needles/LR

Larch needles/FR

Pine needles/LR

Pine needles/FR

1 0.76/3.68 0.74/3.57 1.61/4.90 1.71/6.582 0.81/4.43 0.81/4.95 1.77/6.25 1.89/7.463 0.88/4.39 0.88/4.63 1.91/5.62 –4 0.96/4.63 0.95/5.26 2.14/5/32 –5 1.18/4.17 1.22/4.90 2.43/4.72 –6 0.91/3.18 0.94/3.36 1.61/6.76 –

The data in Table 6.3 show that the costs of food consumption for the

larch-race larvae feeding on pine needles are almost twice greater than the costs of consumption for larch-race larvae feeding on larch needles. The effectiveness of food consumption for the same-race larvae (e.g., for


larch-race larvae) decreases monotonically, when in the sixth instar, with the consumption effectiveness reaching its maximum, q = 0.087, the larch needles are replaced by the pine needles, and the consumption efficiency drops to 0.032 (Fig. 6.10).

The fir-race larvae show a somewhat lower food consumption efficien-cy (except on fir needles) than larch-race larvae, and when fed on pine needles, fir-race larvae die before they reach the sixth instar (Table 6.4).

Fig. 6.10. Efficiency of food consumption by Siberian silk moth larvae.

Food: 1 – Siberian larch needles; 2 – Siberian fir needles; 3 – Siberian spruce needles; 4 – Scots pine needles

Table 6.4. The effectiveness of food consumption by the sixth-instar larvae

of the larch and fir races

food plant consumption efficiency qLR FR

larch 0.087 0.079fir 0.053 0.058spruce 0.036 0.032pine 0.023 –

Larch-race larvae consuming pine needles pass through the entire de-

velopmental cycle and pupate with efficiency q = 0.032. Fir-race larvae consuming pine needles with efficiency q = 0.022 die in the third instar. Presumably, fir-race larvae feeding on pine needles die because in the third instar, the cost of food preparation, a, reaches the value at which


condition (6.6) is not fulfilled any more. Thus, for Siberian silk moth larvae, the boundary between a successful consumer and a bankrupt, between life and death lies between q = 0.02 and q = 0.03.

The approach proposed here was used to introduce parameters of food quality, classify some insect species as monophages, oligophages, and polyphages based on the costs of consumption of the phytomass of different tree and shrub species, and explain why consumption of energy inefficient food causes death of consumers.

The model of food consumption by insects can be used both for analy-sis of the energy balance of the larva in all developmental stages and for analysis of energy processes of food consumption by different instars. In contrast to standard balance calculations of food consumption by insects, this model divides metabolic expenditures into two components. In this way, instead of using purely “accounting” approach to calculations of energy balance of food consumption, we managed to introduce the notion of optimization of food consumption and begin to study ecologi-cal processes related to the choice of food and food plants by insects.

The possibility of evaluating ecological costs of consumption in short-term experiments (lasting a few days) on rearing larvae opens the way to using evaluations of food consumption costs in forest-entomological monitoring.

The model proposed above offers a new approach to describing pro-cesses of food consumption by insects and interpreting results of experi-ments on rearing insect larvae. Simple conversion of the data of numer-ous experiments on rearing the larvae of defoliating insects into the parameters of ecological costs and food consumption efficiency may result in a considerable increase in the available information on the ecological economics of the insects.

This approach may be used to evaluate food consumption efficiency by the animals of different taxonomic groups.

The similarity found between processes of consumption by humans and by animals may suggest further use of the methods of modern eco-nomic theory to solve ecological problems associated with resource consumption and growth of individuals.

6.2 A population-energy model of insect outbreaks Studies of interactions in the “resource (host plant) – consumer (insects)” system usually analyze mechanisms responsible for resistance of plants to insect damage, such as changes in the “carbon/nitrogen” ratio in plants


(Bryant et al., 1983; Tuomi et al., 1984), delayed induced resistance (Wallner, Walton, 1979; Haukioja, 1980; Schultz, Baldwin, 1982), chang-es in the type of interrelationships between growth and differentiation in plants (Lorio, 1986), and compensatory plant growth (Mattson, Addy, 1975). The optimal defense hypothesis is considered to be true, and, apparently, plants can regulate activity and composition of secondary metabolites, by optimizing their energy cost (Bentley, 1977; McKey, 1974; Feeny, 1976; Rhoades, 1979).

Response of insects to the synthesis of defensive chemicals in plants is a less studied aspect. Since the rate of synthesis of defensive chemicals in plants is determined by the level of insect damage, the model proposed by Coley (Coley et al., 1985) suggests that as the population of insects increases, their food becomes less palatable to them. If defense develops slower than insects, food will become unsuitable for the following genera-tion (Baltensweiler, Fischlin, 1988).

Another effect related to food consumption by insects is the effect of a reduction in the reproduction rate of the insects whose parents took part in strong competition for resources – maternal effect, described by a number of authors (Ginzburg, Taneyhill, 1994; Rossiter, 1994).

A more symmetrical approach to the description of interactions between insects and their food plants seems to be needed. If it is assumed that the plant optimizes its defense strategy while interacting with insects, insects must also have an optimized strategy of food consumption, which would determine changes in the population size and other population characteris-tics (mass of individuals, fecundity, etc.), depending upon food quality.

Consider a model of forest insect population dynamics that unites the population and energy approaches. For the sake of simplicity, we will assume hereafter that females constitute ½ of the population, all females are fertilized, and survival of individuals in the population in all phases of seasonal development is not sex-dependent.

Let the mass of an individual be m. We characterize the probability of survival of an individual in the population using the survival function, f(T, Hum, m, …), which is influenced by a great number of factors such as temperature, T, and humidity, Hum, during the insect life cycle, the mass of the individual, m, the abundance of parasites and predators, the quantity and quality of food, etc. The range of values of f(T, Hum, m, …) is between 0 and 1. At f(T, Hum, m, …) = 0, the population dies out; at f(T, Hum, m, …) = 1, all individuals survive. We assume that different factors influence insects independently, and function f(T, H, m, …) can be written as follows: ( , , , ...) ( ) ( , ,...)f T Hum m p m g T Hum , (6.10)


where function p(m) characterizes the probability of survival of an indi-vidual as dependent on its mass.

Consider constraints on the value of p(m) as related to the values of m. At a low mass, the survival of an individual is extremely unlikely. At a high mass, the probability of survival is low, too, as sustenance will require a great amount of critical resources. The simplest function that has the properties described above has one maximum; to the left of the maximum, the function grows monotonically, as the mass of the individ-ual increases, and, then, after reaching the maximum, it decreases mono-tonically.

Any function with such properties can be used for qualitative analysis. We introduce the following function: ( ) exp( )p m A m m . (6.11)

In (6.11), constant A – a normalizing constant – must be such that at any m and λ, condition p(m) ≤ 1 is fulfilled. This constant can be calcu-lated as follows. Find the maximum of expression (6.11):

( ) 0p mm

, exp( ) exp( ) 0A m Am m ,

1m .

If the value of function (6.11) at point 1/m is no more than 1, it will not be more than 1 at other points, either. Hence, by solving ine-quality max(1/ )p p , where maxp is the maximum probability of sur-vival, we obtain the following expression for constant A: maxA p e .

We will assume hereafter that each female in the population produces the same number of offspring. The fecundity of an individual insect usually increases with the increase in its mass (Kondakov, 1974). In the general case, the number of offspring N of one female, all other things being equal, can be written as the function N(m). In the simplest case, the function N(m) can be considered to be linearly related to the mass of the individual, and then N= k (m – m0) (where k is some constant – specific fecundity – and m0 is the lowest mass of the individual that allows it to reproduce).

Let us now write the formula for the average number of offspring of one female, R0. This is the probability of survival of the individual, p(m), multiplied by its fecundity, N(m): 0 0( ) ( ) ( ) ( ) exp( )R m N m p m k A m m m m , (6.12)


R0(m) is the reproduction coefficient per one female. It is quite easy to find the mass of the female at which reproduction coefficient is the highest. At 0 ,m m after differentiation and simple transformations, we find that R0(m) will reach its maximum at the following mass of the

individual: 02* *m m . Hereafter, this value will be called population

mass. By inserting the mass of the individual into (6.12), we can deter-mine the maximum reproduction coefficient per one individual:

0 02max exp 2AkR m (6.13)

Thus, the mass of individuals of the current generation determines the size of the population in the following generation. If the mass of individ-uals is close to m**, the reproduction coefficient will be close to its maxi-mum, which may cause a dramatic increase in the population size.

Returning to equation (6.3), remember that (3.3) can be interpreted as a straight line of budget constraints, and (1 + b) and a as “costs” of bio-mass synthesis and food assimilation. Based on energy “costs” of different aspects of larval ontogeny, one can discuss responses of individual insects to changes in the properties of their food plants. For instance, a decrease in antibiosis of leaves causes a reduction in the “cost” of food assimila-tion. Then, the “cost” of the synthesis of larval biomass remaining un-changed, the intersection point of the straight line of budget constraints and axis р1 will be shifted closer to zero. This will lead to the correspond-ing changes in the biomass of individuals and female fecundity and other transformations in the population.

The straight line (6.3) determines a great variety of possible strategies of food consumption by the larva. Hence, there are two possible extreme types of strategies of food consumption. One strategy is to use the food that has a high assimilation “cost” but a low “cost” of the synthesis of larval biomass. The other, opposite, strategy is to use the food that has a low assimilation “cost” but a high “cost” of the synthesis of larval bio-mass. Suppose that the larva chooses the strategy that enables it to reach the maximum biomass, P, i.e. the point on the straight line (6.3) at which the maximum expression of p1 p2 is reached. Having solved the optimiza-tion problem, p1 p2 → max, the larva gets an optimal strategy of food consumption (for the given “costs” of food assimilation and biomass synthesis), i.e. coefficient of food expenditure for the synthesis of biomass of an individual, q*, is

1 21( ) ( )

4 (1 )q p опт р опт

a b, (6.14)


Hence, the mass of the individual is given by the following expression:

*

4 (1 )Cm

b a. (6.15)

Quantity m* will be called economic mass of the individual. Changes in the quality and quantity of food will cause changes in the economic mass of the individual.

For the sake of simplicity, we consider the case when the “cost” of bi-omass synthesis, (1 + а), and the amount of available food, C, are con-stant. Then, a change in the economic mass of the individual occurs only when the “cost” of food consumption, b, is changed. The “cost” is influ-enced by both changes in the state of the environment (e.g. variations in temperature, humidity, or soil moisture content) and changes in the state of the food caused by the response of food plants to insect damage.

Let us write the equation describing the change in the “cost” of food assimilation caused by these factors as follows:

2 30( ) ,a N a N N N (6.16)

where N is insect population density; a0, , , are some constants. This relationship is shown in Figure 6.11.

Fig. 6.11. Relationship between the “cost” of food assimilation

and population density The first term on the right-hand side of (6.16) describes the constant

component of the “cost” of food assimilation – the value of the “cost”


at a low population density. The second term on the right-hand side of (6.16) describes the effect of antibiosis of the food plant and an in-crease in the “cost” of food assimilation for insects. The intensity of antibiosis is assumed to be proportional to the loss of food resource, which, in turn, is proportional to insect population density. The third term on the right-hand side of (6.16) describes the effect of the loss of food resource after the food plant becomes susceptible to the impact of insects on the “cost” of food assimilation. The last term on the right-hand side of (6.16) describes an unlimited increase in the “cost” of food assimi-lation under high population densities.

The relationship between reproduction coefficient, R0, and mass, m, introduced into expression (6.16) is shown in Figure 6.12.

Fig. 6.12. Relationship between reproduction coefficient and the mass

of the larva The values of mass at which reproduction coefficient is equal to 1 will

be called stationary values. As shown in Figure 6.12, there are two sta-tionary values – on the ascending branch of the relationship (m1) and on the descending one (m2). If the mass of individuals, m, is equal to m1 or m2, in the absence of external effects, the system will be in steady state for an indefinitely long time. In the presence of external effects, two outbreak scenarios are possible.


In the first scenario, individuals in the population have mass m1. If the external conditions change and cause the “cost” of food assimilation to fall and, consequently, the economic mass to increase, the reproduction coefficient grows, and, hence, population size grows, too.

In the second case, individuals in the population have mass m2. If the external conditions change and cause the “cost” of food assimilation to rise and, consequently, the economic mass to decrease, the reproduction coefficient grows, and, hence, population size grows, too.

Using the above assumptions about the type of interaction between the population and the food plant, we have constructed a simulation model of population dynamics. The model is based on the following assumptions:

population size determines the “cost” of food assimilation in ac-cordance with the following expression:

2 3( ) ( ) ;a N h t N N N economically optimal mass of an individual is determined by the “cost” of food assimilation

* ;4 (1 )

Cm

b a

reproduction coefficient is determined by the mass of individuals 0 0( ) ( ) exp( ).R m k A m m m m

In this model, changes in the population size are described as follows: the stable “cost” of food assimilation, b, is determined based on some initial population density N. The effect of random environmental influ-ences is modeled by multiplying the “cost” obtained by some random value – the modifying factor, f. We used the normally distributed modify-ing factor with the average value equal to zero. Variance of the modifying factor is one of the model parameters. The modified value of the “cost”, b', is used to calculate economic mass, m*. Then, the value of the economic mass is used to calculate reproduction coefficient. The density in the following generation is calculated as the density in the current generation multiplied by the reproduction coefficient. The process is cyclic.

A phase portrait showing the relationship between the density in the (N + 1)th generation and the density in the Nth generation can be con-structed for the qualitative analysis of model behavior. Temporal dynam-ics is obtained by running the simulation model at different values of the variance of the modifying factor.

As noted above, there are two possible scenarios of the behavior of the system, which differ in the position of the stationary state. In the first scenario, the stationary value of the mass is to the left of the maximum of


(a) Phase portrait of the system

(b) Population temporal dynamics

Fig. 6.13. The first scenario of the outbreak

reproduction coefficient. The phase portrait corresponding to this sce-nario is shown in Figure 6.13a. As can be seen from the phase portrait, point A is stable.

At low values of the variance of the modifying factor, the system is close to the stable point A. Numerical fluctuations in this case correspond to variations in the modifying factor. At high values of the variance of the


modifying factor, outbreaks occur. The corresponding temporal dynam-ics of the system is shown in Figure 6.13b.

In the second scenario, the stationary value of the mass is to the right of the maximum of reproduction coefficient. The phase portrait corre-sponding to this scenario is shown in Figure 6.14a. As evident from the phase portrait, there is a limit cycle around point A. At low values of the variance of the modifying factor, the system oscillates in the limit cycle close to point A. At high values of the variance of the modifying factor,

(a) Phase portrait of the system

(b) Population temporal dynamics

Fig. 6.14. The second scenario of the outbreak


outbreaks occur. The corresponding temporal dynamics of the system is shown in Figure 6.14b.

The model proposed here describes a number of effects observed un-der field conditions: changes in the mass of individuals during the devel-opment of forest insect outbreaks and changes in food quality caused by the defense response of the plant. For instance, Figure 6.15 shows the data on changes in the mass of the larva during outbreaks of the gypsy moth (Fig. 6.15a) and the Siberian silk moth (Fig. 6.15b).

(a) gypsy moth

(b) Siberian silk moth Fig. 6.15. Changes in the mass of larvae during outbreak (based on the data

reported by Kondakov (1963)) 1–5 – phases of gradation


As can be seen from Figure 6.15, during outbreaks, the mass of indi-viduals changes in different ways. At the peak of gypsy moth outbreak, the mass of the larvae is the lowest. In contrast to that, at the peak of the Siberian silk moth outbreak, the mass of individuals is the highest. The theoretical scenarios of the development of outbreaks described above are in qualitative agreement with these field data.

Thus, by simultaneously studying both population and energy charac-teristics of individuals in a population, one can describe development of outbreaks, take into account the effects of the influences of both modify-ing and regulating factors on changes in the population density, and take into consideration the effects of antibiosis and the loss of resistance of food plants.

***

The possibility of evaluating ecological costs of consumption in short-term experiments (lasting a few days) on rearing larvae opens the way to using food parameters in forest-entomological monitoring. The only difference between the model proposed in this study and standard bal-ance calculations of food consumption by insects is that this model divides metabolic expenditures into two components. However, in this way, instead of using purely “accounting” approach to calculations of energy balance of food consumption, we managed to introduce the notion of optimization of food consumption and begin to study ecologi-cal processes related to the choice of food and food plants by insects.

The approach proposed here was used to introduce parameters of food quality, classify some insect species as monophages, oligophages, and polyphages based on the costs of consumption of the phytomass of different tree and shrub species, and explain why consumption of energy-inefficient food causes death of consumers.

By simultaneously studying both population and energy characteristics of individuals in a population, one can describe development of outbreaks, take into account the effects of the influences of both modifying and regu-lating factors on changes in the population density, and take into consider-ation the effects of antibiosis and the loss of resistance of food plants. An obvious obstacle to using this approach in practical forest management is the technical difficulty of conducting experiments with insects.

139

7 AR- and ADL-models of Forest Insect Population Dynamics

7.1 An ADL-model (autoregressive distributed lag) of insect population dynamics

Three groups of factors influencing forest insect population dynamics are enemies (parasites, predators, and diseases), modifying factors (primari-ly, weather), and food. These factors must be all included into the model describing changes in insect population density.

The classical Lotka-Volterra model (the same mathematical structure named “parasite – host”, “predator – prey”, or “resource – consumer”, depending on what is modeled) only describes the interactions of popula-tions with densities x and y (Bazykin, 2003):

( ) ,

.

dx kx A x bxydtdy cy mbxydt

(7.1)

Model (7.1) is commonly used to describe interactions in the commu-nity that includes insect population and population of its enemy (parasite or predator). This model can also be used to study resource and insect population consuming this resource. However, it is not suitable for




describing effects of weather factors. For describing the influence of weather factors on population dynamics of prey and predator, one cannot introduce weather effects as additive terms of the equations in system (7.1), e.g., in this way:

1

2

( ) ,

,

dx kx A x bxy a T Tdtdy cy mbxy g T Tdt

(7.2)

where T1 and Т2 are optimal temperatures for the existence of x and y populations.

Modification (7.2) of system (7.1) is not correct, as, formally, (7.2) ac-cepts the state in which at x = 0, dx/dt will be less than zero. It is very difficult to explain the meaning of this state, because at x = 0, only dx/dt 0 is possible, which would imply that the modeler agrees with the idea of autogenesis of individuals or that the model has become spatially open and individuals may migrate. Then, however, the model will need to be essentially changed.

It would be certainly more correct to take into account weather effects by using weather dependent coefficients of the system (7.1). In this case, however, instead of two model equations describing population interac-tions, we will need to consider a system of seven nonlinear equations. Also, system (7.1), with the constant values of coefficients, lacks outbreak modes and, hence, is unable to describe actual variations in the popula-tion density of forest insects.

Therefore, new approaches need to be developed for modeling forest insect population dynamics. Particularly, it’s possible to suggest in the frameworks of phenomenological approach that the parameters of the current population density contain all information on regulatory pro-cesses in the ecosystem and effects of external factors. In this case, we do not need to examine external factors influencing the population but only need to analyze the time series of population dynamics of a particular forest insect species.

Let us use a hypothetical example to examine how а phenomenological approach may be employed to model forest insect population dynamics. Consider a simple simulation model of population dynamics. Early-instar larvae are affected by parasitoids; the individuals that have not been killed by egg parasitoids are influenced by weather factors; and the indi-viduals that have managed to reach the adult stage lay eggs. In this case, let us write the model relating density x(s) of forest insect population

AR- and ADL-Models of Forest Insect Population Dynamics 141

at a definite life stage in year s to population density x(s – 1) in the same stage in the previous, (s – 1)th, year, as follows:

( ) ( , ( 1)) ( 1)

( , 1) ( 1) ( ) ( 1) ( 1).W P

x s F W x s x sR m s I s F s F s x s

. (7.3)

According to (7.3), population density х(s) of the current year is de-termined by fecundity R(m, s – 1) of females of generation (s – 1), the value of the sex ratio in the population of generation (s – 1), func-tion FW(s) of weather influence on the population in year s, function Fp(s) of the effects of parasites and predators in year (s – 1), and population density х(s – 1) in year (s – 1).

Model (7.3) assumes that the effects of parasites and predators in year (s – 1) are determined by the prey population density in year (s – 2), i.e. there is a time lag in the parasite – host system. Then

0( 1)( 2)P

FF sx s

, (7.4)

where F0 and are some constants. The function of the effect of weather parameters on population density

in the sth year is assumed to be nonmonotonically dependent on the value of the hydrothermal coefficient (HTC) of W(s) in the current year:

( ) ( )exp( ( ))WF s AW s W s ), (7.5)

where А and are some positive constants. The model assumes that fecundity of females is determined by their

mass m (Kondakov, 1974; Isaev et al., 2001):

0 0

0

[ ( 1) ],( ( 1))

0,r m s m m m

R m sm m

, (7.6)

where r is specific fecundity per female (eggs/g female body mass); m0 is minimal mass of the female that is sufficient for it to lay eggs.

In its turn, the mass of individuals is nonmonotonically dependent on population density (Soukhovolsky et al., 2008):

( 1) ( ( 1)) exp( ( 1))m s B x s x s , (7.7)

where B, , μ > 0 are parameters. Having applied the log-transformation to (7.3), we obtain:

( ) ln ( )

ln ( , 1) ln ( 1) ln ( ) ln ( 1) ( 1).W P

L s x sR m s I s F s F s L s

(7.8)


The sex ratio ( 1)I s is assumed constant for all outbreak stages, and 0ln ( 1) ln constI s I .

