THE IMPLICATIONS OF ASYMMETRIC DISPERSAL FOR METAPOPULATIONDYNAMICS AND CONSERVATION

By

MIGUEL A. ACEVEDO

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2013

c⃝ 2013 Miguel A. Acevedo

2

To my mother who taught me the values of perseverance and hard work

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ACKNOWLEDGMENTS

First and foremost, I would like to thank my advisor Dr. Robert J. Fletcher, Jr, for

believing in my ideas and sharing with me his wealth of knowledge and for providing

invaluable guidance. It was a great honor to be his student. I am grateful to Dr.

Raymond Tremblay and Dr. Elvia Melendez-Ackerman for providing the data analyzed in

Chapter 1. I would also like to thank Dr. J. Cole Smith and Jorge Sefair for their patience

and for their invaluable help with the mathematical formulations in Chapter 3. I would

also like to thank my committee, Dr. Madan Oli, Dr. Katherine Sieving and Dr. Robert

Holt for their help at multiple stages of my degree. I would like to thank Andre Revell,

Conor Egan and Kaan Kerman for their help in the field. This work greatly benefited from

discussions with Dr. Benjamin Bolker, Dr. Severine Vuilleumier, Dr. Christine Miller and

members of the QSE3 IGERT. I am grateful to Dr. Jeff Hostetler, Aida Miro, Raya Pruner,

Nathan Marcy, Dr. Jason Evans, Dr. Kyle McCarthy, Varun Goswami, Divya Vasudev,

Rajeev Pillay, Chris Rota, Kristen Aaltonen, Eva Kneip, Binab Karmacharya, Noah

Burrell, Katie Haase, Luke McEachron, Madelon Van de Kerk, Irina Skinner, Mauricio

Nunez-Regueiro, Brian Reichert, Ellen Robertson, Irina Skinner, Rebecca Wilcox, and

Dr. Jennifer Seavey . Funding was provided by the NSF Quantitative Spatial Ecology,

Evolution and Environment (QSE3) Integrative Graduate Education and Research

Traineeship Program (IGERT) grant 0801544 at the University of Florida and an NSF

Doctoral Dissertation Improvement Grant (DEB–1110441). Funding was also provided

by the School of Natural Resources and Environment and the Department of Wildlife

Ecology and Conservation at the University of Florida.

I would also like to thank my family, specially Melba Torres and Carlos Milan, and

my good friends Dr. Rey Rivera and Dr. Melanie Mediavilla for their support. I am also

particularly grateful to Christine Gutierrez for her invaluable support.

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TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

CHAPTER

1 THE IMPLICATIONS OF ASYMMETRIC DISPERSAL FOR CONNECTIVITYAND PATCH OCCUPANCY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 Study System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.2 Site Occupancy Model and Parameter Estimation . . . . . . . . . . 15

1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.4.1 Detectability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.4.2 Colonization and Extinction . . . . . . . . . . . . . . . . . . . . . . 241.4.3 Asymmetric Dispersal and Metapopulation Modeling . . . . . . . . 26

2 THE PROXIMATE CAUSES OF ASYMMETRIC MOVEMENT . . . . . . . . . . 29

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.1 Study Area and Focal Species . . . . . . . . . . . . . . . . . . . . 312.2.2 Study Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.2.1 Mark-recapture . . . . . . . . . . . . . . . . . . . . . . . . 312.2.2.2 Field experiment . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.3 Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2.3.1 Mark-recapture . . . . . . . . . . . . . . . . . . . . . . . . 372.2.3.2 Field experiment . . . . . . . . . . . . . . . . . . . . . . . 38

2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 OPTIMAL SITE SELECTION STRATEGIES: PROTECTING AGAINST WORST-CASEDISTURBANCE SCENARIOS . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2.1 Network Flow Interdiction Models . . . . . . . . . . . . . . . . . . . 493.2.2 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.2.1 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . 52