Having applied the log-transformation to (7.4), we obtain:

0ln ( 1) ln ( 2).pF s F L s (7.9)

After log-transforming (8.5), we obtain:

ln ( ) ln ln ( ) ( ).WF s A W s W s (7.10)

As values of W(s) are of order 1 and, hence, ln W(s) 0,

ln ( ) ln ( ).WF s A W s (7.11)

Having applied the log-transformation to (7.6), we obtain:

0ln ( ( 1)) ln ln ( 1) .R m s r m s m (7.12)

As m0 is small compared to masses of the larvae (Isaev et al., 2001; Kondakov, 1963),

ln ( ( 1)) ln ln ( 1).R m s r m s (7.13)

The mass of individuals of such species as the gypsy moth decreases considerably as the population density increases (Benkevich, 1984; Isaev et al., 2001). In this case,

ln ( 1) ln ln ( 1) ln ( 1)m s B x s B L s (7.14)

and

ln ( ( 1)) ln ln ln ( 1).R m s r B L s (7.15)

By substituting (7.9), (7.11), and (7.15) into (7.8), we obtain the equa-tion relating the current density of the population to its densities in the previous two years and to the HTC of the current year:

0

0

( ) ln ln ln ( 1) ( 1) ln ( )ln ln ( 2) ( 1)

L s r I B L s L s A W sA F L s L s

or

0 1 2 3( ) ( ) ( 2) ( 1)L s Q Q W s Q L s Q L s , (7.16)

where 0 0 0ln ln ln ln lnQ r I B A F ; 1Q ; 2Q ; 3 (1 )Qare some constants.


Equation (7.16) is a so-called ADL-model (autoregressive distributed lag), which is widely used in different applications (Box, Jenkins, 1970; Stock, Watson, 2011).

01

( ) ( ) ( ) ( ) ( ) .k h

j jX s a j X i j b j Y i j X (7.17)

Having carried out these transformations and knowing the time-ordering operator of effects on population from equation (7.3), which describes effects of different factors on population dynamics, we obtain equation (7.17), which does not contain the effects of the factors, and population densities are determined by the densities measured during the previous surveys and by the current weather. In fact, equation (7.17) is a time representation of a phenomenological model of insect population dynamics, which is similar to the phenomenological model in the phase space described in Chapter 2. If we also assume that external effect opera-tors are commutative, model (7.17) can be used to analyze population dynamics of different species under different sequences of external effects.

Thus, models of population dynamics can be based on equations that are well known in the circuit theory and the automatic control theory and have been thoroughly analyzed by a number of authors (Stock, Watson, 2011; Veremey, 2013; Kim, 2007).

Previously, different authors (Berryman, Turchin, 2001; Royama, 1992; Turchin, 1990; Turchin, Taylor, 1992) described AR models of forest insect population dynamics, but those models did not take into account the contribution of modifying factors (i.e. they were not ADL-models), and no cross-correlation functions of the model series and field data series were used to evaluate the quality of the models.

ADL-models are classified by the values of parameters k and h, which characterize the order of an ADL-model (Stock, Watson, 2011). If values L(s), L(s – 1), L(s – 2), L(s – k), W(s), W(s – 1), …,W(s – h) are known for the ADL(k, h) model, coefficients Qi of equation (7.17) can be deter-mined by treating (7.17) as a regression equation, which is linear relative to variables {Qi}. Thus, to quantify coefficients of this equation, we need the data of the time series of population density { ( )}L s and the HTC time series { ( )}W s .

There are various ways to determine the order of autoregression for a given time series. It appears that the simplest way is to calculate the so-called partial autocorrelation function of the model time series and determine the number of the terms of the partial autocorrelation func-tion whose coefficients are significantly different from zero (Box, Jenkins, 1970).


The quality of an AR- or ADL-model can be evaluated by two parame-ters: the degree to which survey data and model densities agree and the value of the phase shift between the time series of the survey data and the model time series. The agreement between the ranges of the model and field data was evaluated by the coefficient of determination R2 of regres-sion equation (7.17). The phase shift between the model and field time series of population densities was determined by calculating the cross-correlation function of the model time series and the transformed time series { ( )}L s of the field data. If the maximum of the cross-correlation function was at the shift k = 0 and the value of the cross-correlation func-tion f(k = 0) was close to 1, the model and field time series changed syn-chronously and no phase shifts were observed between these time series.

It is important to realize that the autoregressive model proposed here cannot be directly used to predict population dynamics, as in this model determination of the current value of the model time series involves both previous values of this series and its subsequent values. Hence, the past is evaluated based on the future, which certainly contradicts the causality principle. Chapter 9 will describe the use of autoregressive models to make short- and medium-term forecasts.

Thus, we can write the following algorithm of constructing a model of population dynamics of a forest defoliating insect species:

1. Transform the time series of population density to obtain a sta-tionary homoscedastic LTI set;

2. Calculate the partial autocorrelation function of the LTI set and determine the order k of autoregression;

3. Calculate coefficients of the model autoregressive equation of type (7.17) for the order k of autoregression and different values of the order h of moving averages;

4. By using standard methods, evaluate the validity of the ADL(k,h)-model and the significance of its coefficients; exclude ADL-model terms with non-significant coefficients;

5. Calculate values of the model LTI set of population density; 6. Calculate the cross-correlation function of the LTI set of survey

data and model series; if the maximum of the cross-correlation function is close to 1 at the zero shift, the model is correct.

In the sections below, we will describe the use of this algorithm to model population dynamics of different forest insect species and, then, analyze additional parameters of autoregressive models, such as stability of the models and gain margin, robust stability, and response to external impacts.


7.2 A model of population dynamics of the gypsy moth in the South Urals

The gypsy moth (Lymantria dispar L.) is a very interesting species for the theory of population dynamics. This species has a vast range, covering the area between the Asian coast of the Pacific Ocean and the East coast of North America; this is an outbreak species; it is simple to conduct surveys of its populations in the egg phase, and results are very accurate. Therefore, the gypsy moth is a very suitable species for analysis of the effects of modifying (climatic and topographic) factors and regulating factors (parasites and predators, interactions with host plants) on its population dynamics. In addition to that, it is important to control and predict population densities of this species in order to protect tree stands from the damage and even death it can cause: being an eruptive species, the gypsy moth is a very hazardous forest pest. Numerous studies have addressed population dynamics and seasonal development of the gypsy moth in Eurasia and North America (Benkevich, 1984; Berryman, 1995; Bigsby et al, 2011; Campball, 1981; Elkinton, 1990; Fraval, El Yousfi, 1989; Keena, 1996; Khanislamov et al., 1962; Kireeva, 1983; Kondakov, 1963; Liebhold et al., 2000; Liebhold, 1992; Mason, McManus, 1984; Meshkova, 2009; Montgomery, Wallner, 1988; Myers et al., 1998; Régnière et al. 2009; Reineke et al., 1999; Sharov et al., 1996; Waggoner, 1984; Williams et al., 1990).

For the analysis of population dynamics to be effective, a sufficiently large database is needed. In this regard, gypsy moth populations in the South Urals are unique. Sporadic data on population dynamics of this species have been reported since 1948, and since 1956, gypsy moth sur-veys have been regularly conducted in the Chelyabinsk Oblast (Gninenko et al., 2011; Sokolov, 1998; Sokolov, Gninenko, 2002;). Thus, the time series of gypsy moth population dynamics in the South Urals is longer than 60 years. This must be the longest series of continuous surveys of one forest insect species in Russia. Compare: continuous surveys of pine looper (Bupalus piniaris L.) population dynamics in pine forests of South Siberia have been conducted for only 34 years, since 1979 (Palnikova et al., 2002); the continuous time series of population dynamics of other forest insects in Russia are even shorter.

Let us examine an ADL model of gypsy moth population dynamics in the South Urals, following the study by Soukhovolsky et al. (2015) and using the data of long-term surveys carried out by forest protection officers in the Chelyabinsk Oblast. Population surveys were performed


in autumn (September), in the egg phase. The number of egg masses and eggs per mass were counted on constant sample plots (100 trees per plot). For the FSZ, the survey data are available for the period between 1956 and 2012. Between 1973 and 1995, no gypsy moth eggs were found on the sample plots. The time series for the SZ covered the period between 1974 and 2012. In this series, only in 1997, no egg masses were found. In the MFZ, surveys began in 1996. No gypsy moth egg masses were found in 1970, 1971, 1979, 1981–1992, and 1995.

In the years when no gypsy moth egg masses were found, the egg mass density was less than 1 egg mass per 100 trees. Under such ecological conditions, the density of gypsy moth egg masses is about 1 egg mass per 1000–3000 trees (Raspopov, Rafes, 1978). Based on this, we repaired the time series of gypsy moth population density and replaced all zero values of population densities by 0.1 egg tree–1. After this repair, statistical analysis of the time series was performed with no gaps.

The weather data for the mountain-forest zone (northern region) were obtained at the weather station in the town of Verkhniy Ufalei, for the forest-steppe zone at the Timiryazevskoye weather station, and for the steppe zone – at the weather station of the town of Bredy. In our calcula-tions, we used the hydrothermal coefficient (HTC) proposed by Selyaninov.

The time series of surveys were transformed into LTI sets by using the algorithm described in Chapter 2. Briefly, transformation of the time series of population dynamics consists of the following steps: transform the data to the logarithmic scale, isolate a constant component from the log-transformed time series of gypsy moth population dynamics – the trend, and regularize the resulting time series through high frequency filtration. These transformations reduced the variance of population density values and eliminated random fluctuations caused by inaccuracy of survey data and deviations associated with local effects of modifying factors. After the initial time series was transformed into the LTI set, the properties of the { ( )}jL s time series were studied by using conventional methods of correlation and spectral analysis (Anderson, 1971; Kendall, Stuart, 1973; Jenkins, Watts, 1969; Pollard, 1979).

Table 7.1 lists changes in statistical parameters of the time series of gypsy moth population dynamics in different natural zones during se-quential data transformations.

Figure 7.1 shows the time series { ( )}jy t of logarithmic density and the time series { ( )}jL s of gypsy moth population dynamics in the FSZ after detrending and removal of high-frequency oscillations.


Table 7.1. Changes in statistical parameters of the time series of gypsy moth population dynamics during sequential data transformations

Variable

habitatFSZ SZ MFZ

Long-term

annual average

variance

Long-term

annual average

variance

Long-term

annual average

variance

Population density x(t) 286.47 545603.8 418.47 523770.6 54.87 27410.11

Log-transformed population density y(t) = ln x(t)

3.21 8.5849 4.38 6.0516 0.98 8.5264

Trend-free variable h(t) 0.01 7.5625 0.09 6.0025 0.05 8.5264

HF-oscillations -free variable L(t) 0.09 6.5025 0.02 5.6644 0.05 7.2361

Fig. 7.1. Time series of gypsy moth population density in the FSZ (1 – series

{ ( )}jy t ; 2 – series { ( )}jL s ; 3 – linear trend of the density ( ) 0.054 103.96z t t )

As shown in Figure 7.1, for the past 50 years, six gypsy moth outbreaks

with population densities higher than 400 eggs/tree have occurred in the forest-steppe zone of the Chelyabinsk Oblast. In addition to that, a weak linear trend in population density has been observed over this time. The presence of this trend might be related to the high gypsy moth popula-tion density at the peak of outbreak during 2003–2012.

Table 7.2 gives parameters of outbreak cycles of gypsy moth popula-tions in different zones in the South Urals based on survey data.


Table 7.2. Characterization of gypsy moth outbreak cycles in the South Urals (1957–2012)

Outbreak number

Beginning of outbreak,

year

End of outbreak,

year

Duration of

outbreak, years

Log-transformed population density,

eggs/treemaximum minimum

Forest-steppe zone1 Before 1957 1961 > 4 6.24 –2.302 1962 1972 11 6.16 –2.303 1973 1981 9 6.57 –2.214 1982 1995 14 6.14 –2.305 1996 2001 7 3.91 0.966 2003 After 2012 > 9 8.52 *

Average values 10 6.26 –1.46Steppe zone

Before 1976 1984 > 9 2.01 –0.88 1985 2001 7 2.59 –4.69 Since 2002 * > 11 3.93 *

Average values 9 2.84 < –2.7Mountain forest zone

Before 1961 1974 > 14 3.27 –3.46 1975 1983 8 2.00 –3.30 1990 2002 12 4.10 –3.60 Since 2004 * > 9 * –1.78

Average values 11 > 3.1 < 3.0* – The outbreak has not finished yet, and, thus, the parameter cannot be evalu-ated.

During the last 50 years, the average outbreak duration was 10 years,

the average population density in the outbreak peak phase was somewhat higher than 500 eggs/tree, and the average population density in the low-density phase was about 10 eggs/tree (Table 7.2).

Spectral analysis of the LTI sets {Lj(s)} of all natural zones also showed the presence of cyclic oscillations with similar shapes of the spectra (Fig. 7.2).

The maximum of the spectral density function for the FSZ is observed at f = 0.104 1/year, which corresponds to cyclic oscillations with the period of T = 1/f = 9.6 years. For the SZ, no significant temporal trend is observed in the gypsy moth population dynamics, but there are cyclic oscillations characterized by the spectral density function of the time


series {L(s)} (curve 2 in Fig. 7.2). The highest peak of the spectrum for the SZ corresponds to f = 0.06 1/year, or to the average duration of the outbreak cycle T = 17 years.

Fig. 7.2. Spectral density of the time series {L(s)} for the FSZ (curve 1),

SZ (curve 2) and MFZ (curve 3) Surveys in the MFZ have been conducted since t0 = 1960. However,

evaluation of the power spectrum of the time series {L(s)} for the MFZ is not quite reliable (curve 3 in Fig. 7.2), as there were many years when population density in the MFZ was below the detection limit of the surveys, and we cannot be sure that there are outbreak cycles of the gypsy moth in this zone.

Thus, although the durations of outbreaks in different natural zones are similar (10–11 years), the gypsy moth population density in the egg phase is considerably higher in the FSZ than in the SZ and MFZ. These results are consistent with the fact that the areas of the tree stands dam-aged by the gypsy moth are much larger in the FSZ than in the other zones (Ponomarev et al., 2012).

The graph in Figure 7.1 shows that the curve of gypsy moth population dynamics in the FSZ has no phase of stable density: from the low-density phase, with the minimum egg density, it immediately enters the popula-tion buildup phase. This type of population dynamics is described as sustained outbreak (Isaev et al., 2001). Sustained outbreaks develop when


the trees damaged by insects recover quickly enough for the insects to have available food in the same tree stands. During sustained outbreaks, populations do not stay in a high-density or low-density state for a long time, and the time series of long-term population dynamics have cyclic density fluctuations.

As gypsy moth surveys in the South Urals were performed in the egg phase, for modeling the gypsy moth population dynamics, we used the ADL-model that related the LTI density L(s) of the gypsy moth popula-tion in the egg phase in year s to population densities in the same phase in the previous years. The order of the autoregressive equation was deter-mined from the shape of the partial autocorrelation function (Fig. 7.3).

Fig. 7.3. Partial autocorrelation function of the LTI series of gypsy moth

population dynamics in the forest-steppe zone of the South Urals. 1 – partial autocorrelation function (PACF); 2 and 3 – confidence 95% limits of PACF The partial autocorrelation function is significantly different from zero

at the time delay k 2, i.e. the model time series has order 2. The agreement between the model and field data was estimated from

the coefficient of determination R2 of the regression equation (7.17). Determination of the synchrony and phase shift of the model and field time series of population density was based on the calculation of the cross-correlation function of the model time series and the LTI set { ( )}L s of the field data. If the maximum of the cross-correlation function was at the shift k = 0 and the value of the cross-correlation function f(k = 0) was close to 1, the model and field time series changed synchronously and no phase shifts were observed between these time series.

Table 7.3 gives results of calculations of regressive model (7.17) coeffi-cients.


Table 7.3. Calculating the coefficients of the regressive model (7.17) of gypsy moth population dynamics in the FSZ

Models Variables Q Std.Err. t P

ADL-model (7.17)

Free term 1.821/ 1.440

0.641/0.564

2.842/2.555

0.007/0.014

W(s) –0.290/ –0.211

0.102/0.092

–2.843/–2.292

0.007/0.027

L(s – 2) –0.529/ –0.629

0.107/0.105

–4.932/–6.003

0.000/0.000

L(s – 1) 1.215/ 1.323

0.104/0.100

11.675/13.178

0.000/0.000

AR(2)model (the Yule process)

Free term 0.027 0.173 –0.153 0.879

L(s – 2) –0.609 0.107 –5.716 0.000001

L(s – 1) 1.251 0.105 11.900 0.000000

Note. The numerator is calculation for version I; the denominator is calculation for version II. Q is the value of the coefficient, Std.Err. is standard error of the coefficient, t is value of the t-test for the coefficient, р is reliability of the calculat-ed coefficient.

As can be seen from the data in Table 7.3, the reduced length of the

time series {L(s)} used for calculations did not change significantly the values of coefficients of the model equation (7.17). The sign of the coeffi-cient for variable W(s), which characterizes the effect of the weather on population dynamics, is negative, i.e. in the model equation, as the HTC decreases (during dry and warm weather), population density increases, which corresponds to the field observations (Gypsy moth in Ural, 2012). The coefficient for variable L(s – 2) is negative: the higher the density is in the (s – 2)th year, the stronger the regulating factors influence the population in the (s – 1)th year. By contrast, the coefficient for varia-ble L(s – 1) is positive: the higher the population density is in the previ-ous year, the more eggs will be laid in the current year. Thus, the signs of coefficients in the model equation correspond to the biological meaning of the model. The highest value of the coefficient for variable L(s – 1) indicates that among all factors, the egg density of the previous year has the strongest effect on the egg density of the current year.

If we simplify the model (7.17) and use only the data on values L(s – 1) and L(s – 2) to describe dynamics L(s) in the FSZ, this second-order autoregressive model – the Yule process (Kendall, Stuart, 1976) – will model gypsy moth population dynamics accurately enough (R2 = 0.77) (see Table 7.3), but the model time series will show a phase lag of one


year relative to the time series of the field data. Thus, the model does need to take into account weather factors.

Figures 7.4 and 7.5 show results of model calculations of the gypsy moth population density in the FS zone; Figure 7.6 shows the cross-correlation function of the model time series and the transformed time series of the surveys.

For the SZ, the number of significant factors in the model is lower compared to their number in the model for the FSZ (Table 7.4).

Fig. 7.4. LTI series of gypsy moth population dynamics in the egg phase

in the FSZ (1) and model calculations (2)

Fig. 7.5. LTI series and model calculations of gypsy moth population dynamics in the FSZ (1 – LTI series of the data {L(s)}; 2 – model calculation for the train-

ing set – the data for years between 1958 and 1982; 3 – calculated density for the testing part of the time series {L(s)} – the data for years between 1983 and 2007)


Fig. 7.6. Cross-correlation function of the model and LTI series

of survey data for the FSZ

Table 7.4. Calculating the coefficients of the ADL(2,1) model of gypsy moth population dynamics in the SZ

Model variables Q Std.Err. t PFree term 0.311 0.555 0.560 0.580W(s) –0.126 0.123 –1.023 0.316L(s – 2) –0.589 0.168 –3.511 0.002L(s – 1) 1.416 0.167 8.481 0.000

Q is the value of the coefficient, Std.Err. is standard error of the coefficient, t is value of the t-test for the coefficient, р is reliability of the calculated coefficient.

Weather parameters (in this case, HTC) do not influence significantly

and unambiguously gypsy moth population dynamics in this zone (Ta-ble 7.4), and for the first approximation description, a model with two variables, L(s – 1) and L(s – 2), may be used. The field data and model calculations of the gypsy moth population density in the SZ are shown in Figure 7.7.

It is amazing that we managed to construct a sufficiently accurate model of gypsy moth population dynamics based on very few data on the ecosystem. Actually, in addition to the time series of the gypsy moth population dynamics (transformed in a certain way), we only used the HTC time series. Population dynamics and the occurrence of outbreaks of the gypsy moth in the South Urals can be soundly explained based on the notion of the leading role of density dependent factors in regulation of dynamics. Weather factors do influence gypsy moth population densi-ty in the Urals, but this influence is considerably weaker than the effects


of density dependent factors and habitat properties. One of the possible reasons why weather plays such a small role in the model of dynamics of sustained-outbreak population is that weather factors produce divergent effects. Different general hydrothermal conditions may lead to outbreaks following different scenarios. The validity of this assumption is supported by the steady periodicity of outbreaks in the absence of periodicity in weather changes and by the significant influence of weather factors on population dynamics in the FSZ and their non-significant influence in the SZ, with the correlation between density dynamics in these two zones being high.

Fig. 7.7. Survey data and model calculations of gypsy moth population dynamics

in the SZ (1 – LTI series of survey data {L(s)}; 2 – calculation for version I) One of the significant factors in population dynamics of defoliating

insects is the norm of response of the population, which is directly related to both phenology of leaf ontogeny and the phase of population density dynamics. The important role of synchronization of physiological pro-cesses in the food plant and the development of the larvae during out-breaks is well known (Asch, Visser, 2007; Khanislamov et al., 1962). In addition to that, studies of different gypsy moth populations showed a noticeable influence of hydrothermal conditions in the host plant habitat on the maternal effect – morphophysiological and trophic param-eters of the daughter generation of the larvae (Rossiter, 1991) – in the low-density phase and a dramatic decrease in this effect in the early stages and peak of the outbreak (Andreeva et al., 2008; Ponomarev et al., 2012). Thus, differences in the change of the population response norm, which depend on the phenology of the host plant and the phase of densi-


ty dynamics, may also considerably neutralize the effects of weather factors on population density dynamics.

Hence, it would appear that one of the major factors that determine the sustained outbreak type of gypsy moth population in the South Urals must be quick recovery of the food – birch leaves – after the damage caused by gypsy moth larvae and re-establishment of steady phenological cycles of leaf ontogeny. Rapid recovery of the food resource is essential for sustained outbreaks of forest insects. For example, the larch bud moth Zeiraphera griceana (see the next section of this chapter), which has sustained outbreaks in the Swiss Alps, feeds on larch needles, and they, like birch leaves, recover in one or two seasons following the damage caused by insects (Baltensweiler et al., 1977).

It is rather simple to predict behavior of the population for the sus-tained-outbreak species. The model results obtained in this study show that the data on gypsy moth population densities for two years are suffi-cient to make an accurate forecast of the future population density. The contribution of weather factors to the total effects of regulating and modifying factors on gypsy moth population in the South Urals consti-tutes only 10–15%. If sufficiently long time series of gypsy moth surveys are available, the population dynamics of the species may be predicted for the entire outbreak cycle.

7.3 Modeling population dynamics of the larch bud moth in the Alps

Analysis of the larch bud moth population dynamics was based on two survey datasets. One dataset, which describes population dynamics of this species in the Engandine Valley, was collected between 1836 and 1984 (Baltensweiler, 1988). This must be the longest available time series of population dynamics of forest insects. These data have been analyzed by different researchers (Isaev et al., 1984; Liebhold et al., 2006). The other dataset consists of the data on population dynamics of a community of defoliating insects, including the larch bud moth, in different habitats in the Alps (Baltensweiler, 1991).

Zeiraphera griseana Hub. occurs everywhere in Europe where the larch grows. Adults emerge between July and October, their lifespan is about 30 days, and oviposition lasts about 20 days. One female lays between 20 and 180 eggs, with a maximum of 350 eggs, depending on its nutri-tional status (Baltensweiler, 1988).


Figure 7.8 shows the LTI series of population density of the larch but moth in the Engandine Valley.

Fig. 7.8. The LTI series of population dynamics of the larch bud moth

in the Engandine Valley between 1863 and 1984 The spectrum of the LTI series was calculated to evaluate the periodici-

ty of larch bud moth outbreaks (Fig. 7.9).

Fig. 7.9. Spectral density of the LTI series of larch bud moth

population dynamics in the Engandine Valley Parameters of the spectral density of the larch bud moth in the

Engandine Valley show that outbreaks of this species in this region of the Alps occur at a frequency of 0.12 1/year, or with a periodicity of 8.5 years.


The first step in constructing the ADL-model of the larch bud moth population dynamics in the Engandine Valley was to determine the order of autoregression by the shape of the partial autocorrelation function (PACF) of the LTI series (Fig. 7.10).

Fig. 7.10. Partial autocorrelation function of the LTI series of survey data

(1 – partial autocorrelation function (PACF); 2 and 3 – confidence 95% limits of PACF)

The PACF shape suggests that as a model function, we can use a third-

order autoregression AR(3):

0 1 2 3( ) ] ( 1) ( 2) ( 3).L s a a L s a L s a L s (7.18)

Table 7.5 lists calculated parameters of the AR(3) model.

Table 7.5. Parameters of the AR(3) model

parameters а1 a2 a3 a0

AR(3) coefficients 1.973 –1.686 0.542 0.008Error of coefficient of AR(3) 0.080 0.126 0.080 0.056R2 0.937F-test 559.93gain margin 4.53

“Pure” autocorrelation model AR(3) evidently describes very well

changes in the population density of the larch bud moth over more than 100 years (Table 7.5). Figure 7.11 compares the survey data and results of the model calculations.


Fig. 7.11. Comparison of the LTI series of survey data (1)

and AR(3) model data (2) The survey data and the model data are clearly in very good agree-

ment. The description of larch bud moth population dynamics in the Engandine Valley is very accurate indeed, as evident from the high values of the coefficient of determination R2 and the F-test for equation (7.17), which is considered as a regressive equation with unknown varia-bles а0, a1, a2, а3.

The phase synchronism of the model series and the LTI series of the survey data is confirmed by the parameters of the cross-correlation function of these series (Fig. 7.12).

Fig. 7.12. Cross-correlation function of the LTI series of the survey data

and the AR(3) model series


As can be seen in Figure 7.12, at k = 0, the correlation coefficient be-tween the time series analyzed was practically equal to 1, suggesting, first, the phase synchrony of the series and, second, the good agreement be-tween the survey and model data.

Model AR(2) used to model larch bud moth population dynamics in the Engandine Valley describes the survey data very well (with the coeffi-cient of determination R2 = 0.91, which is somewhat lower than R2 for model AR(3), but the time series of the survey and model data are not phase synchronous (Fig. 7.13 and 7.14).

Fig. 7.13. LTI series of model AR(2)

Fig. 7.14. Cross-correlation functions of the LTI series of the survey data

and the series of the AR(2) model


Between the LTI series and the model series of AR(2), there is a 1 year phase delay. Thus, larch bud moth population dynamics can be accurate-ly described only by the third-order autoregression.

Results of modeling larch bud moth population dynamics in the Engandine Valley can be compared with the results of modeling larch bud moth population dynamics in different habitats in Alpine larch forests that had been performed for 27 years, between 1952 and 1979 (Baltensweiler, 1991).

Modeling was based on the data on population dynamics of the larch bud moth in the following habitats (Fig. 7.15):

1) Goms in the Upper Rhone River Valley (Kanton Wallis, Switzer-land) – 1300–1600 m above sea level;

2) Oberengadin (Kanton Graubunden, Switzerland) – the valley situ-ated at altitudes of 1600–1800 m above sea level, a UNESCO bio-sphere reserve;

3) Valle Aurina (Provincia Autonoma di Bolzano-Alto Adige, Italy); 4) Lungau (Land Salzburg, Austria).

Fig. 7.15. Study areas in the Alps: 1) Goms in the Upper Rhone River Valley;

2) Oberengadin; 3) Valle Aurina; 4) Lungau (Google maps data) Fig. 7.16 shows LTI series of larch bud moth population dynamics in

all habitats. Figure 7.17 shows the spectral density function of the LTI series of

larch bud moth population dynamics in Oberengadin, indicating that the transformed series has a cyclic component with a frequency of 0.1 1/year, i.e. with a ten-year periodicity of the dynamics.

To determine the order of k in the AR-model of larch bud moth popu-lation dynamics in Oberengadin, we used partial correlation function (Box, Jenkins, 1974). The order of autoregression was determined from the value of the lag of the last significant term in the partial autoregres-sive function (Fig. 7.18).


Fig. 7.16. LTI series of larch bud moth population dynamics in different habitats in the Alpine forests (1 – Goms; 2 – Oberengadin; 3 – Valle Aurina; 4 – Lungau)

Fig. 7.17. Spectral density of the LTI series of larch bud moth

population dynamics in Oberengadin The shape of the autocorrelation function suggests that this time series

can be modeled by using an AR(2) model, which coincides with the order of autoregression in the gypsy moth model.

To calculate the ADL(2, 1) model, we used weather data of the weather stations closest to the outbreak sites of the larch bud moth (Table 7.6).

As can be seen from the data in Table 7.6, the distance between the outbreak site and the weather station is very short (7 km) for Oberengadin, short (36 km) for Goms, and quite long (100 km or more) for Valle Aurina and Lungau. Therefore, models ADL(2, 1) were con-structed only for Oberengadin and Goms. Coefficients of the regressive model are listed in Table 7.7.


Fig. 7.18. Partial autocorrelation function of the LTI series of larch bud moth

population dynamics in Oberengadin. (1 –PACF; 2 and 3 – confidence 95% limits of PACF)

Table 7.6. Geographical positions of larch bud moth outbreaks

and weather stations closest to outbreak sites

Habitat (H)

Longi-tude,

latitude of H

Height above

sea level of H, m

Weather station

(W)

Longi-tude,

latitude of W

Height above

sea level of W, m

Distance between H and

WOberengadin (Kanton Graubunden, Switzerland)

9.764, 46.428

1600-1800 Samedan 9.873,

46.536 1721 7

Goms (Kanton Wallis, Switzerland)

8.216, 46.477 1560 Grimsel

Hospiz 8.333, 46.572 2161 36

Valle Aurina (Provincia Autonoma di Bolzano-Alto Adige, Italy)

11.983, 46.997 1054 Bolzano 11.354,

46.494 260 100

Lungau (Land Salzburg, Austria)

14.141, 47.587 1075 Salzburg 13.043,

47.800 424 121

As can be seen from the data in Table 7.7, the sign of the coefficient for

variable ( 1)W s , which characterizes the effect of the previous-year weather on the current population size, is negative, i.e. in the model


Table 7.7. Coefficients calculated for the model equation (7.17) of larch bud moth population dynamics, based on survey data collected

between 1952 and 1979 (Oberengadin)

Model variables Model parameters T P Coefficient Std.Err.C 1.60 1.91 0.83 0.41L(s – 1) –0.76 0.13 –6.02 0.00L(s – 2) 1.29 0.12 10.46 0.00W(s – 1) –1.44 1.17 –1.24 0.23

equation, as the HTC decreases (during dry and warm weather), popula-tion density increases. The coefficient for variable ( 2)L s is negative: the higher the density is in the ( 2)s th year, the stronger the regulating factors (parasites and predators) influence the population in the ( 1)s th year. By contrast, the coefficient for variable ( 1)L s is positive: the higher the population density is in the previous year, the more eggs will be laid in the current year. Thus, the signs of coefficients in the model equation correspond to the biological meaning of the model.

The resulting model has the following form: ( ) 1.6 0.76 ( 2) 1.29 ( 1) 1.44 ( ).L s L s L s W s (7.18)

Coefficient of determination 2 0.85R points to the good agreement between model calculations and survey data on the population density of the larch bud moth in the Oberengadin Valley (Fig. 7.19).

Fig. 7.19. Population dynamics of the larch bud moth

(1 – LTI series of the survey data; 2 – model calculations)


Fig. 7.20 shows the cross-correlation function for the LTI series of the field data and the model series.

Fig. 7.20. Cross-correlation function for the LTI series of the field data

and the model series The maximum of the cross-correlation function is at the delay 0k ,

and the value of the cross-correlation function ( 0)f k is close to 1, i.e., the model data series is in phase with the transformed series of the field data.

Similar calculations were done for the time series of the larch bud moth population dynamics in other Alpine habitats (Table 7.8.).

Table 7.8. Coefficients calculated for the model equation (7.17)

of larch bud moth population dynamics, based on the survey data collected in Goms, Valle Aurina, and Lungau

Region

Length of the time

series, years

Coefficients of equation (7.17)* 2R

0Q 1Q 2Q 3Q 4Q

Goms, Switzer-land

12 2.54/ 3.9

–0.79/ 0.21

1.29/ 0.24

–0.01/ 1.99

–2.56/ 2.6 0.82

Valle Aurina, Italy

20 –2.2/ 1.95

–0.96/ 0.17

1.36/ 0.15

1.04/ 1.12

0.87/ 1.17 0.89

Lungau, Austria 20 4.54/

1.8–0.71/0.13

1.21/0.13

–2.82/1.07

–1.6/1.06 0.90

* The numerator shows the coefficient value and the denominator shows the standard error of the coefficient.


The phase shift φ and coefficient of determination 2R for the survey data (Table 7.8) show that even the shortest time series (12 years) is in phase with the actual forecast, which proves the validity of the model for Zeiraphera griseana L. – a sustained outbreak species.

7.4 Simulation models of population dynamics of defoliating insects in the Krasnoturansk pine forest

The Krasnoturansk pine forest (54º16.315´ N, 91º37.757´ E) is a spatially isolated natural environment with pure pine stands about 100 years old. The Krasnoturansk pine forest is not subjected to heavy economic im-pacts, but local residents use it for recreational purposes. Thus, it may be regarded as an ecosystem with natural ecological processes.

A brief forest inventory (prepared in 1979, when the study began) in different habitats of the Krasnoturansk pine forest is given in Ta-ble 7.9.

Table 7.9. Forest inventory of the tree stands in different habitats

of the Krasnoturansk pine forest

Hab

itat Tree stand characterization

Area, ha

compo-sition

age class

average height, m

average diameter,

cm

quality of locality density forest

type*

1 331 10C IV 19.0 23 II 0.8 А2 207 10C IV 17.0 18 III 0.9 B5 335 10C IV 18.0 22 II 0.7 С, D 6 193 10C II, IV 8.0 9 III 0.8 Е7 110 10С IV 20.0 21 II 0.8 Е

* forest types: А – herb-moss pine forest; B – cowberry-moss pine forest; С – lichen-cowberry-moss pine forest; D – herb-lichen-dead litter pine forest; Е – herb-cowberry-moss pine forest.

Annual surveys of the community of defoliating insects, including the

pine looper, the tawny-barred angle, the pine-tree lappet, and two sawfly species, were conducted on five permanent sample plots in the Krasnoturansk pine forest for 35 years (1979–2014) (Palnikova et al., 2014).


The survey data for different defoliating insect species (Table 4.1) were transformed using the algorithm described above, similarly to the time series of population dynamics of other species. The resulting LTI series in pine looper population density in habitat “Lake” is a stationary series with the zero mean (Fig. 7.21).

Fig. 7.21. LTI series of the pine looper population density in habitat “Lake” Figure 7.22 shows the spectral density function of the LTI series of the

pine looper population density in habitat “Lake”.

Fig. 7.22. The spectral density function of the LTI series of the pine looper population density in habitat “Lake”

The spectral density function has a pronounced, although rather

broad, peak at a frequency of 0.0625 1/year, i.e. the pine looper popula-


tion in this habitat shows cyclic fluctuations with an average period of 16 years.

Cyclic oscillation of the LTI series may be caused by either the effect of modifying weather factors or the effects of regulating factors. If the cycles occur because of regular weather fluctuations, firstly, the spectral density function of the weather time series must also have a peak at a frequency close to f = 0.0625 and, secondly, the maximal value of the cross-correlation function between the weather time series and the LTI series must be close to 1 (this maximum may be shifted relative to 0 along the X-axis).

Owing to the cyclic behavior of the time series of population dynam-ics, this time series can be modeled by using an ADL model. To deter-mine the order of the model, let us consider the partial autocorrelation function of the LTI series of the pine looper population in habitat “Lake” presented in Figure 7.23.

Fig. 7.23. Partial autocorrelation function of the LTI series

of the pine looper population in habitat “Lake”. (1 –PACF; 2 and 3 – confidence 95% limits of PACF)

The order of autoregression can be taken to be equal to 2, as suggested

by the data in Figure 7.23. Hence, the value of an LTI series component in some year is related to the values of the LTI series components in the two previous years. Such relations are described by the second-order autoregressive model AR(2) (Box, Jenkins, 1970): 0 1 2( ) ( 1) ( 2).L i a a L i a L i (7.19)

Since we know the values of the L(i) terms in the LTI series of the pine looper population dynamics in habitat “Lake”, coefficients а0, а1, and а2 can be found by considering (8.19) as a linear regressive equation with the known L(i) values and unknown а0, а1, and а2 coefficients.


Quantification of the coefficients of a linear regressive equation is a standard task of regression analysis, and Table 7.10 gives values of coefficients of the AR(2)-model (7.19) for the LTI series of pine looper population dynamics in habitat “Lake”.

Table 7.10. Coefficients of the AR(2)-model (7.19) for the data

of the transformed series of pine looper population density in habitat “Lake”

Statistical parameters Coefficients of equation (8.19)а0 а1 а2

Coefficient values –0.06 1.39 –0.68Errors of coefficient 0.09 0.16 0.16R2 0.78F-test 48.21

The cross-correlation function between the AR(2)-model time series of

the population density and the LTI series peaks at the zero shift between the series analyzed (Fig. 7.24). Hence, these series are synchronous, and the transformation has not affected the dynamics of the population density time series.

To take into account possible weather effects, we tested the ADL(2,1) model that included a weather parameter – seasonal normalized HTC determined from the data of the Opytnoye polye weather station, which is situated in Minusinsk, about 80 km away from the sample plot.

Fig. 7.24. Cross-correlation function between the AR(2)-model time series

of the population density and the LTI series


Table 7.11 gives the values of parameters of the ADL(2, 1) model for the LTI series of pine looper population density in habitat “Lake”.

Table 7.11. Coefficients of the ADL(2, 1) model for the LTI series

of pine looper population density in habitat “Lake”

Statistical parameters Coefficients of equation

а0 a1 a2 a3

Coefficient values 0.42 1.21 –0.55 –0.44Errors of coefficient 0.41 0.16 0.16 0.40R2 0.74 F-test 26.44

Comparison of Tables 7.10 and 7.11 shows that the accuracy of the

model is not increased when a supplementary, weather, term is intro-duced into it. Moreover, the cross-correlation function, which character-izes coherence between the LTI series of survey data and the ADL(2, 1) model series, reaches its maximum at the cross-correlation delay Δt = –1 year (Fig. 7.25), i.e. the model series shows a phase lag of one year relative to the LTI series of survey data.

Fig. 7.25. Cross-correlation function between the LTI series of the pine looper

population in habitat “Lake” and the ADL(2, 1) model series Even more impressive is the agreement between the values in the LTI

series and the AR model series for the pine looper population in habitat “Dune”, where the highest population density is 13 insects per tree. Data transformation, similar to the transformation of the time series of pine


looper population dynamics in habitat “Lake”, resulted in the AR(2)-model of the LTI series of the pine looper population in habitat “Dune” (Table 7.12).

Table 7.12. Coefficients of the AR(2)-model for the LTI series

of pine looper population density in habitat “Dune”

Statistical parameters Model coefficients

а(0) а(1) а(2)Coefficient value 0.036 1.603 –0.793Error of coefficient 0.080 0.116 0.114R2 0.927F 171.65

As suggested by the values of the coefficient of determination R2 and

F-test (Table 7.12), the second-order autoregressive model AR(2) de-scribes very well the LTI dynamics of pine looper population density in habitat “Dune”. Figure 7.26 compares the dynamics of the LTI series and the AR(2)-model series.

Fig. 7.26. LTI series (1) and AR(2) model series (2) for pine looper population

in habitat “Dune” As coefficient of determination R2 for the AR(2)-model of pine looper

population in habitat “Dune” is very close to 1, we may conclude that weather factors do not make any significant contribution to the popula-tion dynamics of the species in this habitat.


Similar conclusions are suggested by analysis of the time series of D.pini population dynamics (Fig. 7.27 and 7.28, Table 7.13).

Fig. 7.27. Spectral density of the LTI series of D.pini population dynamics

in habitat “Dune”

Fig. 7.28. The LTI series (1) and AR(2)-model (2) for the D.pini population

in habitat “Dune” Table 7.13. Coefficients of the AR(2)-model for the transformed time series

of D.pini population density in habitat “Dune”

Statistical parameters Coefficientsа0 а1 а2

Coefficient value 0.007 1.518 –0.771Error of coefficient 0.077 0.124 0.125R2 0.890F-test 109.00


Thus, both the time series of gypsy moth population dynamics in the South Urals, characterized by regular outbreaks and wide variations in population density – between values close to zero and several thousand eggs per tree – and the time series of pine looper and pine-tree lappet population dynamics in the south of Middle Siberia, with population densities that only once, during 34 years of surveys, rose above the value of 1 larva per tree, show the same type of relationship between popula-tion densities of the adjacent years: positive correlation of the current year density to the previous year density and negative correlation of the current year density to the density of the year before the previous year. We interpreted the negative correlation between population densities in the ith and the (i – 2)th years as a reflection of the effects of parasites and (or) predators on the population in the ith year. Unfortunately, it is very difficult to estimate the effects of parasites and predators on a low-density population, but the fact that the same trend relates the population densities of three adjacent years, irrespective of their absolute values, is impressive.

Nevertheless, the models of the population dynamics of the gypsy moth, an outbreak species, and the pine looper, which stayed in the stable sparse state during the entire survey period, differ in the effects of weath-er on population dynamics. For the gypsy moth, the effect of the weather on population density is certainly significant, while for the pine looper population, which stays in the stable sparse state, no weather effect is observed. Chapter 8, which will approach the outbreak model as phase transition, will discuss in more detail possible mechanisms of the influ-ence of weather, as a modifying factor, on population dynamics as related to the current population density.

7.5 Modeling and predicting population dynamics of the European oak leaf-roller

The data on the European oak leaf-roller (Tortrix viridana L.) population density in the pupal phase in Moscow and the Moscow Oblast were reported (Golubev, Bagdatyeva, 2013). The authors also proposed a model describing population dynamics of this species. For the forecast, they used the following logarithmically linear model:

3

1

( )lg ( ) lg ( 1) lg .( 1)

j

jj

x i jx i x i ax i j

(7.20)


Equation (7.20) describes the relationship between population densi-ties of the adjacent years x(i) and x(i – 1) with the reproduction coeffi-

cient averaged for four seasons ( )( 1)x i j

x i j. In accordance with (7.20), the

current population density of the model species is determined through population densities in the previous four years. The authors verified the model by calculating coefficients aj from the data on population density for the period between 1962 and 1985.

The quality of the forecast in model (7.20) was evaluated by the vari-ance of the difference between the logarithms of the predicted and actual population densities (Golubev, Bagdatyeva, 2013). It would seem, though, that the synchrony between the phases of the model and field time series is more important for the forecast of population dynamics. Based on the high synchrony between the model and field time series, one can predict trends in population dynamics and assess the risk of an outbreak.

Model (7.20) does not explicitly estimate the influence of the weather on variations in population density, and it is vaguely assumed that popu-lation dynamics of the species under study is only influenced by regulat-ing factors. Moreover, according to the model, the European oak leaf-roller population “remembers” the regulating effects for four years, but it is not clear what factors are responsible for such “long” memory of a species that has one generation per year.

Let us consider an alternative ADL-model of the European oak leaf-roller population dynamics, based on the data reported by (Golubev, Bagdatyeva, 2013) and additional data on the daily temperature and amount of precipitation in the study region for the period between 1962 and 1985.

The first step in constructing the model was to transform the time se-ries of the field data into the stationary time series, which can be studied by using spectral and correlation analysis. Analysis showed that the time series studied was characterized by the logarithmic population density decreasing linearly over time. To be able to correctly use spectral and correlation analysis techniques in the further data processing, it is neces-sary to get rid of such components of the time series as the trend and high frequency noise. Figure 7.29 is a graphical representation of sequen-tial transformation of the field data time series.

For the resulting LTI series L(i), we calculated its power spectrum, characterizing the frequency components and periodicity of population density fluctuations (Fig. 7.30).


Fig. 7.29. LTI series of population dynamics of the European oak leaf-roller

before (1) and after isolation of the trend (2)

Fig. 7.30. Spectral density function of the LTI series of the European oak

leaf-roller population density The European oak leaf-roller population shows cyclic fluctuations with

an average period of about 10 years (Fig. 7.30). The spectral density peak is, however, rather broad, suggesting deviations from the “rigid” 10-year periodicity. The presence of the periodic component in the spectrum of the time series implies that the process is non-Markovian, and, hence, the forecast density must be related to the densities of several previous years. Therefore, let us consider an AR(k)-model of the European oak leaf-roller population dynamics. The order k of autoregression was deter-mined by using a conventional method of calculating partial autocorrela-tion function of LTI series (Fig. 7.31).

In the autocorrelation function, the first two terms are significant (Fig. 7.31), and, thus, the model autoregressive series will have the or-der k = 2: 0 1 2( ) ( 1) ( 2).L i L a L i a L i (7.21)


Fig. 7.31. Partial autocorrelation function of LTI series of the European oak leaf-roller population density (1 – PACF, 2 – limits of the 95% confidence

interval of PACF)

Fig. 7.32. LTI series of the field data (1) and the AR(2)-model (2)

of the dynamics of the European oak leaf-roller population density Figure 7.32 shows the LTI series of the field data and the AR(2)-model

series. To estimate the synchrony between the time series of the field data and

the model time series, we calculated the cross-correlation function of these series (Fig. 7.33).