5

3.2.2.2 Connectivity assessment . . . . . . . . . . . . . . . . . . 543.2.2.3 Disturbance: simulating a worst-case scenario . . . . . . 553.2.2.4 Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2.2.5 General model assumptions . . . . . . . . . . . . . . . . 57

3.2.3 Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2.4 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.3.1 Roseate Terns: Step-by-Step . . . . . . . . . . . . . . . . . . . . . 593.3.2 Skipjack Tuna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.3.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4.1 Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.4.2 Conservation Planning for a Worst-case Scenario for Connectivity 653.4.3 Uncertainty and Worst-case Scenario Planning . . . . . . . . . . . 67

APPENDIX

A MODEL SUMMARIES AND ADDITIONAL FIGURES . . . . . . . . . . . . . . . 68

B CONNECTIVITY ASSESSMENT: MATHEMATICAL DETAILS . . . . . . . . . . 73

C DISTURBANCE OPTIMIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . 74

D PROTECTION OPTIMIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

E MODEL ITERATIONS FOR WEIGHTED LIFE EXPECTANCY OPTIMIZATION 80

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6

LIST OF TABLES

Table page

1-1 Model selection of occupancy models predicting colonization and extinctionof Lepanthes rupestris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2-1 Model selection for each experimental treatment . . . . . . . . . . . . . . . . . 42

A-1 Parameter estimates and standard errors (SE) for the multi-season occupancymodel that resulted in the best fit . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A-2 Parameter estimates and standard errors (SE) for the model that included asymmetric dispersal kernel (Ssym) as a site covariate . . . . . . . . . . . . . . . 69

E-1 Model iterations showing the interaction between disturbance and protectionstages when pT = [3773, 2467, 350, 172] . . . . . . . . . . . . . . . . . . . . . . 80

7

LIST OF FIGURES

Figure page

1-1 Various approaches to model patch connectivity . . . . . . . . . . . . . . . . . 13

1-2 Calculation of the δ parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1-3 Distribution of δij in the Lepanthes system . . . . . . . . . . . . . . . . . . . . . 21

1-4 Partial relationships between extinction, colonization and connectivity measures(Ssym and Sasym) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2-1 Observed inter-patch transitions of C. vittiger in a 30× 30 m plot . . . . . . . . 32

2-2 Diagram showing the study design . . . . . . . . . . . . . . . . . . . . . . . . . 34

2-3 Linear regression analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2-4 Partial relationships of the best fit model explaining the rate of movement inthe area treatments of the field experiment . . . . . . . . . . . . . . . . . . . . 41

2-5 Movement rate (± SE) at the patch level in the experimental treatments. . . . . 45

3-1 Schematic diagram of the three stages of the modeling process: assessment,disturbance and protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3-2 Spatial network representation of Roseate Tern colonies in northeast USAand Skipjack Tuna in the Western Pacific . . . . . . . . . . . . . . . . . . . . . 54

3-3 Example of the iterative process between the disturbance and protection modelsfor a protection budget of u = 2 and allowing b = 2 patches to be disturbed . . 60

3-4 Sensitivity of the optimal life expectancy in the protection stage given changes(ϵ) in probability of staying in the same patch (�qii ). . . . . . . . . . . . . . . . . . 61

3-5 Optimal life expectancy (z) given all possible protection budgets (u) and numberof patches allowed to be disturbed (b) . . . . . . . . . . . . . . . . . . . . . . . 62

A-1 Spatial representation of patches (boulders and tree trunks) sampled for Lep-anthes rupestris occurrence in the Luquillo Experimental Forest. . . . . . . . . 69

A-2 Theoretical distributions of the parameter δij . . . . . . . . . . . . . . . . . . . . 70

A-3 Predicted partial relationships between colonization, extinction and patch areafor both phorophytes for the best-supported model . . . . . . . . . . . . . . . . 71

A-4 Partial relationship between patch area and detectability . . . . . . . . . . . . . 72