The cross-correlation function peaked at a delay of k = 0 (Fig. 7.33), suggesting the absence of a phase delay between the LTI series of the field data and the AR(2)-model time series. The good agreement between the AR(2)-model of the dynamics and the LTI series of the field data is indicative of the small possible contribution of modifying (e.g. weather) factors to changes in the population dynamics of the European oak leaf-roller. Nevertheless, the quality of the model is somewhat enhanced if we


Fig. 7.33. Cross-correlation function of the LTI series of the field data

and the model time series of the European oak leaf-roller population density

Table 7.14. Parameters of the ADL(2,1) model of the European oak leaf-roller

model parameters Variables

year (i – 1) year (i – 2) HTC (may) L0

coefficient 1.10 –0.68 0.65 –0.58error of coefficient 0.18 0.19 0.50 0.56R2 0.70 F 13.88

use the ADL(2, 1) model incorporating the May HTC of the forecast year. Table 7.14 gives the results of calculations.

Weather parameters that have not been taken into account in this model may certainly influence the European oak leaf-roller population density, but, anyway, the contribution of the weather to the change in the population density of this species is rather small, no more than 10–15% of the total variance of population density.

7.6 Gain margin of the AR-models of forest insect population dynamics

Autoregressive models, in general, and second-order autoregressive mod-els, in particular, have been thoroughly studied, and the authors do not think there is any need to describe the methods of analysis and parameters


of autoregressive models from the very beginning; detailed descriptions can be found in the available literature (Bolshakov, Karimov, 2007; Veremey, 2013; Gaiduk et al., 2001; Dorf, Bishop, 2004; Kim, 2007).

The condition for the asymptotic stability of an n-order discrete sys-tem described by the difference equation 1 2( ) ( 1) ( 2) ... ( 1)x j a x j a x j bg i (7.22)

is the inequality

( ) 1,z j j = 1, …, n,

where z(j) are the roots of the characteristic equation (7.22). For the second-order AR model, the characteristic equation D(z) has

the following form: 2

1 2( ) .D z z a z a (7.23)

The second-order autoregressive equation is used to describe diverse processes in the chain theory and the control theory (ссылки). It is well known that the stability region of the AR(2)-model is determined by a set of inequalities for coefficients of the AR(2)-model (ссылки):

1 2

2 1

2

1,1,

1 1.

a aa a

a (7.24)

It follows from (7.24) that for the stable AR(2)-model, coefficients of the autoregressive equation a1 and a2 must lie inside a triangular region in the {a1, a2} plane.

The methods of the analysis of forest insect population dynamics and construction of ADL-models presume that the type of the model is known and that model parameters are time invariant. However, real populations and the environments in which these populations exist cannot be modeled exactly: they may change unpredictably, and different perturbing factors may affect them.

Systems will always deviate from an ideal model, for a number of rea-sons:

changes in the parameters of the population and the environment caused by various factors; properties of the population and the environment that have not been taken into account in the model; delayed responses in the population – environment interaction that have not been taken into account in the model;


changes in the stable state of the population; errors of population surveys; and unpredictable external impacts.

For the ADL models of forest insect population dynamics, gain margin of the model and its robust stability are important parameters for evalu-ating changes in the population state under various external transfor-mations and characterizing the population itself. Hence, as the states of the environment and the population of forest insects are indefinite, we need to evaluate, firstly, the model gain margin and, secondly, its robustness.

Gain margin characterizes the closeness of a point to the boundaries of the stability region. The gain margin of discrete systems is evaluated by the Mikhailov criterion and the Mikhailov hodograph

exp( )( ) ( ) | , , ,z jD j D z (7.25)

where ( )D z is the higher-power z normalized characteristic polynomial of the system.

The system with some characteristic polynomial is stable if the Mikhailov hodograph (7.25), with variable ν changed between –π and π, beginning on the real line, passes sequentially across 4n quadrants, mov-ing around point z = 0 counterclockwise (Gaiduk et al., 2011).

Let us examine the autoregressive equation for the model of the larch bud moth in Oberengadin:

( ) 1.44 ( 1) 0.90 ( 2) 0.03.L i L i L i (7.26)

The characteristic polynomial ( )D z for equation (7.26) has the follow-ing form:

2( ) 1.44 0.90 0.D z z z

After the transition from variable z to the new complex variable ν: 11

z , we obtain a characteristic equation relative to ν:

2( ) 1.44 0.90.j jD j e e (7.27)

To plot the Mikhailov hodograph, we isolate the real component Р(ν) and the imaginary component Q(ν) in equation (7.27).

( ) cos2 1.44cos 0.90,( ) sin2 1.44sin .

PQ

(7.28)


Then, in the {P(ν), Q(ν)} plane, we plot a hodograph curve for the variable values between –π and π, each point of this curve being val-ues P(ν0) and Q(ν0) at some value ν = ν0. The gain margin evaluated by the Mikhailov criterion is the radius of the circle with the center at point z = 0, which can be inscribed around zero of the Mikhailov hodo-graph.

Although the calculation looks complicated, the gain margin is calcu-lated by using a simple program (Table 7.15) in the MATLAB package.

Table 7.15. Listing of the program in the MATLAB package for calculating gain

margin of the autoregressive model (Gaiduk et al., 2011)

Dz = [1 a[1] a[2];Dz = Dz/Dz(1); nu = (-pi: pi/(100*length(Dz)):pi;j = sqrt(-1); z = exp(j*nu); GM = polyval (Dz, z);eta = min (abs(GM));disp ([‘gain margin eta =‘ num2str(eta)])

To evaluate the gain margin, one should upload the program from Ta-

ble 7.15 in the MATLAB package and perform only one operation: intro-duce values of the autoregressive model coefficients into line 1. By defini-tion, the gain margin 0 , and the lower the value, the more likely the system will be “upset” and lose its stability under external impacts.

In this book, we use the model of larch bud moth population dynamics to examine the methods for evaluating the gain margin and the robust stability of ADL-models of forest insect population dynamics and discuss ecological interpretation of results obtained.

The parameters of the stability and gain margin for the larch bud moth in different habitats in the Alps are given below, in Table 7.16.

Conditions of (7.24) are met in all habitats (Table 7.16), and the AR(2)-models of the larch bud moth for all habitats are stable. The gain margin for model equations for the larch bud moth in all habitats is sufficiently large – larger than 1.

Is there a relationship between average population density and gain margin? Figure 7.34 shows the relationships between the average densi-ty <x> of various species in habitat Oberengadin and the gain margin of the AR(2)-models for these species.


Table 7.16. Coefficients of the AR(2) model and their ratios for defoliating insects in habitat Oberengadin in the Alps

Parameters of AR(2) model

species

Z. diniana

P. laricis

O.autum-nata

E.dura-tella

S. lari-cana

T. sal-tuum

P. aeri-

ferana average popu-lation density <x>

58.31 1.54 0.65 0.88 0.07 0.07 0.01

а1 1.44 1.31 1.26 1.41 1.16 1.19 1.42 а2 –0.90 –0.67 –0.77 –0.84 –0.68 –0.59 –0.82 а1 + а2 0.54 0.64 0.49 0.57 0.47 0.60 0.60 а2 – а1 –2.34 –1.98 –2.03 –2.25 –1.84 –1.78 –2.25 R2 0.95 0.80 0.80 0.89 0.72 0.68 0.87 F-test 208.47 41.69 40.81 84.08 27.15 21.91 71.41 Gain margin 1.33 1.01 1.04 1.25 0.77 0.72 1.27

As we can see from Figure 7.34, there is linear dependence between

logarithm of population density and gain margin for all species in habitat (except P. aeriferana).

Fig. 7.34. Relationships between the average density <x> of various species in

habitat Oberengadin and the gain margin of the AR(2)-models for these species It is interesting to analyze the relationship between the gain margin

of the species and coefficient of variation V of its density, which can be regarded as an indirect indicator of the risk of outbreak (Fig. 7.35).


Fig. 7.35. The relationship between the gain margin of insect species

in Oberengadin and the coefficient of variation of their density Thus, values V and <x> are monotonically related to (Fig. 7.34

and 7.35), suggesting that for forest insect populations with the same order of autoregression (e.g., for AR(2) series), population density fluctuations will increase with an increase in .

***

All time series of forest insect population dynamics analyzed in this book and elsewhere are adequately described by AR- or ADL-models. The reasons why the “past” effectively influences the current and the future population densities may be associated with the delayed response of parasites and predators to the increase in prey population densities, reactions of host plants, which may be exhibited as a change in the “cost” of food consumption, maternal effect, etc. In any case, phenomenological analysis of the time series of forest insect population dynamics offers an estimate of the characteristic time of population response to the impact of regulating factors.

Based on the approach used in this study, one can quantify the effect of the weather on forest insect population density. For a number of Europe-an insect species, the contribution of weather factors is very small, and an AR model can effectively account for up to 95% of the total variance in population densities of the model species. For the species in severer habitats (the Urals, Siberia), weather factors can account for no more than 10–15% of the total variance in population densities. The most obvious explanation of these differences in the susceptibility of insects to climate effects is “tougher” weather conditions in the Urals and Siberia, with their highly continental climate (greater temperature differences, shorter plant growing season, winter cold, etc.).

183

8 Modeling of Population Dynamics and Outbreaks of Forest Insects as Phase Transitions

8.1 Models of phase transitions for describing critical events in complex systems

ADL models of forest insect population dynamics describe the occur-rence of outbreaks as the consequence of delay of feedbacks regulating population dynamics. However, a different approach can also be used to describe quantitative and qualitative changes that occur in forest insect populations as they reach outbreak densities and then recover their low-density state.

In accordance with the phenomenological model of population dy-namics, there are two phases of insect population development: the stable low density state and outbreak. During the phase of the stable low density state, the population density is the lowest possible and the insects colo-nize only some of the hosts. The outbreak occurs when the pest popula-tion density exceeds some critical value хr, and the insects colonize all suitable host trees in their habitat (Isaev et al., 2001; Kondakov, 1974).

In Chapter 2, we showed that possible states of forest insect population can be described with potential function G(х) – the function inverse to the probability of insects surviving in a population of density x, de-pending on the effects of various regulating and modifying factors. For




a particular insect population, the general form of this dependence is certainly unknown, but one can assume that stable states of the popula-tion are characterized by the minimum values of potential function G(х) (Soukhovolsky, 2004).

The condition ( ) minG x may be regarded as an extremal principle. Such extremal principles are often used in descriptions of complex sys-tems, together with kinetic equations that describe the dynamics of microscopic variables of the system. To describe a complex system, n equations of dynamics are replaced by a certain function relative to which it is assumed that the system tends to reach the states in which the value of this state function is the lowest. Then, there are no more than 3–4 variables in the model of a physical system, whatever the number of the system components is.

There are various approaches to the construction of ecological models, based on using extremal principles (Fursova et al., 2003). These ap-proaches are good for finding stable states of the system but not for describing the dynamics of the processes occurring in it (Brout, 1967; Landau, Lifshits, 1964). Nevertheless, simplification of the models of ecosystems by using this model approach seems a promising tactic.

In this chapter, for constructing models of population dynamics, we will consider processes occurring in insect population and leading to a dramatic increase in population density or to an equally dramatic density decrease as processes analogous to phase transitions in physics.

In physics, the notion of phase is used as an integrated indicator of the state of a certain substance or a physical system (Bruce, Cowley, 1984). A very simple example is three phases of water (solid, liquid, and gase-ous). A change in the state of the system under certain external impacts is called a phase transition. The phase transition occurs when the external impact reaches a certain critical level. For example, the water heated to its boiling point, Тс = 100 С, changes its state from liquid to gaseous, and when a piece of iron is cooled to the critical temperature – the Curie point, Тс = 770 С, it generates its own magnetic moment.

In accordance with P. Ehrenfest’s classification, for physical systems there are (Dmitriev, 2004)

first-order phase transitions, such as boiling, which involves a dis-continuous change in such parameters of the physical system as the density of the matter and concentration of components, and second-order phase transitions, such as superconductivity, superfluidity, and magnetization, when density and internal energy do not change, but their derivatives with respect to the external field change discontinuously.

Modeling of Population Dynamics and Outbreaks of Forest Insects 185

It has been assumed that phase transitions can be described using a cer-tain thermodynamic potential, G, and condition minG is an optimiza-tion principle, regulating the process of phase transition (Landau, Lifshits, 1964). The thermodynamic potential of a complex system depends on a large number of factors, and in most cases, the general form of func-tion G is unknown. However, in the case of second-order phase transi-tions, the number of factors that influence the behavior of the system close to the point of phase transition is usually considerably reduced, suggesting that function G depends on just two variables: the so-called order parame-ter, q, characterizing certain general properties of the system under study, and an external variable such as temperature T (Landau, Lifshits, 1964). The phase transition, expressed as a change in the order parameter, begins when the value of the external variable becomes less (or more – depending on the particular system) than a certain critical value.

The models of phase transitions incorporate the universality principle, in accordance with which processes of phase transitions depend on just a few basic properties of the system, such as dimensionality, the number of com-ponents of the order parameter, and distance dependence of interactions in the system (Bruce, Cowley, 1984). Simple static models of phase transitions are time independent. This assumption significantly simplifies the model: the description of the system only needs stable minima of function G.

The idea of universality of phase transitions in systems with one order parameter and homogeneous spatial structure simplifies and unifies the construction of models of phase transitions. Even without knowing the exact form of function G, one can describe the process of phase transition and find critical values of the external factor and values of the order parameter in stable states of the system.

In this study, we have constructed models of forest insect outbreaks as phase transitions in forest ecosystems.

8.2 Population buildup and development of an outbreak of forest insects as a first-order phase transition

Insect population density changes continuously both in space and in time. Temporal fluctuations around the x1 density of the population in the stable sparse state and fluctuations between х1 and х2 in the outbreak phase can be studied separately. The characteristic times of such fluctua-tions and their amplitude are rather small (2–3 seasons), and, thus, they


do not produce any significant effect on forest ecosystems. By contrast, density fluctuations in the phases of outbreak development and decline often lead to the death of the tree stand, and, therefore, analysis of mech-anisms of such fluctuations is crucial to understanding ecological pro-cesses observed in forest ecosystems.

Transitions from the stable sparse state to the outbreak phase will be treated as first-order phase transitions, expressed as discontinuous changes in the density of forest insect populations from х1 to х2 and back.

To describe 1 2x x transitions of the population, let us determine the normalized population density m as follows:

,r

r

x xIx x

(8.1)

where х is insect population density; xr is the critical value of population density.

It follows from (8.1) that at ,rx x 0.I If the population density is low and ,rx x 1I . By contrast, outbreak conditions are character-ized by the density ,rx x and then 1.I

Phase transition models include the function of state, G, which is gen-erally assumed to be dependent on a large number of various parameters of the system under study. However, function G can sufficiently accurate-ly be related only to two variables: some integrated parameter of the state of the system – the so-called order parameter q – and the external varia-ble z. Dependence ( , )G f z q may be presented as a power series of the order parameter (Landau, 1937; Landau, Lifshits, 1964).

The simplest way to take into account the effects of external factors in models of first-order phase transitions is to treat them as a constant external field, h (e.g., ambient temperature T or hydrothermal coeffi-cient – the ratio of the total seasonal precipitation to the seasonal average air temperature). Then, the phase transition equation will include a bilinear term describing tree stand interactions with this field (Patashinsky, Pokrovsky, 1982). Let us write the formula for potential function G(z, I), using Landau’s equation, as follows:

2 3 40( ) ( ) ,cG I G a z z I cI bI VIh (8.2)

where G0, a, b, c, V are some constants; z is a regulating factor (normal-ized infestation of insects by natural enemies); zc is the critical value of the regulating factor; 0h W W is the external field characterizing the effect of modifying factor W on survival of the insect population; W0 is the constant characterizing the optimal value of the modifying factor.


In (8.2), the control variable z ( 0 z ) characterizes the effect of regulating factors (parasites, predators, food quality) on the survival of individuals in the insect population. The higher the value of z, the less the regulating factors influence the population. Stable (or metastable) states of the system correspond to the minima of potential function G. These

minima can be derived from standard conditions 0GI

and 2

2 0GI

.

First, let us examine the case when the external field h = 0, i.e. modify-ing factors do not produce a significant effect on changes in insect popu-lation density. Then, extrema of function G(I) are derived from the condition

2 32 ( ) 4 0.rG a z z I cI bII

(8.3)

Equation (8.3) has three solutions:

I = 0 and 2 4 ( )

.2

rc c ab z zI

b (8.4)

Solution I = 0 corresponds to the local maximum of function G(I), and the other two solutions correspond to the local minima of function G(I), i.e. to two stable states of the population (Fig. 8.1, curve 1).

At point I = 0, the value of function G(I) = G0, which is higher than the values G(I1) = G(I2) at points I1 and I2. Value G0 characterizes the height of the potential barrier between two stable states. Values of the func-tion G(I) close to I1 (or close to I2) are usually characterized as a potential well, and the depth of the potential well is determined as the difference between the values of G0 and G(I1) (or G(I2)). Point I = 0 is the point of phase transition from the state with the lowest population density and negative I to the state with the highest population density (outbreak) with positive I. In fact, the description of the properties of the theoretical function G(I) in (8.2) fully corresponds to the description of empirical potential function in Section 2.4 of this book. Hence, an explanation of the form of the empirical potential function, describing forest insect population dynamics, may be based on the idea that population processes are first-order phase transitions expressed as discontinuous changes in population density.

If we now assume that modifying factors influence population dynam-ics and h 0, after differentiating (8.2), we obtain

32 ( ) 4 0.cG a z z I I Vhq

(8.5)


Fig. 8.1. A general representation of the normalized potential function characterizing an outbreak of forest defoliating insects as a first-order

phase transition (curve 1 is the case when external field h = 0; curves 2 and 3 – field h > 0 and h(3) > h(2))

Let us examine different cases of solving equation (8.5). 1. If external field h = 0, the solution is obtained from the equation

presented above, (8.4). 2. Let z > zc, and field h be not strong. Then, we can neglect terms I2

and I3 and obtain 2 ( ) 0ca z z q Vh (8.6) and

( , ) .2 ( )c

VhI h za z z

(8.7)

It follows from (8.7) that in this case, the normalized population densi-ty I(h, z) > 0, and population enters the outbreak phase.

2. To solve equation (8.5) in the case when field h is weak, but z < zc, let us present it as follows: 0( , ) ( ) ( , ),I z h I z I z h (8.8) where 0( )I X is the solution to equation (8.2) in the case when external field h = 0 and .I I


Then, expanding equation (8.8) into series, we obtain

2

3 32

0

3( , ) ( ) ...2 4 ( ) 216

( ) .4

cc c

c

Vha Vh b aI h z z zb a z z ba z z

VhI za z z

(8.9)

Equation (8.9) characterizes the order parameter I close to the point of phase transition, when external field h is weak.

Let us use susceptibility value Ih

as a parameter characterizing

the influence of the external field on the order parameter. It follows from (8.2) that

2 .2 ( ) 12r

I Vh a z z bI

(8.10)

Substituting the equilibrium values of the order parameter into (8.10), in the absence of the field, we obtain a formula for the susceptibility of the population to the influence of the external field in the regions below and above the critical values of regulating factors zr:

at ; at .4 ( ) 2 ( )r r

r r

V Vz z z za z z a z z

(8.11)

As can be seen from equation (8.10), close to zr, susceptibility tends to infinity. At both very low and very high population densities of parasites and predators, the external field has little influence on changes in popula-tion density, and only as the density approaches the critical value zr, susceptibility increases dramatically. However, the susceptibility to the effects of external modifying factors of the population in the stable sparse state, according to (8.11), is half the susceptibility of the population in the outbreak phase.

Hence, when various factors influence population dynamics, the pri-mary effect is a weather-independent density increase, while weather effects will be manifested only after the initial population density in-crease. If the effect of the weather is strong, the equilibrium value of the order parameter is not dependent on the population density, suggesting complete suppression of the outbreak as a phase transition.

To understand the dynamics of population processes at the low and high population densities, it is important to describe density fluctuations close to stable (or at least metastable) population states I1 and I2 and fluctuations between states I1 and I2.


Let us introduce the values of the characteristic size of the habitat oc-cupied by the population, l, and the characteristic amplitude of popula-tion density fluctuations, d. Then, the characteristic time of migration of individuals, d, can be expressed in the following way (Klimontovich, 2015):

2

.dld

After introducing the characteristic size of the habitat and amplitude of fluctuations of the population, analysis of temporal and spatial density fluctuations and spatial structure of the population will differentiate between two types of population density fluctuations: high-frequency small-scale fluctuations and low-frequency large-scale fluctuations. In the case of high-frequency small-scale fluctuations, when 1d and

1kl (where k is the wave number), population density will change little in space and time, and the population will always stay close to one of the unstable states.

Large-scale low-frequency density fluctuations, when 1d and 1kl , characterize the transition from one stable state to another stable

state, and the characteristic time of these fluctuations (the so-called Kramers time) (van Kempen, 1985) corresponds to the average time between two adjacent outbreaks.

Let us assume that the value of the order parameter, I, changes over time t, and at the initial time of examination of the system, value I(t = 0) fluctuates close to value I2. If the amplitude of these fluctuations, (t), is small (i.e. 2( ) (0) ( )t G G I ) and the potential barrier G(0) is rather high, the probability of the population jumping from state I2 to state I1 (i.e. outbreak) is low.

Figure 8.2 shows a curve of pine looper population dynamics in Ger-many (1958–1989) in the phase of low-density state (the long-term annual average density of the larvae in the survey years was 1.62 larva per tree). Figure 8.3 presents the spectrum of this time series.

The characteristic frequency of fluctuations around the long-term an-nual average density of 1.62 pupae per m2 is 0.4 1/year (Fig. 8.3), i.e. the characteristic time of high-frequency population density fluctuations

cf1

2.5 years.

If the effect of regulating factors becomes weaker and the external field h > 0 becomes stronger, I will tend to zero, the depth of the left-hand local maximum will decrease proportionally to the value of the external


field, and at some time tr, the amplitude of fluctuations of the normalized population density (tr) will become comparable with or even greater than the (0) ( ( )rG G I t difference. In this case, the system will jump into the stable (or metastable) state with normalized density I1, and an out-break will occur. The change of the sign of the external field and (or) an increase in the effect of the regulating factors (such as an increase in the impact of parasites and a decrease in the amount of the available food and worsening of its quality) will lead to the backward jump of the popu-lation, outbreak suppression, and return of the system to the stable state.

A typical example of such fluctuations is changes in pine looper popu-lation density in Thuringen, which were analyzed in Chapter 2 (Schwerdfeger, 1968).

Fig. 8.2. Pine looper population dynamics in Germany (1958–1989)

in the low-density phase

Fig. 8.3. The spectral density of pine looper population time series


Fig. 8.4. The density distribution of the pine looper population

during 60-year surveys in Thuringen The histogram of the density distribution of survey data shown in Fig-

ure 8.4 is bimodal: in 15% of the total counts, pine looper population density was more than 3 pupae per m2, but in 70% of the total counts, population density was no more than 0.5 pupa per m2. Between these density values, which can be interpreted as the densities in the stable sparse state and in the outbreak phase, the proportion of the densities between 1 and 3 pupae per m2 (the density close to the critical one) was rather small.

The peak of the density spectrum of pine looper population in Saxony is at f = 0.086 1/year (Fig. 2.5), and this is considerably lower than the frequency of density fluctuations of the pine looper population in the stable sparse state: f = 0.4 1/year. Thus, there are two types of population density fluctuations, which characterize fluctuations close to the lower level of population density and fluctuations associated with the jumps between two states of the population.

8.3 Possible mechanisms of the development of forest insect outbreaks

The model describing population dynamics of forest insects as the first-order phase transition can suggest three mechanisms for the develop-ment of outbreak: a fluctuation mechanism, a field mechanism, and a tunnel mechanism. Below we characterize each of them.

The fluctuation mechanism is related to population density fluctua-tions close to the stable sparse state with low density x1. If the current


population density deviates from stable state x1, the effects of the negative feedbacks (i.e. the effects of regulating factors – parasites, predators, food “cost”) will in some time return the population to the state with densi-ty x1. Thus, the population will be always in the stable sparse state. If, however, the magnitude of fluctuations is sufficiently great and negative feedbacks are weak, several developments are possible, with increasing population density and occurrence of outbreak.

If the left-hand potential well G1 = Gr – G1 (see Chapter 2.4) is not deep, the population can “vault” the potential barrier and appear in the region of the effect of the state with density x2. Then, an outbreak occurs. If the right-hand potential well G2 = G2 – Gr is not deep either, the fluctuating population is highly likely to vault the potential barrier again and return to the state with density x1.

If G1 G2, and these values are comparable with the magnitude of fluctuations, the population will fluctuate with a rather high frequency between x1 and x2, and sustained outbreak will develop (Isaev et al., 1984). This type of outbreak may occur regardless of external modifying factors such as weather.

If the condition G1 G2 is fulfilled but the depth of potential wells is great compared to the magnitude of fluctuations, the population will be transferred from the low-density state to the high-density state and back rather seldom, as considerable fluctuations of population density will occur infrequently. The population will also fluctuate between x1 and x2, but it will stay in each of the states for quite a long time, and fixed out-break will occur (Isaev et al., 1984). A characteristic example of a species with fixed outbreak is the poplar leaf blotch miner Lithocolletis populifoliella Tr. (Tarasova et al., 2004). The long time period of very high population density of the poplar leaf blotch miner was followed by an equally long time period of very low population density of the insect, when there were very few mines of the poplar leaf blotch miner on Populus nigra L. leaves in Krasnoyarsk.

Another type of population dynamics that can be related to population density fluctuations corresponds to the scenario in which condition

2 1G G is fulfilled but potential well G1 is shallow. Then, popula-tion mainly stays in the state with density x2. However, an occasional strong fluctuation can shift the population into the state with very low density x1 and then quickly return it to the state with density x2. When the population is in the state with density x1, it will be extremely difficult to find insects representing it during field surveys, and the species will seem to have vanished from the ecosystem.


The “field” mechanism of the occurrence of outbreaks is related to the effects of external modifying factors, mainly weather. One of the minima of potential function may disappear, and population will be shifted to the state with the unimodal potential function (Fig. 8.1). This is characteristic of weather-dependent outbreaks. In terms of the time series models described in Chapter 7, this type of outbreak (outbreak proper) will be described by models ADL(k, m) at k > 1 and m > 0.

The periodicity of the outbreaks of this type will be determined by the probability of weather changes in the period of low population density.

Finally, the specific tunnel type of population dynamics can be ob-served when population density increases dramatically by the value comparable with the width of the potential barrier in the bimodal poten-tial function. This may be a possible development when the food “costs” drop dramatically (see Chapter 6), and the mass and fecundity of the females in the population increase. Unfortunately, we do not know of any “tunnel” outbreaks among defoliating insects, but “tunnel” outbreaks of bark beetles, which develop on wind-fallen trees or on the trees weakened by outbreaks of defoliating insects, have been observed very frequently.

8.4 Colonization of the tree stands by forest insects as a second-order phase transition

Parameters important for understanding tree – insect interactions, such as the amount and quality of the available food (see Chapter 6) and the level of colonization of the trees in the stand by insects, are not explicitly included in the model of the outbreak as a first-order phase transition. This is certainly a serious limitation of the model of the outbreak as a first-order phase transition, and the optimization approach to describ-ing population dynamics proposed here should be further improved. Thus, to describe colonization of trees by insects during an outbreak as an ecological analog of phase transitions in physical systems, the same steps should be taken as those proposed above for the description of changes in insect population density as a first-order phase transition: introduce the order parameter, q, identify the external factor whose effect causes phase transition, and introduce an equation relating these varia-bles. However, during colonization of the trees by insects, no discontinu-ous change occurs in the percentage of the colonized trees in the stand, and, thus, we propose a model of a second-order phase transition to describe tree – insect interactions (Landau, Lifshits, 1964).


Forest insect populations are characterized by both the density of x individuals (absolute colonization) and the distribution of individuals over the area, specifically, the distribution of individuals on sample units: trees, model branches, etc. In entomological monitoring, the simplest and most frequently used indicator of spatial distribution of insects in sample units is the relative colonization A – the fraction of sample units with individuals of the insect species studied (Monitoring…, 1965). To de-scribe colonization of the tree stand by insects, let us define the order parameter, q, as the value related to the relative insect colonization, A, of sample units in the tree stand:

1 ,n kq An

(8.12)

where kAn

is the relative insect colonization of sample units (trees,

forest plots, etc.), n is the number of sample units, and k is the number of sample units colonized by insects.

It follows from (8.12) that 0 1q . The value q = 0 of the order pa-rameter corresponds to the outbreak phase, when insects occur in all sample units (trees, plots, etc.). Characteristic values of the order parame-ter for the sparse stable population phase are 0 1q . In this case, insects populate some of the sample units.

As the external parameter, corresponding to the temperature in physi-cal systems, we chose population density x. In accordance with the phe-nomenological theory of forest insect population dynamics, an outbreak (i.e., phase transition) occurs when the population density exceeds a certain critical value of x. When the sparse stable population phase is replaced by the outbreak phase, the value of the order parameter is re-duced to zero, i.e. all trees in the outbreak site are colonized by the insects.

Dependence ( , )G f x q may be sufficiently accurately presented as a power series of the order parameter (Landau, 1937; Landau, Lifshits, 1964). For describing the second-order phase transitions, power series will have only even degree terms: 2 4

0 ,G G Aq bq (8.13), where b = const, and А is the linear function of external variable x – population density.

In this case, the order parameter (8.12) will describe the state of the tree stand. The external factor will be population density, x. Coefficient A in (9.13) will be written as linearly dependent on population density: ( ).rA a x x (8.14)


It follows from (8.14) that if population density is higher than the criti-cal value x = xr, A > 0.

The value of the order parameter for the outbreak phase and the sparse

density phase will be found from equation 0Gq

, characterizing the

minimum value of function G:

32 ( ) 4 0.rG a x x q bqq

(8.15)

Equation (8.15) has two solutions:

q = 0 and 2 ( ) .2ra x xqb

(8.16)

Solution 0q describes the outbreak phase. The second solution is true for the population density below the critical value, xr. This solution characterizes a sparse population.

At 0,q the value of the ecological risk, 1 0 ,G G while at

( ) ,2ra x xqb

2

2 0( ) .

4ra x xG Gb

Hence, 1 2 ,G G and the ecological

risk for populations of non-outbreak insect species that have sparse stable densities is lower than the ecological risk for outbreak species. The num-ber of non-outbreak species should be significantly greater than the number of outbreak ones. This conclusion is in good agreement with the data of forest insect surveys. In boreal forests of Eastern Siberia only 74 out of 315 economically important species of forest insects were capable of reaching outbreak levels (23.5%) (Isaev et al., 2001).

Knowing insect population densities provided by insect surveys and relative estimates of insect colonization of tree stands in different phases of insect outbreaks, we can verify the model proposed here by determin-ing the relationship between x and q2 in the{x, q2} plane. Based on mod-el (8.2), at 0q , points characterizing population density must be locat-ed on the abscissa, to the right of point xr. For x < xr, in accordance with (8.7), there must be a negative linear relationship between q2 and

density x. Point x = a/2b of the intersection of line 2 ( )2ra x xqb

and the

abscissa gives the value of xr. Figure 8.5 shows the data of surveys that demonstrate the relationship

between q2 and x for populations of the pine looper Bupalus piniarius L. in Altai, Minusinsk, and North Kazakhstan pine forests between 1962 and 1988.


Fig. 8.5. Relationship between parameter q2 and the density of pine looper populations in different outbreak phases in Altai, Minusinsk, and North Kazakhstan pine forests between 1962 and 1988 (Palnikova et al., 2002).

(1 – stable state; 2 – outbreak) Pine looper surveys were performed in spring, during the pupal stage,

on 1 1 m plots (Monitoring…, 1965). The survey data are clearly in good agreement with the theoretical model (Fig. 8.5). For population densities below the critical value, there is a significant linear relationship between population density and the squared order parameter. For popu-lation densities higher than the critical value, the order parameter is close to zero. The critical density xr for the pine looper is 5.65 pupae per plot, and this is in good agreement with the data reported in the literature – 6 pupae per plot (Monitoring…., 1965).

Figure 8.6 shows the relationship between parameter q2 and population densities of the Siberian silk moth Dendrolimus sibiricus Tschetv. In different phases of the outbreak cycle. Calculations were done using Y. P. Kondakov’s data (Kondakov, 1974).

Here we also have a good agreement between survey data and theoreti-cal equation (8.16). The critical value xr calculated from equation (8.16) is equal to 657 larvae per tree, and the number given in the handbook on monitoring forest pests for the critical density of the Siberian silk moth population is 650 larvae per tree (Monitoring…, 1965).

Compare model (8.16) with the Poisson model describing the relation between the relative and absolute estimates of insect colonization and based on the assumptions that all sample units are similar in their prop-erties, individual insects choose their food randomly, and the interaction between individuals is weak if any. In this case, the distribution of insects among sample units can be described by the Poisson model (Jeffers,


1978). According to this model, the proportion q = 1-A of sample units without insects (i.e. the analog of the order parameter in model (8.12)) is expressed by the following formula:

q = 1 – A = exp(–x). (8.17)

Fig. 8.6. The relationship between parameter q2 and the density of the Siberian

pine moth Dendrolimus sibiricus Tschetv. populations in different phases of the outbreak cycle. (1 – stable state; 2 – outbreak)

It follows from (8.17) that at x → 0, i.e. at low population density, q → 1,

that is, there are very few sample units with insects. At x → ∞, q → 0, i.e. all sample units are colonized by the insects. It also follows from (8.17) that the type of relationship between relative and absolute colonization must be similar for all insect species, as in (8.17) there are no species-specific constants.

According to (8.17), as the population density grows, relative coloniza-tion of tree stands increases rapidly. At x = 3 individuals/sample unit, q 0.05 and А 0.95. Obviously, if the distribution of insects among sample units is random, at a population density above 3 insects per sample unit, there will be very few sample units without insects.

Thus, analysis of survey data on the relationship between the relative and absolute colonization by insect populations can provide a basis for choosing between model (8.16) and model (8.17). If the results of the analysis suggest that the equation of the relationship between the relative and absolute colonization is significantly different from the Poisson model (8.17), this will be indicative of interactions between insects and their host plants as well as between phytophagous insects consuming the phytomass. These interactions may be expressed as cooperative coloniza-


tion of trees. Parameters of these interactions can be estimated from the values of constants B and xr of model (8.17).

Statistical analysis showed that the survey data and calculations based on model (8.17) disagree. Both survey data and results of calculations based on model (8.16) suggest that for the Siberian silk moth, population density of 3 insects per tree corresponds to relative colonization of about 4% of the trees and for the pine looper population – about 45% of the trees. Calculations based on model (8.17) suggest that insects must be present on 95% of the trees.

Figure 8.7 shows survey data and model values of the squared order parameter calculated from models (8.16) and (8.17) for estimating the distribution of Homoptera insects (Lepyronta coleopterata L., Philaenus spumarius L., Cinara nuda Mordv., C.hyperophila Koch.) on trees of middle-taiga plain forests of the Krasnoyarsk Territory (Bulanova et al., 2008).

Fig. 8.7. Order parameters determined from the survey data and calculated

from models (8.16) and (8.17) for insects in the forests of the Yemelyanovskii District of the Krasnoyarsk Territory. 1 – calculations from model (8.17),

2 – calculations from model (8.16), 3 – survey data Results of calculations using the model of phase transitions (8.16) are

in good agreement with the survey data, whereas results of the Poisson model (8.17) and field measurements considerably disagree. Table 8.1 compares values of relative colonization of the trees by insects of various taxonomic groups (Tarasova et al., 2008) determined from the survey data and calculated from models (8.16) and (8.17), based on the highest values, x(max), of absolute colonization by insects observed during surveys.


Table 8.1. Relative colonization of trees estimated using survey data and models (8.16) and (8.17)

Order of insects x(max) Relative colonization А

Data model (8.16) model (8.17) Heteroptera 0.24 0.14 0.139 0.213* Coleoptera 0.25 0.21 0.219 0.221 Lеpidoptera 0.46 0.37 0.371 0.369 Hymenoptera 0.1 0.10 0.101 0.095 Homoptera 8.7 0.08 0.084 1.000*

* the value is significantly (р = 0.95) different from the survey data. The survey data and results of calculations using model (8.16) are very

similar. On the other hand, the calculation of relative colonization based on the Poisson model (8.17) is in some cases significantly different from survey data, and the higher the values of absolute colonization the greater the differences between the calculations of relative colonization based on model (8.17) and the survey data. At low population densities, calcula-tions from both model (9.16) and model (8.17) are in good agreement with the survey data. This good agreement may be accounted for by the fact that at low population densities (x < 0.5), the exponent in the right-hand side of (8.16) can be expanded into Taylor series to linear terms. Then, from (8.16) we obtain

2 1 2 .q x (8.18)

This may suggest that at low population densities, both models pre-dict a linear relationship between relative colonization and population density. As the population density grows, expansion (8.18) becomes incorrect, and the Poisson model gives too high calculation values of relative colonization, while model (8.16) remains correct even at high densities.

It is evident from the data presented here (Fig. 8.5–8.7, Table 8.1) that parameters characterizing the relationship between the relative and absolute colonization by insects (model (8.16)) are species-specific and vary both between different groups of insects and within one systematic group in different regions. Model (8.17) is less sophisticated. Our analysis showed that calculations based on interpretation of the insect outbreak as a second-order phase transition suggest a good agreement between the model of phase transitions and field data.


8.5 Risks of elimination of the population from the community

The population whose density is close to the critical level хс may be eliminated from the community. To assess the risks of elimination, one needs to know the values of хс, which can be determined by using the model of second-order phase transitions.

The model variable describing the state of the population is z, which shows the difference between population density in the stable state, х1, and the current population density, x:

1

1 1

0, ,, ,

x xz

x x x x (8.19)

where 1x is population density in the stable sparse state and x is popula-tion density determined by surveys.

2 40 ( ) .cG G a z z q bq (8.20)

The minimum of function G(q) is determined from the standard equa-tion:

2

20, 0.G Gq q

(8.21)

After simple calculations, we obtain values of the order parameter, q1 and q2, with which the highest values of function G(q) are achieved:

0, , 1,

( ), , 2.2

c

jc c

z z jq a z z z z j

b (8.22)

It follows from (8.22) that at z < zc, there is a linear relationship be-tween squared order parameter q2 and control variable z:

20 1 ,q W W z (8.23)

where 0 ,2 caW zb

1 .2aWb

At q = 0 0

1c

WzW

and, hence,

1 .c cx x z (8.24)


Thus, if model (8.20) is in good agreement with the field data, regres-sion equations relating the values of density х and order parameter q in the survey data showing population density x below density x1 can be used to evaluate parameters W0 and W1 of equation (8.23) and then obtain an evaluation of the lower critical density xс from (8.24).

Equation (8.20) includes variable z, which is determined as the differ-ence between population density x1 in the stable state and the current population density. To calculate critical density xс from equations (8.20)–(8.24), one needs to know the value of x1. However, even if the population is continuously monitored in its natural habitats and there are data on the long-term population dynamics, it is difficult to determine x1 from these data directly, as the time series of population dynamics, even when the population is in the stable sparse state, show considerable fluctuations of population density (Fig. 8.8).

Fig. 8.8. Time series of the pine-tree lappet population dynamics

in the Krasnoturansk pine forest (habitat “Bald Hill”) Studies of population dynamics of forest defoliating insects at low-

density levels were conducted in the Krasnoturansk pine forest (south of the Krasnoyarsk Territory, 54º16.315’N, 91º37.757’E). The species ob-served in this study were the pine-tree lappet Dendrolimus pini L. and the tawny-barred angle Semiothisa liturata Сl. Annual surveys of populations of these species have been performed on five constant sample plots in different habitats in the Krasnoturansk pine forest since 1979 (Tarasova, 1982; Palnikova et al., 2002).


To evaluate the population density x1 in the stable sparse state, we use two values: long-term annual average density of the population in

a particular habitat, 1

1[ ] ( )n

iE x x i

n, and the median Vе[x] of the time

series of the density dynamics of this population. Table 8.2 gives values of E[x] and Me[x] based on the survey data on

long-term population dynamics of the pine-tree lappet Dendrolimus pini L. in five habitats in the Krasnoturansk pine forest.

Table 8.2. Evaluations of density х1 and coefficients of the model equation (8.23)

of pine-tree lappet population dynamics

Habitat Parameter Estima-ted х1

Coefficients of equation (9.23)W0 W R2 zc xc

Terrace Mе[x] 0.05 0.0025 0.0697 0.99 0.036 0.014 Е[x] 0.42 0.066 0.1598 0.83 0.413 0.007

Dune Mе[x] 0.08 0.0063 0.1243 0.99 0.051 0.029 Е[x] 0.36 0.0677 0.2136 0.88 0.317 0.043

Narrow Plane

Mе[x] 0.05 0.0025 0.07 1.00 0.036 0.014 Е[x] 0.16 0.0192 0.1534 0.97 0.125 0.035

Lake Mе[x] 0.05 0.0025 0.0706 0.99 0.035 0.015 Е[x] 0.08 0.0066 0.1207 0.99 0.055 0.025

Top Hill Mе[x] 0.05 0.0025 0.09 1.00 0.028 0.022 Е[x] 0.18 0.0262 0.1778 0.97 0.147 0.033

For all habitats in Table 8.2, the medians of the population dynamics

time series have very comparable values (only the value of Mе(x) in habitat “Dune” is somewhat higher), but the long-term annual average densities of the pine-tree lappet populations differ between habitats, and the average density in the “Narrow Plane” habitat is five times higher than the average density in the “Lake” habitat.

Based on the evaluations of parameter х1 for different habitats listed in Table 8.2, we calculated coefficients W0 and W1 of regression equations, describing, according to (8.23), relationship between deviations z from the stable state х1 and the squared order parameter q2, and then, calculat-

ed values 0

1c

WzW

and 1 .c cx x z

Figure 8.9 shows a typical relationship between the squared order pa-rameter q2 and z, based on the data of surveys of D.pini populations in habitat “Terrace”.


Fig. 8.9. A relationship between the squared order parameter q2 and deviations z

from the stable state х1 based on the data of surveys of D.pini populations in habitat “Terrace” (1: z < zc; 2: z zc)

Fig. 8.10. Relationship between the time series of population dynamics and

the values of the lower critical density of D.pini population in habitat “Top Hill” (1 – annual survey data; 2 – the lower critical density based

on the median evaluation Me[x]) The survey data and model equation (8.22) are in good agreement

(Fig. 8.10). Results of calculation of critical densities for the pine-tree lappet populations in different habitats are listed in Table 8.2. No matter which parameter, median Me[x] or average density E[x], is used, the


values of the lower critical density хс for different habitats are quite com-parable: хс = 2 larvae per 100 trees.

Figure 8.10 shows the relationship between the current densities of the D.pini populations and the critical densities calculated by using Me[x] for one of the habitats (“Top Hill”).

Figure 8.11 shows the relationship between the current densities of the tawny-barred angle population and the critical density calculated by using Me[x] for habitat “Lake”.

Calculations of the lower critical density of the tawny-barred angle population are given in Table 8.3.

Fig. 8.11. Density dynamics and the lower critical value for the tawny-barred

angle in habitat “Lake”

Table 8.3. Parameters of the equations of the relationship between the order parameter and the tawny-barred angle population density

(evaluation of х1 was based on the values of the medians Mе[х] of population dynamics time series in the habitats)

Habitat Estimated х1 Parameters

W0 W1 R2 zc xc

Terrace 0.05 0.0025 0.069 0.995 0.036 0.014Dune 0.05 0.0024 0.070 0.984 0.034 0.016Narrow Plane 0.049 0.0021 0.061 0.995 0.034 0.014

Lake 0.05 0.0266 0.760 0.970 0.035 0.015Bald Hill 0.05 0.0025 0.090 0.990 0.028 0.022


For all habitats in Table 8.3, the critical density values хс of tawny-barred angle populations are very similar to each other. The coefficient of deter-mination R2 for all time series of the data is very close to 1, suggesting good agreement between the field data and the model proposed in this study.