8

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

THE IMPLICATIONS OF ASYMMETRIC DISPERSAL FOR METAPOPULATIONDYNAMICS AND CONSERVATION

By

Miguel A. Acevedo

May 2013

Chair: Robert J. Fletcher, JrMajor: Interdisciplinary Ecology

Variations in movement have broad implications for many areas of ecology and

evolution. For instance, in metapopulation theory, movement plays an important role

promoting (meta)population persistence over time. Most metapopulation modeling

approaches describe patch connectivity using pair-wise Euclidean distances resulting in

the simplifying assumption of a symmetric connectivity pattern. Symmetric connectivity

may be rarely observed in nature where organisms move responding to environmental

cues or advection sources through heterogeneous landscapes. Assuming symmetric

dispersal when movement is asymmetric may result in biased estimates of colonization,

extinction and persistence, which has important implications for management and

conservation. Here I studied the implications of asymmetric connectivity for patch

colonization and extinction. I leveraged a long-term, time-series on colonization-extinction

dynamics in the wind-dispersed orchid Lepanthes rupestris and found that models

that accounted for these asymmetries had better fit. The implications of asymmetric

connectivity to metapopulation dynamics may be contingent on the driving mechanism

behind them. I used a combination of an observational study of individual movements

and translocation experiments to test for possible mechanisms of directed movement

in the cactus-feeding insect, Chelinidea vittiger and found that variations in patch size

were the most important driver of directed movements in this species. I also developed

9

a novel site selection model that optimally selects patches that will best preserve

connectivity given a worst-case disturbance scenario.

10

CHAPTER 1THE IMPLICATIONS OF ASYMMETRIC DISPERSAL FOR CONNECTIVITY AND

PATCH OCCUPANCY

1.1 Introduction

Variation in movement has broad implications for evolutionary biology (Kawecki

and Holt, 2002), community ecology (e.g., Salomon et al., 2010; Tilman et al., 1994),

and population dynamics (Armsworth and Roughgarden, 2005; Revilla et al., 2004;

Wiegand et al., 1999). In metapopulation ecology, movement is a fundamental process

for metapopulation dynamics and local population persistence over time (Hanski,

1998). Often, the focus on movement emphasizes dispersal from natal environments

or previous breeding locations (Colbert et al., 2001). Here, we broadly use the term

dispersal to reflect these movements and related movements that generate variation in

emigration, immigration, and colonization and extinction rates (sensu Vuilleumier et al.,

2010).

Most metapopulation approaches describe colonization-extinction dynamics

following the area-isolation paradigm (Dias, 1996; Hanski, 1998; Pellet et al., 2007).

Under this paradigm, extinction is negatively related to patch area assuming that

population size increases with patch area; (Hanski, 1998), and colonization is negatively

related to isolation from other patches. Patch isolation is often quantified using patch

distance-dependent connectivity measures (i.e., the inverse of isolation) that assume

the probability of colonization declines with distance to surrounding occupied patches,

which act as sources of propagules. To do so, these measures typically use an assumed

dispersal kernel weighted by the occupancy state and area of surrounding patches

(Hanski, 1994).

While these connectivity metrics have proven useful, these metrics make assumptions

regarding the movement behavior of the species. For instance, a negative exponential

function of Euclidean distance is commonly assumed in many patch connectivity

measures (Hanski, 1998; Moilanen and Nieminen, 2002). Assuming a kernel based

11

on Euclidean distance results in the simplifying assumption of symmetric dispersal in

which the likelihood of an individual dispersing from patch i to patch j is the same as the

likelihood of an individual dispersing from patch j to i . Nevertheless, several factors in

nature often cause an asymmetric pattern of dispersal, such as spatial heterogeneity

(e.g., Ferreras, 2001; Prevedello and Vieira, 2010) and advection sources (e.g., wind,

ocean and marine currents; (Keddy, 1981; Treml et al., 2008).

An asymmetric pattern of dispersal may also arise due to variations in patch or

inter-patch attributes (Figure 1-1). At the patch level, variation in patch area may result

in a greater likelihood of dispersal towards large patches because active dispersers

may better detect or prefer large patches. For passive dispersers, large patches may

simply act as large targets or because these are large targets for passive dispersers (i.e.

target effects; (Gilpin and Diamond, 1976; Lomolino, 1990); Figure 1-1a). Quantitatively,

these asymmetries due to patch level variations have been traditionally incorporated

in connectivity measures by weighting a dispersal kernel by patch area. Inter-patch

attributes, such as advection sources (e.g., wind, ocean or river currents), may result

in a greater likelihood of dispersal in the direction of the advection source than in the

opposite direction (Figure 1-1d). That is, the effective distance in the direction of the

advection source may be less than the effective distance in the opposite direction

(Figure 1-1d).

Connectivity measures currently used in metapopulation modeling generally lack

a formal way to incorporate this kind of asymmetry in effective distance. Moilanen and

Hanski (1998) provide ways to account for habitat type-specific effective distances (h);

however, the effective distance measure is still symmetric (i.e. dhij = dh

ji ; Figure 1-1c).

Other alternate parameterizations employ least-cost paths (LCP) between patches

(Chardon et al., 2003). Even though they have been shown to be better descriptors of

effective distance than Euclidian distance (Sawyer et al., 2011), in most applications

these LCP are also symmetric (LCPij = LCPji ; Figure 1-1b).

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Figure 1-1. Various approaches to model patch connectivity. Patch connectivity inIncidence Function Models (a) takes into consideration variations in patcharea. Least cost paths (b) replace Euclidean distance by the least cost pathbetween two patches. Other proposed measures (c) scale Euclideandistance depending on habitat type (or land-cover/use type). Even thoughthese modifications, dij = dji in all these modeling approaches, which makesthem inapplicable to model organisms that disperse asymmetrically drivenby advection sources (d).

Recent metapopulation theory hypothesizes that failing to acknowledge dispersal

asymmetries leads to an overestimation of patch connectivity, resulting in biased

estimates of colonization and extinction (Bode et al., 2008; Vuilleumier et al., 2010;

Vuilleumier and Possingham, 2006). There are very few empirical tests on these

hypotheses, in part because we lack an adequate way to quantitatively incorporate

asymmetric dispersal due to inter-patch attributes in metapopulation models.

Here, we leverage a long-term, time-series on colonization-extinction dynamics

in the wind-dispersed orchid Lepanthes rupestris to test for asymmetric dispersal.

We assess for the role of asymmetries for colonization and extinction dynamics by

extending the connectivity function of Hanski (1994) to account for dispersal asymmetry

due to wind advection. We expected asymmetric dispersal would be prevalent in this

13

metapopulation and that such asymmetries would alter colonization and extinction

estimates relative to models assuming dispersal symmetries. Given the wind-dispersed

nature of the species, we also expected that target and rescue effects will play a crucial

role in the estimation of colonization and extinction probabilities.

1.2 Methods

1.2.1 Study System

Orchids, such as Lepanthes rupestris, that grow on rock (lithophytic habitat) and/or

trees (epiphytic habitat) are appropriate to study metapopulations with asymmetric

dispersal because they often live in spatially discrete ephemeral habitats and are

passively dispersed by directed sources such as wind (e.g., Snall et al., 2005;

Tremblay et al., 2006). Moreover, many epiphytic and lithophytic orchids are subject to

colonization and extinction dynamics due to their small population sizes and stochastic

reproductive success driven, in part, by dispersal and pollinator limitation (Ackerman,

1995; Olaya-Arenas et al., 2011; Tremblay, 1997).