***

The use of methods of condensed matter physics in ecology provides new opportunities for more than just the theory of forest insect population dynamics. The description of critical events in ecosystems, such as insect outbreaks, fires, wind-thrown trees (Isaev et al., 2010), and forest succes-sions (Isaev et al., 2012; Isaev et al., 2014; Soukhovolsky et al., 2014) integrates different ecological processes related, first, to qualitative changes in the systems under study and, second, to the influence of the external modifying factors on the system (similar to the influence of the temperature on physical objects – magnetic materials, ferroelectric mate-rials, liquid helium, and superconductors).

Of course, the model of phase transitions is universal because it does not deal with the temporal dynamics of the model processes but only describes the stable and metastable states of the system. Thus, the model of phase transitions is not suitable for describing temporal dynamics of population density. On the other hand, ADL models, which describe temporal dynamics of ecological processes, do not contain critical pa-rameters characterizing qualitative changes in the state of populations. By combining the dynamic ADL models and the static models of phase transitions, one can describe various aspects of ecological processes occurring in insect populations.

207

9 Forecasting Population Dynamics and Assessing the Risk of Damage to Tree Stands Caused by Outbreaks of Forest Defoliating Insects

9.1 Methods of forecasting forest insect population dynamics

Attacks of forest defoliating insects that cause tree damage and death are a natural ecological process leading to acceleration of material cycling in forest ecosystems, tree species successions, and generation of new tree stands, which get involved in biosphere processes. On the other hand, the damage and death of tree stands that are economically and recreationally valuable to people is most undesirable.

Therefore, the forest entomological service monitors changes in popu-lations of dangerous defoliating insects, determines the pending danger from different defoliating insect species to forests, conducts surveys of populations of these species, makes forecasts of pest population dynam-ics, and assesses the risk of damage to the forest in the case of an increase in the density of defoliating insect populations (Isaev, Kondakov, 1998).




The risk R(t0, t1) of damage to the tree stand by defoliating insects by time t1, calculated at time t0 (t0 < t1), will be determined as follows:

0 1 1 1( , ) ( ( )) ( ( )),R t t p x t LS x t (9.1),

where 1( ( ))p x t is the probability that at time t1, the insect population will have density x(t1), higher than the critical value xr; 1( ( ))LS x t are possible losses caused by insects, which are determined through such parameters as the number of trees killed by insects or the mass of the needles (leaves) consumed by the insects by time t1, when the insect population reaches density x(t1).

It follows from (9.1) that to make predictive estimate of the risk of tree stand damage, we, first, need to obtain the predictive value of population density, x(t1), at time t1 and then, make a forecast of losses in the tree stand. Evaluation of losses must include both economic losses – the death of commercial tree stands – and ecosystem service losses, related to carbon sequestration, release of oxygen, maintenance of a definite soil moisture content, etc.

The probability of an outbreak and evaluation of the outbreak-caused losses are certainly interrelated but not strictly correlated and, thus, need to be predicted separately.

Pest control measures become unnecessary when the population den-sity declines and the insects no longer migrate to other regions owing to the effects of weather factors, parasites and predators, and impairment of food quality.

Classification of the types of forecast can be based on the value of the Δt01 = t1 – t0 difference. If Δt01 = 1 year, this is a short-term one- year forecast. If Δt01 = 2–5 years, this is a medium-term forecast. If Δt01 10 years, this is a long-term forecast. Short- and medium-term forecasts are necessary for forest protection decision making. The long-term forecast is an instrument for deciding whether a particular defoliat-ing insect species needs to be included in the system of forest entomolog-ical monitoring, even if there are no records of this species at the time when the decision is made to begin monitoring the outbreaks.

In the sections below, we will discuss in detail the objectives of fore-casting outbreaks and risks of damage to tree stands caused by defoliating insects.

As we mentioned in Chapter 7, ADL-models of population dynamics can incorporate all available survey data. In these models, the calculated value of population density at time t is determined by both past and future population densities. Forecasts are only based on the past data.

Forecasting Population Dynamics and Assessing the Risk of Damage 209

Thus, to be able to use ADL-models for forecasting, we need to modify the methods.

Forecast methods using AR(k) models are the closest to the model ap-proach. In this case, it is assumed that modifying factors have no consid-erable influence on population density, and the current population density is determined by its density in k previous years. If the forecaster has a sufficiently long time series of survey data on population dynamics for n years (we will call this time series a training set), after transfor-mation of the survey data, the shape of the partial autocorrelation func-tion of the training set can be used to determine the order of autoregression k and calculate coefficients {a0, a1, …ak} of the autoregres-sive equation, as we described in Chapter 7.

After calculating coefficient of the AR-model for the training set con-sisting of n values, one can calculate the predictive population density in the (n + 1)th year:

00

ˆ( 1) ( ).j n k

jj

L n a a L n j (9.2)

Then, using the ˆ( 1)L n value and the values of population densities in the previous k years, one can calculate the predictive density value in the (n + 2)th year, using the values of coefficients from (9.2):

0 12

ˆ ˆ( 2) ( 1) ( 1 ).j k

jj

L n a a L n a L n j (9.3)

By continuing these calculations recurrently, one can obtain sequential predictive values for r years.

One can always perform calculations based on (9.2) and (9.3), but it is not clear whether this forecast will be effective, how the length of the training set will influence the reliability of the forecast, and what forecast-ing error we will get depending on the value of r.

To evaluate the effectiveness of the forecast, let us choose the LTI se-ries of survey data of length (n + r), in which the first n members will be used as a training set and the remaining r members will be used as a test of the forecast of length r, performed by applying a recursive procedure to (9.2) and (9.3). We will introduce two indicators of the forecast effec-tiveness: the function of the cumulative squared deviations from the known density value in the test set and the function of the cumulative phase shift between the predictive and testing time series. An effective forecast of length r will be characterized by a sufficiently small value of deviations of predictive series values from the testing series values and by


the absence of a phase shift between the predictive and the testing time series.

To better understand the proposed approach, let us use it to examine gypsy moth population dynamics in the South Urals, which was also analyzed in Chapter 7. We took training sets of different lengths (12, 17, 23, and 25 years), and for each of the training sets, we calculated coeffi-cients of the AR(2)-model and errors of coefficients (Table 9.1).

Table 9.1. Coefficients of the AR(2) model for different lengths

of the training set

Coefficients of the AR model

Length of the training set, n12 17 23 25

a0 1.880 0.902 0.750 1.025a2 –0.627 –0.733 –0.757 –0.785a1 0.929 1.304 1.467 1.433s0 0.805 0.393 0.403 0.429s2 0.244 0.205 0.176 0.200s1 0.290 0.210 0.181 0.202R2 0.549 0.751 0.783 0.712

The accuracy of describing the LTI series with the AR(2)-model, which

is characterized by the value of the coefficient of determination R2, is approximately the same when the length of the training set is equal to or greater than 17. When, however, the length of the training set is 12, R2 = 0.55. That is, the AR(2)-model can describe only about 25% of the total variance of survey data.

Figure 9.1 shows the comparison of the survey data collected between 1959 and 1989 with the LTI series estimated for the training set of the length n = 17 (i.e., between 1959 and 1975) and the LTI series predicted for 1976–1989.

As can be seen from the data in Figure 9.1, the use of the AR(2)-model gives an acceptable forecast of population density only for 1 year. How-ever, the cross-correlation function of the predictive and testing time series for the years between 1976 and 1989 does not show any phase shift for the entire period (Fig. 9.2).

Thus, the AR(2)-model shows low accuracy and short length of the forecast of the LTI series amplitude, but the predictive time series effec-tively describes the phase of cyclic oscillations of gypsy moth population. The low accuracy of this forecast is not surprising: in Chapter 7, we showed that the current density of the gypsy moth population is deter-


mined both by the population density and by the weather conditions of the two previous years. The AR(2)-model, however, does not take into account weather conditions. It is interesting, though, that even with the low amplitude accuracy of the forecast, the forecast of the outbreak phase is very accurate.

Fig. 9.1. Predictive estimation of gypsy moth population dynamics

in the South Urals. 1 – survey data; 2 – calculation based on the AR model for the 17 year-long training set, between 1959 and 1975; 3 – prediction

for the testing sample for the period between 1976 and 1989

Fig. 9.2. Cross-correlation function between the predictive

and the testing values of the LTI series The accuracy of the forecast can be enhanced by taking into account

both regulating factors (population density of the previous years) and modifying factors (weather). Let us study the possibility of predicting the density dynamics of the population whose density in year (n + 1) changes


depending both on its densities in k previous years and on weather conditions of m previous years. As stated in Chapter 7, the density dy-namics of such a population is modeled by using an ADL(k, m)-model. In this case, for calculating population density in year (n + 1), the fore-caster needs to have not only the data on the population density for the last k years (we assume that they are available), but also the data on weather parameters in year (n + 1). The forecast, however, is made in year n, when the weather parameters (such as HTC) of the following year are not known yet. Moreover, to make a weather forecast for the follow-ing year is an equally (or even more) complicated task than to make a short-term forecast of population dynamics.

Thus, for short-term forecasting based on the ADL-model, we propose the following procedure:

1. Let the forecaster have the time series {L(j)} of the data on popula-tion density for n last years. The order of autoregression, k, is determined by the number of significantly non-zero coefficients of the partial auto-correlation function of the {L(j)} series. Then, the data on population density L for n years and weather parameters of those years are used to determine the order of the ADL(k, m) model and calculate coefficients of this model:

01 0

( 1) ( 1 ) ( )k m

j lj l

L n L a L n j bW n l . (9.4)

It is clear from (9.4) that in year n, in order to forecast population den-sity in year (n + 1), one needs to know weather parameters W(n + 1), but they are obviously not known in year n.

2. Generate a set of possible weather scenarios {W(j, n + 1)} for the fol-lowing season. This task can be fulfilled in various ways: by calculating conditional probabilities of weather parameters in the following year based on the weather data of the previous years; by determining the possible range of weather parameters and studying a certain set of weath-er parameters covering the entire possible range of W values; by con-structing a Monte Carlo generator of the future weather.

3. For each of the scenarios generated, W(j, n + 1), calculate popula-tion density L(j, n + 1) in the following season, by using equation (9.4).

4. Calculate the relative frequency function of L(j, n + 1) values and determine the statistical parameters of the forecast: the type of the rela-tive frequency function of weather parameters, the median or the mean value (for the unimodal relative frequency function of weather parame-ters), standard deviations or quantile estimations of weather parameters.


Based on these estimations, a decision is made on possible protective measures (the decision-making problem will be discussed below).

To estimate the effectiveness of the short-term forecast, let us examine the retrospective procedure, in which a known time series of population density of length N (N > n) is used to isolate the first n members of the series as a training set and then, for predicting the (n + 1)th member, the procedure described above is performed. The squared difference between the predictive and the observed densities is used to characterize the effectiveness of the n-base short-term forecast. Then, the (n + 1) member is incorporated into the training set, and the short-term generation procedure is repeated for year (n + 2). The averaged estimation of the forecast quality is done from a set of forecasts for (N – n) members of the retrospective time series.

To better understand the proposed procedure, let us use it to make a short-term forecast of gypsy moth population density in the South Urals. Figure 9.3 shows the LTI series of the survey data and the ADL(2, 2) model based on these data.

Fig. 9.3. Survey data on gypsy moth population density for 20 years (1)

and calculations based on the ADL(2, 2) model (2) The data on the HTC of the first 20 survey years were used to esti-

mate the minimum, median, and maximum HTC values: HTC(min) = 1.15, HTC(med) = 4.57, and HTC(max) = 6.21. For each of the three HTC values, we made one-year-ahead forecasts of the density. Fig-ure 10.4 shows these predictive densities for 14 years, based on three HTC values.

This forecast can be used to estimate trends in population density changes for at least one year. To simplify the predictive model, let us calculate average predictive values of densities, based on the three fore-cast time series (Fig. 9.5).


Fig. 9.4. Survey data (1), one-year-ahead forecasts based

on the minimum HTC (2), the median HTC (3), and the maximum HTC (4)

Fig. 9.5. Survey data (1) and an averaged forecast for one year (2)

Figure 9.5 shows that the simplest version of the forecast, which is

based on the HTC value averaged by the time of the training set, gives a sufficiently correct short-term forecast of population density. For some of the years, when the population density was increasing, the forecast values are practically equal to the survey data.

To construct a forecast of population dynamics employing the model proposed above, we used a standard procedure: we chose a training set of some length n from the field data series, and these data were used to calculate coefficients of the AR(2)-model; then, by using the model verified in this way, we calculated the values of population density for the period between year (n + 1) and some year r. The quality of the forecast was estimated by several parameters: the agreement between the model and field calculations of population density; the value of the phase shift between the forecast and field time series; the minimal length of the training set; and the maximal length of r at which the forecast was in good agreement with the field data. The greater the value of r, the better


the quality of the long-term forecast, even if we take into account that the future weather is unknown and it is impossible to use the ADL(2, 1)-model presented above or any other ADL-model.

Figure 9.6 shows results of calculations based on the predictive mod-el (9.2) for the European oak leaf-roller population. The training set of the time series of the species population dynamics was 11 years, between 1964 and 1975, and for the forecast, we used the time series segment between 1976 and 1984.

Fig. 9.6. Survey data, the training set, and the predictive set of the transformed

population density. 1 – the transformed time series of survey data; 2 – the model time series based on the data from the training set (years 1–11) of the trans-

formed time series of the European oak leaf-roller population density; 2–3 the test model for the data in years (12–23), based on the model for the training set

For the first three forecast years, model (9.2) gives a very good agree-

ment with the survey data; then, there is a one-year time shift, but the forecast data are in rather good agreement with the field data (Fig. 9.6). Figure 9.7 shows the data on the agreement between the forecast and field values of population density.

Good agreement between the field and the forecast data was found at rather high population densities. At low densities, the forecast somewhat overestimated the density values. However, for practical purposes, such inaccuracies are irrelevant if there is no phase shift between the field and forecast time series and if the date of the population density peak can be predicted.

A more detailed analysis could certainly provide an estimate of the op-timal length of the training set, but our calculations suggest that future changes in the density of the model insect population can be predicted quite reliably.


Fig. 9.7. Correspondence between the field data and the model calculations

for the forecast model of the population density of the European oak leaf-roller Thus, ADL-models are not suitable for making short- and medium-

term forecasts, in which the data on known population densities for n years should be used to evaluate future population densities for the following r years (r > 1): as the length of the forecast is increased (i.e. as r is increased), the values of coefficients of the ADL-model become less accurate not only because the weather for several future seasons is not known but also because the AR terms in the model (9.2) will characterize not the survey data but their model estimates, whose accuracy may be low.

Therefore, as the forecast based on the ADL-model is unreliable, one may try a forecast based on the AR-model. If this forecasting method is used, no knowledge of the future weather is needed, but the length of the forecast time series is limited.

When insect-extermination measures were employed in the previous season, it is a special forecasting task to suggest continuing pest control or stopping extermination because the population has left the state with high density х2 and will further decrease its density without any addition-al control measures. In this case, population dynamics cannot be predict-ed by using AR models, which do not take into account additional effects on the population. A useful parameter for evaluating the state of the population may be normalized infestation of insects by their natural enemies, z (see Chapter 5). If z is low, extermination measures should be continued. If z is high, population density will be decreased under the impact of parasites, and no other control measures will be necessary.

Food quality may be used as another indicator of the state of the popu-lation. The cost of food can be determined in a short-term experiment


(4–5 days) on rearing the larvae, following the procedure described in Chapter 6. If the cost of the food estimated in the experiment is high, one can expect pest population density to decrease without any addition-al impacts.

9.2 Long-term forecast of population dynamics of defoliating insects

The purpose of long-term forecasts is not to predict when outbreaks will begin in the decades to come. Practically speaking, nobody needs a fore-cast of the outbreak that will occur in 30 or 50 years. Theoretically speak-ing, such long-term forecasts have dubious reliability. In this study, we define the ‘long-term forecast’ as a procedure intended for estimating the general properties of population dynamics of an insect species, such as outbreak risk, periodicity, and intensity.

Long-term forecasts may also be important for studying the insect spe-cies that currently do not reach outbreak densities and, thus, their popu-lation dynamics does not need to be predicted in short-term forecasts.

As an instrument of long-term forecast, one can use description of population dynamics with potential functions (see Chapters 2 and 8). As noted in Chapter 2, potential function G(x) for the bistable system with two stable (or metastable) values of population density, х1 and х2, is characterized by the presence of two local minima of G(x) values (poten-tial wells) (Fig. 9.8).

Fig. 9.8. The general shape of the potential function,

G(x) for the bistable system


A theoretical model of forest insect population dynamics with two sta-ble (or at least metastable) states was described in Chapter 8 as a first-order phase transition. This model suggests that when G(x1) G(x2), the population will be mainly in state with density x1. The probability of jumping to state with density x2 will depend on the effects of external modifying factors and fluctuations in population density under the impacts of regulating factors. The probability that the population will be ejected beyond the critical value, xr, will depend on the height of the left-hand potential barrier 1 1( ) ( )rG G x G x and the degree of susceptibil-

ity of the population 1dGdx

in state with density x1 to changes in

population density. The degree of susceptibility can be approximately

estimated as 11

1r

Gx x

. The higher 1, the lower the probability that

under the impacts of modifying and regulating factors, the population will be able to overcome the potential barrier. If under the impacts of external “fields” or fluctuations, population density exceeds xr, the popu-lation will tend to approach a new stable (or metastable) state, x2. An outbreak will occur if the value of x2 is much higher than the critical density for this population, which can be estimated, for example, from the model of second-order phase transitions (see Chapter 8). If the value

2(2, ) rx r x x is low, the population will merely increase its size. The probability of returning from state x2 to state x1 will be determined by the height of the right-hand potential barrier 2 2( ) ( )rG G x G x and the degree of susceptibility of the population in state x2 to external impacts and the rate of its density fluctuations.

Thus, based on the shape of potential function for a particular popula-tion of forest defoliating insects, one can estimate the probability of the population leaving the low-density state, the risk of outbreak develop-ment, and the chance that the population will return to the stable low-density state. However, for some of the defoliating insect species, it is difficult to quantify the heights of potential barriers and degrees of sus-ceptibility at which first-order phase transitions – population buildups or outbreaks – occur. A more reliable approach is to base the long-term forecasts on ordinal scales of the heights of potential barriers and degrees of susceptibility. Table 9.2 gives a classification of the risks of occurrence of defoliating insect outbreaks as dependent on the heights of potential barriers and degrees of susceptibility determined in ordinal scales.

Figure 9.9 shows potential function G(x) for the pine looper popula-tion in Thuringen.


Table 9.2. Classification of the risks of occurrence of defoliating insect outbreaks as dependent on parameters of potential functions

potential barrier

suscep-tibility

maximum density

the risk of leaving the stability region

the risk of out-break development

High low High minimal very low

High high High existent Outbreaks are rare but possible

High low Low minimal Almost noneHigh high Low existent minimalLow high High high high

Low low High existent Outbreaks are possible but rare

Low high Low high Only density buildups are

possible

Low low Low low Only infrequent density buildups

are possible

Fig. 9.9. Potential function G(x) for the pine looper population in Thuringen From the graph in Figure 9.9 we can see that the population has two

stable states: the density at the low level, characterized by the broad minimum with densities between 1min exp( 4) 0.02x pupa per m2 and

1max exp( 1) 0.37x pupa per m2 and the density at the high level, 2 exp(2) 7.4x pupae/m2. 1 2( ) 6.7 7.5 ( ) 12G x G x , suggesting

that the probability of the population being in the state with density x1 is higher than the probability of the population being in the state with


density x2. The left-hand potential barrier of relative height (1, ) 13G r is too high for population densities of the pine looper to increase fre-quently.

Figure 9.10 shows the potential function of the pine-tree lappet popu-lation in the Krasnoturansk pine forest (habitat “Dune”).

Fig. 9.10. Potential function of the pine-tree lappet population

in the Krasnoturansk pine forest (habitat “Dune”) The shape of the potential function of the pine-tree lappet population

differs considerably from the shape of the potential function for the pine looper. The deep minimum G(х1) = 1.7 of the potential function indicates that the pine-tree lappet population mainly stays in the state with the LTI density of –0.2. The high potential barrier (1, ) 30G r indicates that the population rarely moves to the upper metastable state with L = 0.6.

Figures 9.11, 9.12, and 9.13 show potential functions for populations of the larch bud moth in Switzerland, the gypsy moth in the South Urals, and the pine looper in South Siberia.

The parameters of the potential functions of the defoliating insect populations are given in Table 9.3.

The absolute values of survey data cannot be compared, as population densities were measured in different units (Table 9.3). Nevertheless, the values of x1 (population density in the stable sparse state) and xr (critical density) are quite similar for all the species. It would be certainly more correct to compare relative parameters such as the heights of potential barriers G and degrees of susceptibility . As mentioned above, the height of potential barrier G1 and degree of susceptibility 1 are im-portant parameters, which characterize the probability of the population leaving the stability region. The lower G1 and the higher 1, the higher


Fig. 9.11. Potential function for the larch bud moth population in Switzerland

Fig. 9.12. Potential function for the gypsy moth population in the South Urals

the probability that the system will be hurled from the left-hand potential well to the right-hand one, i.e. the probability of a population density increase or an outbreak. The data in Table 9.3 suggest that the larch bud moth and the gypsy moth have the highest probabilities of outbreak development. The pine looper population in Thuringen has a somewhat lower probability of leaving the stability region. However, the degrees of susceptibility to changes in population density in the stable sparse state are similar for these three species.


Fig. 9.13. Potential function for the pine looper population in South Siberia

Table 9.3. Parameters of the potential functions

of the defoliating insect populations

Para-meters

Species

Pine looper in Thuringen

Pine-treelappet

in Siberia

Pine looper in Siberia

Larch bud moth in

Switzerland

Gypsy moth in the South

Uralsdensity

unit pupae per m2 larvae per tree

larvae per tree

larvae per kg needles eggs per tree

x1 –1 –0.3 –1.5 1 –1x2 2 0.7 2 4 6.5xr 0.9 0.3 1 2 0.5

xr – x1 1.9 0.6 2.5 1 1.5X2 – xr 1.1 0.4 1 2 6G(x1) 7 2 5 4 5G(x2) 12 16 40 3 6G(xr) 20 33 43 12 15

G1 13 31 38 8 101 7 52 15 8 6.7

G2 8 17 3 9 92 7 42.5 3 4.5 1.5

For the pine looper and pine-tree lappet populations in the pine stands

of South Siberia, the probability of leaving the stability region is very low, as the heights of the potential barriers G1 for these populations are considerably higher than the heights of the potential barriers for the populations of the pine looper in Saxony, the larch bud moth, and the gypsy moth. On the other hand, the degrees of susceptibility of the gypsy


moth and pine looper populations to an increase in population density in the stable sparse state are rather high, suggesting that short-duration strong weather changes can expel these populations from the stability region, but at low х2 values, short-term density increases are more likely than outbreaks.

For the outbreak species in Table 9.3, the heights of potential barriers G2 are nearly the same, but the degrees of susceptibility 2 of the gypsy moth and the larch bud moth are much lower than that of the pine looper in Thuringen, suggesting that on average, the pine looper can stay at the peak of the outbreak longer than the other two species.

In the previously proposed classification of population dynamics and types of outbreaks, every insect species was assigned to one type of dy-namics only (Isaev et al., 2001). Based on the type of the population phase portrait, one can assign the outbreak to one of the four types: the out-break proper, the fixed outbreak, the sustained outbreak, and the reverse outbreak. If the types of population dynamics and outbreaks are classified by using potential functions, all differences in population dynamics can be characterized through gradual changes in potential function parame-ters. Moreover, the gradual classification method can be used to charac-terize intermediate types of population dynamics and outbreaks and describe the situations in which a population in some habitat changes the type of outbreak.

9.3 Assessment of the maximum risk of damage to tree stands caused by insects

An important parameter in studies of population dynamics of species that briefly and infrequently build up their numbers to outbreak levels is the maximum population density in the phase of outbreak peak, as this value determines the maximum damage done to trees by insects. The time series terms cannot be conveniently used to ask questions about mechanisms of population dynamics and probable density variations that are different from the maxima observed during surveys. However, such calculations can be done, and to estimate the distribution of the maxima, we will use the model of double exponential distributions of extremes of random sequences (Leadbetter et al., 1989):

max( ) 1 exp( exp( )),mF x x (9.6)

where хmax is the value of the local density maximum of the time series.


After transformation of (9.6), we obtain a linearized equation that in the model of double exponential distributions describes the relationship between max max1 ( ) ln( ln( ))W F x x and the value of the local maxi-mum of population density, хmax.

max ,W A Bx (9.7)

where А and В are some constants. It follows from (9.7) that if the distribution of the time series extremes

is described by the double logarithmic model, there must be a negative linear relationship between W and xmax. Figure 9.14 shows the relation-ship between W and хmax for the maxima of the time series of pine looper population dynamics in Thuringen.

Fig. 9.14. Distribution of maxima of pine looper population density, xmax

Figure 9.14 clearly shows that the relationship between the values of W

and local maxima is very accurately (with the coefficient of determina-tion R2 = 0.963) expressed by linear equation (9.7). Similar relationships are also observed for population dynamics of other defoliating insects (Fig. 9.15 and 9.16).

If equations characterizing the relationship between the local maxi-mum of population density and the probability of the occurrence of the actual state with this density are correct, the probability of the occurrence of this state can be formally evaluated for any хmax, by substituting the хmax value into equation (9.7). Moreover, the model places no limitations on the maximum possible value of хmax: according to (9.7), any value is possible, even though the probability of the occurrence of the actual event is vanishingly small. This conclusion is, however, not quite correct, and the maximum population density is actually limited.


Fig. 9.15. The distribution of density maxima, хmax,

of the larch bud moth population in Switzerland

Fig. 9.16. The distribution of density maxima, хmax,

of the gypsy moth population in the South Urals

9.4 Modeling and forecasting of eastern spruce budworm population dynamics

The eastern spruce budworm (Choristoneura fumiferana Clemens) is one of the most destructive insects in the forests of Canada. Population dynamics of this insect has been the subject of much research (Blais, 1983; Candau et al., 1998; Candau, Fleming, 2005; Gray, MacKinnon, 2006; James et al., 2011; Magnussen et al., 2005; Royama et al., 2005). The data on the population dynamics of the eastern spruce budworm have been included in the Global Population Dynamics Database. In this section, we analyze population dynamics of the eastern spruce budworm employing the methods proposed in the present study. We use survey data on population of this insect collected in New Brunswick (Canada,


67 W, 47 N) between 1945 and 1972 (Miller, McDougall, 1973; www.imperial.ac.uk/cpb/gpdd).

As the methods have been described in detail in other chapters of this book, we present results of the analysis of eastern spruce budworm population dynamics in a graphical form, with brief comments.

Between 1945 and 1970, one outbreak was observed, with population density increasing by a factor of 900 – from 0.061 to 54.5 larvae m–2

(Fig. 9.17).

Fig.9.17. Population dynamics of eastern spruce budworm

(New Brunswick, Canada, 67 W, 47 N) After transformation, we obtained a stationary series of eastern spruce

budworm population dynamics (Fig. 9.18).

Fig. 9.18. The transformation of initial time series of population dynamics:

LTI-series of eastern spruce budworm population dynamics


Using the LTI-series of population dynamics data, we constructed a phase portrait of the population (Fig. 9.19).

Fig. 9.19. Phase portrait of eastern spruce budworm population dynamics

(1 – the trajectory of 1945–1965 Yrs. gradation; 2 – the beginning of next gradation)

The outbreak cycle of the eastern spruce budworm includes all phases

described in Chapter 2, and in 15 years, the population returns to actually the same state. The shape of the phase trajectory suggests that the critical density of the population is 4 < xr < 50 larvae m–2.

Periodicity of eastern spruce budworm population dynamics can be estimated using the LTI-series spectrum (Fig. 9.20).

Fig. 9.20. Spectral density of eastern spruce budworm population tine series


This procedure is correct, as the LTI-series is stationary. The maxi-mum of the function of spectral density is reached at a frequency of 0.038 1/yr. (period of oscillations T = 26 years) (Fig. 9.20). However, the peak of the spectrum is broad, and density variations may range be-tween 9 and 26 years.

To construct the AR model of eastern spruce budworm population dynamics, we need to estimate the order of the AR model by calculating the partial autocorrelation function of LTI-series (Fig. 9.21).

Fig. 9.21. Partial autocorrelation function of the LTI series

of eastern spruce budworm population dynamics. (1 – PACF; 2 and 3 – confidence 95% limits of PACF)

Outside the limits of the 95% confidence interval, there are two values

of the partial autocorrelation function, which correspond to delays 1 and 2, suggesting that the order of the AR function is 2 (Fig. 9.21). Know-ing the order of autoregression, we can calculate coefficients of the auto-regressive equation (Table 9.4).

Table 9.4. Characteristics of AR(2)-model of eastern spruce

budworm population dynamics

Characteristics of AR(2)-model a1 a2 a0

Coefficients 1.65 –0.75 0.05Errors of coefficients 0.13 0.12 0.11R2 0.95 F-test 193.7

The signs of the coefficients of the autoregressive function correspond

to the description of the model in Chapter 7: coefficient a1 > 0 and coeffi-


cient a2 < 0. The value of the coefficient of determination, R2, is very close to 1, suggesting that regulating factors almost completely describe popu-lation density variation, while the contribution of modifying factors (such as weather) is non-significant. Figure 9.22 illustrates the good agreement between the data calculated using the AR(2) model and survey data.

Fig. 9.22. Survey data on eastern spruce budworm population density (1)

and calculations based on the AR(2)-model (2) from Table 9.4 The model series and the LTI-series of survey data are synchronous,

proving the validity of the model (Fig. 9.23).

Fig. 9.23. Cross-correlation function between the predictive

and the testing values of the LTI series


Indeed, the maximum of the cross-correlation function is achieved at k = 0, i.e., there is no phase delay between the survey data series and the model series. The AR(2) model can be used to make a short-term forecast of eastern spruce budworm population dynamics (Fig. 9.24).

Fig. 9.24. The forecasting of eastern spruce budworm population dynamics;

1 – survey data, 2 – the AR(2)-model time series based on the data from the training set; 3 – forecast for 3 years

The short-term forecast (for 2–3 years) based on the AR(2) model gives

an adequate estimate of the future population density. In order to estimate the values of х1 (population density in the stable sparse state), xr (critical density of population), and x2 (density in the outbreak phase) using the LTI-series data, let us determine potential function G(x) (Fig. 9.25).

Fig. 9.25. Potential function of eastern spruce budworm population


Based on the minimum and maximum of the potential function, we obtain х1 0.05 larvae m–2, xr 9 larvae m–2, and x2 55 larvae m–2 of the eastern spruce budworm population. Unfortunately, the survey data do not contain values of the relative abundance of the population, and, thus, we cannot estimate the relationship between the squared order parameter and population density (a theoretical model is described in Chapter 8) and determine critical density xr. Nevertheless, we can estimate an ap-proximate shape of this relationship (Fig. 9.26).

Fig. 9.26. The equation of connection between population density and square

of order parameter q; critical population density xr = 9 larvae per m2; 1 – stable state of population; 2 – the outbreak phase

Thus, survey data should include not only the average population den-

sity but also relative abundance (the percent of sample units with insects of the species of interest). Then, the critical density of the population can be determined regardless of the estimates based on the phase portrait and potential function.

***

To predict population dynamics of defoliating insects is a much more complicated task than to model defoliating insect populations: to predict future is more difficult than to “predict” the already known past (which is the objective of modeling). However, it is not impossible to predict population dynamics of defoliating insects.

The short-term forecast of population dynamics of the species whose coefficient of determination, R2, for the AR model is close to 1 (suggest-


ing small contribution of modifying factors to the variance in population density) can be made by using an AR model based on the data of previ-ous surveys.

To make a forecast of population dynamics of the species whose coef-ficient of determination for the AR-model is significantly lower than 1, with the coefficient of determination for the ADL model close to 1 (sug-gesting rather large contribution of modifying factors to the variance in population density), one would need weather forecast for 1 or 2 seasons. Obviously, if the authors were able to predict weather, they would not study insect population dynamics. However, analysis of ADL models of defoliating insect populations shows that an approximate forecast of the future weather based on the data on the weather of the past years (fortu-nately, long-term time series of weather data are now available) enables rather exact estimation of the phase (but not amplitude) of the predicted time series of defoliating insect population dynamics. Knowing the current phase of population development is imperative for decision making while controlling population dynamics of damaging insects.

Thus, autoregressive models are promising tools for making short-term forecasts of the state of the population.

As to making long-term forecasts (for decades or, better, centuries) of population dynamics, it is a convenient occupation for two reasons. Firstly, it is perfectly safe for the current scientific reputation of the forecaster and, secondly, these forecasts need not be exact. In forecasts, we are free to use such words as “may”, “might”, or “it is not unlikely”. Speaking of the methods developed for long-term forecasting of forest insect population dynamics, the methods of condensed matter physics may be used (as it was described in Chapter 9) to qualitatively describe ecological processes associated with tree – insect interactions under specific environmental conditions. It is not unlikely, though, that more complex models of phase transitions, including several order parameters characterizing the state of the system, might be used.

However, to construct AR- and ADL-models, potential functions, and models of phase transitions for describing colonization of tree stands by insects, the analyst needs to have sufficiently long time series of popula-tion density. Therefore, monitoring insect populations is very important for ecological forecasting.

233

10 Global Warming and Risks of Forest Insect Outbreaks

10.1 Climate change and forest insect outbreaks in the Siberian taiga

Global climate change is an issue that is widely discussed now. Climate is the most important factors determining the dynamics of insect populations. The outbreaks of various insect species in recent years associated with to climate changes (Aukema et al., 2008; Carroll et al., 2004; Harrington et al., 2001; Kausrud et al., 2012; Logan et al., 2003; Preisler et al., 2012; Raffa et al., 2008; Vanhanen et al., 2007). Climate change has a direct impact on physiological processes in insects (Bale et al., 2002; De Sassi et al., 2012), and indirectly affect the population dynamics in relation to climate impacts on the protective system of plants (Awmack, Leather, 2002; Raffa et al., 2008; Uniyal, Uniyal, 2009). It is assumed that the effect of climate impacts on the development of forest insect outbreaks will intensify in the coming decades (Bentz et al., 2010; Logan, Powell, 2001).

In Siberia, climate change may both cause the major tree species of the taiga forests to shift ranges and lead to broadening and shifting of the areas where populations of the species currently defined as outbreak ones




and the species that now show stable population dynamics will be able to reach outbreak levels. The Siberian taiga is composed of vast woodlands dominated by one tree species. In Siberia, development of outbreaks may be catastrophic for forests, resulting in degradation or even death of one of the two remaining globally important woodlands.

The forested region of the Yenisei River valley can be divided into three zones: zone of fir Abies sibirica Ledeb., the zone of Scott pine Pinus sylvestris L. in the Angara River valley, and the zone of larch Larix sibirica Ledeb. in South Evenkia. All these forests grow in rather harsh climate, with very low winter temperatures and a short growing season. However, ironically enough, the harshness of the climate is a factor decreasing the risk of the occurrence of outbreaks in the taiga forests, and defoliating insects can thrive in some, but not all, of the taiga regions (Epova, Pleshanov, 1995). The prevalence of the same tree species over a vast area is a factor favoring the development of outbreaks of specific pests (primarily the Siberian silk moth Dendrolimus superans sibiricus Tschetv.) that cover the areas of several hundred thousand hectares (Isaev et al., 2001).

Therefore, there is a need to assess the risks of outbreaks of certain defoliating insects in taiga forests that may be caused by possible climate change. Unfortunately, there are no analogs to Siberian forests growing in the conditions similar to those expected in Siberia due to global climate change. Thus, the only approach that can be used to assess the risks of outbreaks of defoliating insects in Siberia is mathematical modeling. Below we discuss methods for assessing risks of outbreaks of defoliating insects based on the methods of constructing ADL-models of population dynamics and models of outbreaks as first-order phase transitions developed in the previous chapters.

In Siberia, climate is a significant factor in population dynamics of forest insects. Population dynamics of the pine looper Bupalus piniarius L. is a good example. In the Krasnoyarsk Territory, the pine looper as a species occurs in pine forests over a vast area from the Angara River valley ( 58 N) to the center of Tuva ( 52 N), but population dynamics of the insect varies depending on climate zone. In the pine forests in the Angara region, no outbreaks of this species have occurred over the past 100 years. Two outbreaks, in 1944 and 2014, occurred in the pine forests at Krasnoyarsk. Five outbreaks have occurred in the south of the Krasnoyarsk Territory since the late 1930s. Finally, in the pine stands of Tuva (the Balgazyn forest), the pine looper outbreak could have occurred in the 1950s although no recorded data are available (Palnikova et al., 2002).

Global Warming and Risks of Forest Insect Outbreaks 235

Table 10.1 lists long-term annual average values for the climate zones of the Krasnoyarsk Territory, located at different latitudes along the meridian 90°.

Table 10.1. Climatic parameters in different habitats in the Krasnoyarsk

Territory and the frequency of occurrence of pine looper outbreaks

Weather station location

Geographic coordinates

Winter(November–February)*

Summer(June–

August)**

Number of outbreaks

for the last 70 years/outbreak

probabilityN E ТW (TW) ТS (TS)

Bogu-chany 58.3836 97.4536 –21.5 10.5 16.9 3.8 0/0

Krasno-yarsk 56.0167 92.8706 –14.2 8.8 16.9 3.8 2/0.028

Minu-sinsk 53.7098 91.7154 –9.7 7.8 25.4 4.7 5/0.071

Kyzyl 51.7165 94.4366 –25.9 6.5 18.8 3.8 1/0.014***

* TW – average winter temperature, (TW) – standard deviation; ** TS – average summer temperature, (TS) – standard deviation. *** exact data unavailable.

The data in Table 10.1 suggest that the absence of the outbreaks in the

north and south of the pine looper range could be attributed to low winter temperatures (deviations from the long-term winter average temperature in the Angara region reaching 2 ( ) 40TW TW C ), and frequent outbreaks in the south of the Krasnoyarsk Territory could be associated with a rather moderate winter temperature and an elevated average summer temperature.

The vital temperature range for insects (approximately between +3 and +40 °C) characterizes the climatic niche of the population of a given species; the upper sublethal temperature range for insects is between +40 and +50 °C, the lower sublethal temperature range is between +3 and –10 °C, and the lower lethal temperature is below –10 C (Ushatinskaya, 1957). While studying the effects of the low temperatures on the survival of insects, one should certainly remember that in winter, the temperature under a snow layer may be considerably higher than the air temperature (Kolomiyets, 1961). However, these are qualitative evaluations, and it is unclear what quantitative changes in climatic parameters will cause significant changes in the type of population dynamics of forest insects.


10.2 Stress testing of insect impact on forest ecosystems under different scenarios of climate changes in the Siberian taiga

Procedure of the quantitative stress testing to assess the effects of climate changes on the population dynamics of forest insects can involve the use of ecological stress tests based on the data of past years (back testing) and stress tests based on model scenarios of changes in forest ecosystems. However, for quite a number of insect species, there is a lack of sufficiently abundant historical data on their population dynamics. In Siberian taiga forests, the longest continuous datasets for insect population in the same area have been collected for the pine looper, whose population has been measured in surveys since 1979 (Isaev et al., 2015). Tests based on stress climate scenarios can be used to model population response to the effects of climatic factors different from the effects of modifying factors, which are observed now. First, however, one needs to construct a model of population dynamics of the particular forest insect species.

In the present study, modeling of forest insect population dynamics is based on the notion that population density can be changed by two types of factors – regulating and modifying (Isaev et al., 2001). The effect of the regulating factors (intra-population competition, parasites, and predators) depends on the current density of the population while the effect of the modifying factors (mainly weather) does not depend on the current density of the population. However, parasite and predator population densities and, especially, functions of the effects of these regulating factors cannot be evaluated during field observations of the insect population, and, thus, very few of such models can be verified.

In the analysis that follows, we use ADL models previously discussed in Chapter 7:

01 0

( ) ( ) ( ),n m

j kj k

L i a a L i j b W i k (10.1)

where L(i) is LTI-density of population in season i; W(i) is weather parameters of the seasons, n is the order of the AR-component, m is the order of DL-components, а0 is the random component with the mean value equal to zero, and aj, bk, c1, с2 are coefficients.

Objects used to construct models of population dynamics and stress tests were time series of population dynamics of the pine looper B. piniarius L. in pinewoods growing in the Krasnoyarsk Territory.


Surveys of pine looper population were conducted by the authors in the Krasnoturansk pine forest (south of the Krasnoyarsk Territory) on various sample plots (habitats Dune, Narrow Plane, Hill Top) every year between 1979 and 2015. Detailed data on the pine looper population on this territory, landscape and assessment parameters of the sample plots, population dynamics and interactions of the pine looper with its entomophagous parasites have been published elsewhere (Tarasova, 1982; Palnikova et al., 2002; Palnikova et al., 2014; Isaev et al., 2014; Isaev et al., 2015, Palnikova, Soukhovolsky, 2016).

Weather conditions in the habitats mentioned above were characterized by the time series of the hydrothermal coefficient (HTC) – the ratio of the amount of precipitation over a certain period to the average air temperature for this period. For the pine looper model in year i, we used HTC values of September in year (i – 1) and May in year i.

As the data of the long-term surveys of the model species population density and weather parameters during the survey periods are known, model (10.1) can be regarded as a multiple linear regression equation with (n + 1) series of independent variables {L(i)} and m series of independent variables {W(i)} and with (n + m + 1) dependent variables – coefficients of equation (10.1). To estimate the order of the AR-com-ponent, n, of model (10.1), we used partial autocorrelation function (PACF) of the series {L(i)} (Box, Jenkins, 1974). To estimate multiple linear regression parameters, we used MLS in the Statistica 6 package. The coefficient of determination R2 of the linear regression equation and the cross-correlation function of the LTI series and the calculated model series characterize the level of agreement between the survey data and the model.

The models of population dynamics of the pine looper and gypsy moth were used to carry out computational experiments for evaluating the effects of climate changes on population dynamics of these species of forest insects. To perform calculations for the ADL-model, we conducted a special computational experiment: a value was arbitrarily selected from the available series {L(i)} of long-term survey data, and the selected value of population density and (n – 1) values following the selected one were used as initial values of population density. A previous study showed that in Middle Siberia, pine looper population in the larval stage in the current year could be influenced by the weather of May of the current year and September of the previous year (Palnikova et al., 2002). Based on this, we used the available weather data for the regions where we observed the insect populations studied to calculate the means and variances of September and May HTC. The weather data from weather


stations located closest to insect habitats were used to estimate distribution functions, means, variance, and partial autocorrelation function, PACF(k), of HTC series. As none of the PACF(k) values at k 1 differed significantly (with p = 0.95) from zero, we assumed that the HTC values of different years were independent. The distribution of HTC values was adequately described by the normal distribution function. In computational experiments, the ADL-model was constructed by generating random values of HTC with predefined means and variances. For the ADL-model with fixed coefficients of AR-components and generated HTC values with predefined means and variances, we calculated 10,000 10-year prediction time series. For the 100,000 realizations, we calculated histograms of distribution of population LTI-densities.

Figure 10.1 shows partial autocorrelation function of the LTI-series of pine looper population dynamics in habitat Dune, and Figure 10.2 shows the LTI-series of the pine looper population dynamics and the model series of the dynamics of this population.

Fig. 10.1. Partial autocorrelation function (PACF) of the LTI-series of pine

looper population dynamics in habitat Dune; 1 – PACF, 2 – boundaries 2s ( 1s n – standard error of partial autocorrelation (Box, Jenkins, 1970))

The order of the ADL(n, m)-model, n, for the LTI-series of pine looper

population dynamics is 2 (Fig. 10.1). Table 10.2 shows coefficients of models of pine looper population dynamics in different habitats in the Krasnoturansk pine forest.


Fig. 10.2. Pine looper population dynamics in habitat Dune;

1 – LTI-series of population surveys; 2 – ADL(2, 2)-model of dynamics; 3 – critical density Lr of population

Table 10.2. Evaluations of model parameters of pine looper population

dynamics in different habitats in the Krasnoturansk pine forest for the period between 1979 and 2014

Habitat Model variables*

R2 а0

September HTC

May HTC L(i – 2) L(i – 1)

Hill Top 0.17(0.32)

0.13(0.21)

–0.27(0.20)

–0.81(0.11)

1.60(0.11) 0.937

Dune 0.19(0.23)

0.18(0.15)

–0.36(0.15)

–0.90(0.10)

1.72(0.10) 0.958

Narrow Plane 0.06(0.36)

–0.17(0.23)

0.15(0.23)

–0.65(0.15)

1.41(0.14) 0.855

* Values in parentheses are errors in evaluating model parameters; italics denote values below the significance level p = 0.05.