Lepanthes rupestris is a small, wind-dispersed orchid (leafs 1.3 4.3 cm, shoots

15cm in height and flowers of < 6 mm) commonly found along the riverbeds of the

Luquillo Mountains in Puerto Rico (Ackerman, 1995). This patchily distributed orchid

anchors its roots to the substrate and roots are often covered by moss living on the

surfaces of trees or rocky boulders. The seeds are microscopic with a mean dispersal

distance of 4.8 m (Tremblay, 1997). Given the small size of its seeds, little is known

about the fate of Lepanthes seeds after dispersal and the presence of seed banks in this

species has been debated. Whigham et al. (2006) found that temperate orchid seeds

in experimental conditions were viable even after 7 years; however, Lepanthes seeds

are expected to die if they fall in the river. Given the flood dynamics of tropical rivers

(Johnson et al., 1998), the effect of seed banks in the metapopulation dynamics of this

orchid is likely negligible.

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A permanent plot for the study of the metapopulation dynamics of L. rupestris

was established in Quebrada Sonadora in the Luquillo Experimental Forest (LEF; 18◦

18 N, 65◦ 47 W) in 1999. This permanent plot is composed of 1000 occupied and

unoccupied boulders and tree trunks (patches hereafter; Tremblay et al., 2006). Most

patches (920) were mapped to within approximately 10cm (x, y, z position) relative

to the center of each patch using metal rulers, a sighting compass and a clinometer

(Figure A-1). The presence/absence of L. rupestris was censused twice a year from

1999–2008 (17 censuses). Additional descriptions of the site and sampling procedure

can be found in Tremblay et al. (2006). Patch size (i.e., total moss area), was estimated

as the perimeter occupied by the individuals in a patch using a 150 cm2 mesh. The

average wind direction was calculated from daily measures from the nearest weather

station located (∼1km from study site) in El Verde Biological Station (Ramırez and

Melendez-Colom, 2003).

1.2.2 Site Occupancy Model and Parameter Estimation

Incidence function models (IFM; Hanski, 1994; Moilanen and Nieminen, 2002)

are probably the most common occupancy modeling approaches used to estimate

colonization and extinction dynamics in a metapopulation context. Although IFMs have

been applied successfully to a wide range of species, they make two assumptions

that are difficult to meet in most applications. First, they assume that the species

is always detected where it is present (i.e. perfect detection). This is rarely the

case, where an observed absence may be either a true absence (the population is

locally extinct) or simply that the species was not detected, which may result in an

overestimation of extinction (Moilanen and Nieminen, 2002). Second, IFMs assume that

the metapopulations are at a Markovian pseudo-equilibrium in which the occupancy

status of each patch at time t is given only by the patch status at time t − 1. This

assumption is difficult to test and some studies argue that it is appropriate to assume

nothing about the equilibrium state of the metapopulation (Erwin et al., 1998; Pellet

15

et al., 2007). In this study, we applied an alternate method, a dynamic occupancy model,

in which these two assumptions are relaxed (MacKenzie et al., 2003).

The dynamic (i.e., multiple-season) occupancy modeling approach resembles

Pollocks robust design (Pollock, 1982) in that there are two types of sampling periods.

Primary periods are used to estimate colonization and extinction parameters. The

population is assumed to be open between these primary periods. Within the primary

periods, sites are surveyed multiple times (secondary sampling). The population is

assumed to be closed (no immigration, emigration, births, or deaths; see Rota et al.

(2009) for an assessment of this assumption) between these secondary sampling

periods, which are used to estimate detection probabilities. For each secondary

sampling period, there are three occupancy possibilities: presence, absence (which may

correspond to imperfect detection or an actual absence), or missing data (MacKenzie

et al., 2003).

Our primary sampling periods are years (1999-2008, n = 10). The secondary

sampling periods consisted of two censuses that were performed each year. One

census was conducted at the beginning of the year (January-February) and the second

in the summer (July-August). This model formulation allows the system to be open

to colonizations and extinctions during the wet season (August-December). Tropical

storms are common throughout the wet season causing flash floods, which may be

responsible for most local extinctions and anomalous strong winds, which may increase

the magnitude of dispersal events potentially resulting in more local colonizations.