Coefficient of determination R2 for the ADL-models of the pine looper in habitats Dune and Hill Top is higher than 0.9 (Table 10.2), i.e. the model can take into account more than 90% of the survey data variance. The ADL-model for habitat Narrow Plane is somewhat less accurate (R2 = 0.855).

Another measure of model accuracy was the condition of synchronicity of temporal changes between the LTI-series of data and the model series. To evaluate synchronicity, we used the function of cross-


correlation r(k) between the LTI-series of survey data and the model series. In synchronous series, the maximum of the cross-correlation function r(k) must be reached at a shift of k = 0, and the value of the cross-correlation function r(0) must be close to 1 (Jenkins, Watts, 1969). The functions of cross-correlation between the LTI-series of pine looper population densities and model series comply with these requirements (Fig. 10.3).

A

BFig. 10.3. The functions of cross-correlation between the LTI-series and model

series of pine looper population dynamics (A – Hill Top; B – Dune) The calculated values of coefficients of ADL-models were used to

perform stress tests to assess possible effects of climate changes on the occurrence of insect outbreaks. The risks of the effect of forest insects on


the forest can be associated with either a higher frequency of outbreaks or an increase in population density at the outbreak peak. At constant values of coefficients of AR(2)-components L(i – 1) and L(i – 2), the spectral density of LTI-density of population L(i) is determined by the values of these coefficients (Anderson, 1971; Kendall, Stewart, 1973). Thus, we assume that a change in weather conditions will not lead to a change in the frequency of outbreaks, although the absolute population densities may change considerably. Then the risk of outbreaks will be determined by the values of local maxima of population density. Therefore, in order to evaluate pine looper population density, we conducted simulation experiments, using model (10.1) with the coefficients of the AR-part of the model given in Table 10.2 and values of HTC varied between the shift S = –0.7 and the shift S = +2 relative to the current value of HTC0.

Figure 10.4 shows the distribution functions F(L) of the LTI density of the population obtained in simulation experiments with the model of the pine looper in habitat Dune at different HTC means and with the HTC variance characteristic of the last 40 years. These distribution functions are juxtaposed with the line of the critical LTI-density Lr, which characterizes the population density at which the outbreak takes place (Isaev et al., 2001). For the pine looper in the larval stage, the conversion of the species critical density used by the Russian Forest Protection Service (Monitoring…, 1965) results in Lr = 1.5.

Fig. 10.4. Distribution functions of LTI-densities of pine looper model populations in habitat Dune at different HTC values: 1 – (HTC0 – 0.7);

2 – HTC0, 3 – (HTC0 + 1), 4 – (HTC0 + 2); critical density Lr = 1.5


In Figure 10.4, the value of the distribution function FS(Lr) at the point of intersection with the critical density line Lr = 1.5 is the probability that at a given value of the shift S in HTC, the LTI-density of the population might be below the critical value of Lr. However, it will be more convenient to introduce a parameter indicating the risk of the effect of the pine looper on the forest under changing climate – probability at risk, PaR, of outbreak occurrence:

( ) 1 ( ).S rPaR S F L (10.2)

Table 10.3 presents PaR values for model populations of the pine looper obtained in simulation experiments.

Table 10.3. PaR values for pine looper populations

in different simulation experiments

Habitat Shift S relative to the current HTC value–0.5 0* 0.5 1 1.5 2

Hill Top 0.14 0.03 0.02 0.01 0.01 0.01Narrow Plane 0.06 0.03 0.01 0.01 0.01 0.01Dune 0.15 0.06 0.02 0.02 0.01 0.01

* S = 0 corresponds to long-term average HTC values from 1975 to 2010. In computational experiments with HTC values below the currently

observed ones, at decreasing S values (i.e. under drier and hotter weather conditions), the PaR value of outbreak increased (Fig. 10.5).

Fig. 10.5. PaR values versus values of the shift S relative to the current HTC

value for pine looper populations in sample plot Dune


A decrease in the long-term average HTC value causes a considerable (approximately three- or four-fold) increase in PaR values (Fig. 10.5). In experiments with positive S values (i.e. under cooling rather than warming conditions), the calculated PaR values are usually no more than 0.05, that is, under these climatic conditions, the risk of the occurrence of outbreaks is minimal.

In Middle Siberia, by 2050, warming may lead to a 2 °C increase in the summer average temperature and a 20% increase in summer precipitation (Tchebakova et al., 2002). However, the validity of these predictions is questionable. Moreover, changes in the type of the population dynamics of defoliating forest insects and in PaR values may be determined by changes in the variance of the average value rather than by shifts in the averages (Isaev et al., 1997; Soukhovolsky et al., 1996). Outbreaks may be triggered by a dramatic rise in summer temperatures together with a decrease in precipitation in just one season.

As climate forecast models contain objective uncertainty, to evaluate possible shifts in PaR values for the pine looper in Middle Siberia, we considered different scenarios of climate changes: absolute shifts in average temperature, relative shifts in precipitation, and variances of precipitation and temperature values relative to the climatic parameters observed for the last 40 years. For each of the scenarios, we conducted simulation experiments similar to ones described above and calculated PaR values (Table 10.4).

The data in Table 10.4 suggest that under various climate change scenarios – an average temperature elevation, a 10% change in precipitation, and a 25% increase in temperature and precipitation variances – PaR values in one of the habitats (Narrow Plane) do not increase considerably. In another habitat (Dune), a decrease in HTC caused by the temperature elevation and precipitation decrease leads to an about 30% increase in PaR: from 0.07 to 0.09. Thus, temperature shifts reaching +4 °C and changes in precipitation of up to 10% of the current values should not substantially increase the risks of the insect impact on the forest. However, when we consider a scenario for habitat Dune, in which the temperature shift is more than +5, the amount of precipitation is 20% lower than its current average value, and temperature and precipitation variances are 25% increased, the risk of outbreaks increases by more than 60%, the PaR value reaching 0.115. This risk level suggests that insects are capable of affecting the forest significantly, but climate in Middle Siberia is not expected to change so substantially.


Table 10.4. Risks of pine looper outbreaks under different scenarios of climate changes in Middle Siberia

Climate scenarios PaRAverage

temperature shift, C

Precipitation shift, %

Multiplier of variance

shiftHill Top Narrow

Plane Dune

0 0 1 0.041 0.037 0.0712 0 1 0.038 0.038 0.0794 0 1 0.039 0.039 0.0690 –10 1 0.041 0.039 0.0702 –10 1 0.047 0.039 0.0714 –10 1 0.042 0.039 0.0740 10 1 0.040 0.037 0.0822 10 1 0.044 0.038 0.0784 10 1 0.037 0.037 0.0730 0 1.25 0.045 0.041 0.0722 0 1.25 0.043 0.040 0.0904 0 1.25 0.047 0.039 0.0590 –10 1.25 0.053 0.041 0.0642 –10 1.25 0.057 0.040 0.0894 –10 1.25 0.056 0.039 0.0770 10 1.25 0.041 0.038 0.0702 10 1.25 0.041 0.038 0.0584 10 1.25 0.039 0.041 0.0845 –20 1.25 0.053 – 0.115

10.3 Risks of outbreaks of forest insect species with the stable type of population dynamics

At the present time, most of the forest insect species show a stable type of population dynamics. For instance, in forest ecosystems at Lake Baikal, only three species of wood-feeding insects and seven species of defoliat-ing insects out of the 315 forest insect species detected by 2000 have major outbreaks (Isaev et al., 2001). Thus, the question arises: Will cli-mate change induce currently stable defoliating insects to reach outbreak levels?

Let us assume that a species with the stable type of population dynam-ics is close to the stable state with some density х1. If the population of the species with the stable type of dynamics departs from stable state х1,


the negative feedback (i.e. the effect of parasites and predators) will quickly return it to the stable state. Negative feedback regulation is gen-erally based on the magnitude of deviation of population from stable state х1, 1( ) ( ) .x i x i x The value of the negative feedback regulating population density close to х1 is proportional to the value of ( )x i :

( )d x k xdt

, (10.3)

where k is coefficient of proportionality characterizing the strength of response of population to deviation from the stable state.

The greater k, the stronger the effect of the negative feedback will be and the quicker the population that has deviated from the stable state will return to it. If for the species with the stable dynamics the negative feed-back value is small, the strong and quick effects of climatic factors may lead to outbreak. If a species is characterized by strong negative feedback with a short time delay in population response to the external impact, the risk of outbreaks of this species caused by climatic factors is expected to be very low.

Let us now look into the possibility of evaluating the negative feedback by the shape of potential functions. As noted in Chapter 2, the value of the negative feedback can be determined by using the value of deriva-

tive Gx

in the left-hand potential well. If Gx

and 1rG G are large

(i.e. the potential well is deep and its walls are steep), one can assume that the negative feedback rather firmly regulates population dynamics at the level close to х1. If these conditions are not fulfilled, external factors or even strong fluctuations (as noted in 10.2 and 10.3) may cause popula-tions with the currently stable dynamics to reach outbreak levels.

To analyze the shapes of potential functions of populations in the sta-ble sparse state, we studied population dynamics of the pine looper Bupalus piniarius L. in different habitats in Scotland, England, the Neth-erlands, and Germany. The detailed data on the time series of pine looper population dynamics are freely accessible in the database (NERC Centre for Population Biology, Imperial College, the Global Population Dynam-ics Database) at www.imperial.ac.uk/cpb/gpdd. Table 10.5 gives densities of the time series of pine looper population dynamics.

If stable populations are truly systems with the effective negative feed-back, potential functions of such populations must have rather large

values of derivatives Gx

. The greater the value of Gx

, the stronger the

negative feedback.


Table 10.5. Characterization of pine looper population dynamics

Series number/ survey time

Stage of the life cycle

at the time of survey**

Minimum density,

individu-als/unit

mean density, individu-als/unit

maximum density,

individu-als/unit

9382*/Scotland1953–1989 P 0.00008 0.53 1.26

9381/Scotland1953–1989 P 0.00025 3.46 39.80

9232/England1953–1989 P 0.20 5.00 31.60

2730/Netherlands1951–1964 A 0.01 1.12 2.71

2729/Netherlands1951–1964 L 0.12 2.7 5.40

2728/Netherlands1950–1964 L 0.80 10.16 25.60

10061/Germany1958–1989 L 1.13 1.62 2.25

9691/Germany1880–1940 P 0.001 1.40 31.60

10120/Germany1950–1969 P 0.115 1.70 7.10

* – number of the series in the base www.imperial.ac.uk/cpb/gpdd; ** life cycle stages: P – pupa; L – larva, A – adult.

Figure 10.6 shows the LTI-series of pine looper population dynamics

No. 2729 in the database used here, for the Netherlands, 1951–1964, and Figure 10.7 shows the LTI-series pine looper population dynamics No. 9381, for Scotland, 1953–1989.

The potential function of the pine looper population in the Nether-lands is unimodal, with no peak characterizing increased population density. Thus, no outbreaks were observed in that habitat. The potential function of the pine looper population in Scotland is bimodal, suggesting outbreaks of the species in that habitat.

We calculated the steepness Gx

of the left-hand potential well of po-

tential functions by using approximate equality 1

1ln lnr

r

G GGx x x

(Fig. 10.6). Table 10.6 lists values for pine looper populations analyzed here.


Fig. 10.6. Potential function G(x) of pine looper population

(the Netherlands, 1951–1964)

Fig. 10.7. Potential function G(x) of the pine looper population

(Scotland, 1953–1989) For outbreak populations, 10.5, and for populations with stable

dynamics, 16.5 (except time series 2730) (Table 10.6). In the { , G} plane (Fig. 10.8), one can distinguish the ranges of values of the left-hand potential well parameters for both the outbreak populations and the populations with stable dynamics.

For outbreak populations, the left-hand potential well is not deep and its wall is not steep (Fig. 10.8). The notions developed in Chapter 8 suggest that with such properties of the potential function, sufficiently strong fluctuations or an increase in the parameters of the external po-tential field will cause first-order phase transitions – outbreaks. The effects of the weather factors on the development of outbreaks can be


quantified by using the values of susceptibility, h

and ( )Gh

, of

potential functions to the external field, h.

Table 10.6. Characterization of the potential functions of pine looper populations in different habitats of pine forests in Europe

country number in the database survey years type of popula-

tion dynamics* G

Germany 9691 1881–1940 OUT 7.3 14.5 Netherlands 2728 1950–1964 OUT 10.5 10

Germany 10120 1950–1969 OUT 6 6Scotland 9381 1953–1989 OUT 3.1 6.1

Netherlands 2730 1951–1964 ST 9.3 4.7 Scotland 9382 1953–1989 ST 17.5 32.3 England 9232 1953–1989 ST 17 33.9 Germany 10061 1958–1989 ST 53 10.7

* OUT – outbreaks; ST – stable population dynamics.

Fig. 10.8. The ranges of values of the left-hand potential well parameters for the outbreak populations and the populations with stable dynamics

***

Model experiments described above suggest that moderate warming (no more than 4 °C in summertime) does not cause a significant increase in the risk of insect impact on the forest, but severer warming and lower precipitation in Middle Siberia may lead to a considerable increase in the risk of outbreaks of the pine looper – the major pest of pine forests.

Model experiments described above certainly need to be improved. Among other things, it would be good to consider the effects of possible


changes in coefficients of the autoregressive term in (10.1), which may cause changes in the LTI-series spectrum and in the frequency of occurrence of outbreaks. Then, instead of using PaR to assess the risk of damage done to the forest by one insect species, we should introduce the product of PaR and outbreak frequency.

Nevertheless, the proposed approaches and indicators of the risk of insect impact on tree stands seem to be useful for estimating the probability of outbreaks of certain insect species in the forest under changing climatic conditions and for assessing risks of insect impact under climate change in particular regions.

251

Conclusion Successful scientific research must, on the one hand, generalize and formalize real-world processes and, on the other hand, provide the basis for choosing the future directions of research.

Based on these well-known theses, in conclusion, we would like to, firstly, sum up the main results presented in this book and, secondly, discuss possible objectives for future research in forest ecology and the theory of insect population dynamics.

1. What have we managed to show while describing population dy-

namics of forest defoliating insects? 1.1. Physicists know very well the principle of complementarity pro-

posed by N. Bohr in 1927 (Kuznetsov, 1968). In accordance with the principle of complementarity, physical items could be separately ana-lyzed in terms of contradictory properties, like behaving as a wave or a stream of particles. Each of these ways can be used to describe physical items from different standpoints. The use of the principle of complemen-tarity, however, is not limited to quantum physics, and it can be em-ployed to describe various processes. In this book, we used various ap-proaches to describing population dynamics of forest defoliating insects. Analysis of different ways of describing insect population dynamics showed that the use of time series and autoregressive models is effective in modeling and short-term forecasting of dynamics time series, and descriptions employing potential functions can be useful for long-term forecasting of population dynamics. While ADL models are a convenient instrument for forecasting dynamics, by using phase portraits and poten-tial functions to describe current processes in the populations, one can estimate critical population densities, when the properties of the popula-tion undergo a qualitative change.



252 Conclusion

1.2. In this book, we show that such critical events in ecosystems as defoliating insect outbreaks can be described as ecological analogs of phase transitions in physical systems. By approaching outbreaks as phase transitions, we managed to define insect outbreak more accurately, make a coherent description of the effects of regulating and modifying factors on populations, which is impossible by using classical ecological models such as the Lotka-Volterra model.

1.3. We introduced the principle of complementarity of ecological and economic methods for describing ecological processes. Although it was actually a case study of the process of food consumption by defoliating insects, we showed how the methods of economics can be used to de-scribe ecological processes and developed methods of estimating ecologi-cal analogs of costs – major parameters in economics. In our opinion, results obtained in this study are very important as they open the way for describing “economics” of living organisms and evaluating the efficiency of individuals’ activities in populations.

1.4. We think that results of this study can be used for more than de-scribing forest insect population dynamics. It would be very interesting to use the methods of analysis proposed in this book for describing processes occurring in populations of insects damaging agricultural and industrial crops, which are more economically important pests.

2. What are the possible new directions for developing the theory of

population dynamics of defoliating insects? 2.1. We think that an important theoretical problem is to take into ac-

count the noncommutativity of the effects of regulating and modifying factors on population dynamics and to use an operator approach to the description of population dynamics. Now we will try to translate this pile of terms into plain English.

A standard discrete simulation ecological model describes the transfer of the population from the state with density х(t) at time t into the state with density x(t + 1) at time (t + 1) under the impacts of regulating factors (predators or parasites) or modifying factors (weather). In the case of several factors sequentially influencing the population, a natural question is whether the final state of the population will depend on the order in which these factors will produce their effects. If these factors do not influence the population simultaneously, there are two possible scenarios of the consequences of their sequential influence:

the effects are commutative if the results are the same, in whatever order the factors influence the population;

Conclusion 253

if the result of the effects on the population depends on the order in which different factors influence it, these effects are noncom-mutative.

The impacts of several commutative factors on the insect population can be presented as a product of functions of the impacts of different factors. If, however, the effects are noncommutative, this way of presen-tation is not correct, and the order of the impacts needs to be taken into account. The order, though, is often unknown. Thus, we have a problem of correctly describing forest insect population dynamics under noncommutative impacts of different factors.

2.2. All approaches and models described in this book treat popula-tions as point objects in space. This approach does not even allow defin-ing the problem of description and modeling of outbreak sites. It is not that researchers do not understand the importance of this problem but that there is no correct mathematical language for describing these pro-cesses. Analytical models based on pioneering studies performed by A. N. Kolmogorov (Kolmogorov et al., 1937) and R. Fisher (Fisher, 1937) that are currently being developed are too simple to take into account characteristics of population dynamics and tree – insect interactions. On the other hand, simulation lattice models (cellular automata etc.) intro-duce too much arbitrariness into the description of spatial dynamics of outbreak sites, which originates from the arbitrariness of the basic as-sumptions of these models rather than from the complexity and diversity of processes occurring in actual forest ecosystems. In our opinion, devel-opment of optimization models describing the spatial distribution of the individuals in the population over the area and description of the “movement” of the outbreak site boundaries by employing such physical models as the model of viscous fingers (Schröder, 2001), the Kardar–Parisi–Zhang model (Lebedev, 2010), and the FitzHugh – Nagumo model (Anishchenko et al., 2003) will be useful for developing more realistic models of the spatial dynamics of defoliating insect outbreak sites.

2.3. Invasions of forest defoliating insects have been the subject of much recent debate. We believe that for describing these processes, it is necessary to develop the theory of species population dynamics close to the boundaries of their ranges that will simultaneously describe both the invasion of forest insects into new habitats and the expulsion of some of the native species from their habitats. In Chapter 8, we proposed a model of disappearance and appearance of the population in a certain region, which is based on interpreting these processes as ecological second-order phase transitions. Unfortunately, field data are lacking for further devel-

254 Conclusion

opment of this method, as forest entomologists focus on outbreaks of defoliating insects, during which it is easy to conduct population surveys, rather than on studying the populations with low densities, which would involve a lot of effort.

2.4. Another important future objective is to make a detailed analysis and forecast of post-outbreak processes in forest stands – reforestation after the damage done by defoliating insects, assessment of the risks of occurrence of xylophagous insect outbreaks and risks of fires in sites infested by defoliating insects. Forecasts of these processes in combina-tion with estimation of areas of outbreak sites under different forest protection strategies will be used to assess possible losses caused by outbreaks and make a sound choice of the forest protection strategy.

2.5. In addition to the problems listed above (mainly ecological and mathematical ones), there is an issue of developing methods for making optimal decisions regarding the control of the state of forest defoliating insect populations and the state of tree stands. This problem is at the interface between ecology, economics, the theory of organization, and social psychology; it includes assessment of forecast reliability, calcula-tions of the cost-effectiveness of protective measures, development of incentive schemes for forest protection officers, and the search for the solution to a so-called agency problem, which is well known in the theory of financial management (Brigham, Gapenski, 2001).

To find a solution to the agency problem is not an easy task (which is clearly seen in economics), but forest management objectives should include optimization of the decision making process. One of the ways to solve the agency problem is to develop new, more accurate methods of short-, medium-, and long-term forecasts of population dynamics of forest defoliating insects. It may be necessary to analyze the effectiveness of certain organizational decisions such as separation of the bodies re-sponsible for forecasting from those responsible for making decisions.

255

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284 References

285

Index ADL-models, 143 Allee density, 26 Autoregressive models, 143 Bistable system, 217 Coherence of population dynam-

ics, 79, 83, 94 Colonization, 194 Conditional probability, 54 Consumption efficiency, 114 Cross-correlation function, 88, 93,

94 Data repair, 20 Eastern spruce budworm, 225 Energy balance, 113 First-order phase transitions, 185 FitzHugh – Nagumo model, 106 Food consumption, 113 Forecasting, 207

Gain margin, 179 Global climate change, 233 Global Warming, 233 Gypsy moth, 65, 96, 123, 145, 221,

225 Hydrothermal coefficient, 74 Lamerey stairs (diagram), 17, 39 Larch bud moth, 34, 43, 44, 155,

221, 225 Long-term forecast, 217 Lyapunov stability, 26 Mikhailov hodograph, 178 Modifying factors, 3 Necessary weather conditions, 54 Negative feedback, 12 Oak leaf-roller, 172 Optimal consumption model, 114 Order of an ADL-model, 143 Outbreaks, 5



Parasite – host interactions, 101 Partial autocorrelation function,

150, 155 Phase portrait, 24 Phenomenological theory, 10 Pine looper, 37, 45, 55, 83, 86, 94,

105, 191, 197, 222, 234, 237, 246

Population-energy model, 127 Potential barrier, 218 Potential function, 47, 48, 49, 50,

185, 201 Potential well, 45

Regularization, 22 Regulating factors, 3 Reproduction coefficient, 25 Second-order phase transitions,

194 Siberian silk moth, 6, 16, 61, 197 Stress testing, 236 Sufficient weather conditions, 54 Synchronization, 68 Weather effects, 53

286 Index

forest insect population dynamics, outbreaks, and global warming effects

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