Even though some colonizations and extinctions may happen in the dry season, our

conclusions are based on relative comparisons (symmetric vs. asymmetric models; see

below) and the potential violation of the closure assumption will affect both instances

equally, with minimal effect in the final relative comparisons. Given the small size of

the orchid, we expected variation in detectability especially in patches with small areas

16

of moss where it may be difficult to discern between moss and a small L. rupestris

individual.

The effects of area (Aj ) and patch connectivity (Si ) were added as covariates

to model colonization, extinction and/or detection probabilities. We calculated patch

connectivity in two ways, one of which treats the distance effect in the dispersal kernel

as symmetric and a second that adjusts effective distance to allow for asymmetric

movement. Patch connectivity with a symmetric dispersal kernel was calculated using

the general approach applied in many Incidence Function Models (Moilanen and

Nieminen, 2002):

S sym

i =

N∑j =i

exp(−αdij)Aj (1–1)

where N is the total number of patches in the landscape, 1/α is the average

dispersal distance of the species, dij is the Euclidean distance between patches i and

j and Aj is the area of the target patch j (Hanski, 1994, 1998). Note that the effective

distance between patches in this model is symmetric (i.e. dij = dji ). This formulation

can result in asymmetric patch connectivity due to variation in patch area, but not due

to variations in effective distance among patches (i.e. inter-patch attributes). Also note

that the connectivity measures we employed do not incorporate occupancy data as is

traditionally done in some IFMs measures of isolation where the occupancy state of

patch j is incorporated as a binary indicator variable to restrict potential re-colonization

only from previously occupied sites. A binary incorporation of occupancy state assumes

perfect detection, which we found is not the case in our system (Figure A-4). Given that

occupancy state was not incorporated in either symmetric or asymmetric measures of

connectivity, this modification did not affect the general conclusions.

We modified Equation 1–1 to incorporate the effect of average wind direction in the

estimation of connectivity. Patch connectivity with an asymmetric dispersal kernel was

calculated as:

17

Sasym

i =

N∑j =i

exp(−α dij

1− δij

)Aj (1–2)

where δij describes the difference between the angle of wind direction and the angle

between patches i and j in radians with respect to the horizontal axis (Figure 1-2). This

modeling approach resembles the strongly asymmetric distance-dependent theoretical

model from (Vuilleumier et al., 2010) in which δ is incorporated as a modifier inside the

exponential decay function. For simplicity, δij is scaled by π to constrain δij ∈ [0, 1],

δij =|θij − θwind |

π. (1–3)

Figure 1-2. Calculation of the δ parameter. The δij parameter is calculated as thedifference of the angles between patches i and j, and the angle of winddirection with respect to the horizontal axis

For both measures of patch connectivity (Ssym and Sasym), connectivity decreases

with increasing distance between patches. In Sasym when δij = 0, the effective distance

between patches is the same as Euclidean distance, and hence Sasym = Ssym; however,

18

as δij increases, the effective distance between patches increases resulting in decreased

connectivity when compared to Ssym. This modification allows an asymmetric dispersal

kernel because the effective distance between patches in the direction of wind is less

than the effective distance between the same patches in the opposite direction (against

wind direction; Figure 1-1d).

The distribution of the δ parameter will depend on the spatial arrangement of the

patches with respect to the angle of the advection source. For instance, if patches are

randomly arranged in space, we will expect a uniform distribution of δs (Figure A-2a,

A-2b). In contrast, if patches are spatially aligned parallel to the angle of the advection

source (e.g., along a river when wind advection is going downstream), we will expect

a bimodal distribution (Figure A-2c, A-2d). The bimodal nature of the distribution is

expected because the difference in the angles was scaled by π, and thus δij + δji = 1.

As the difference between δij and δji increases, effective distance becomes more

asymmetric, and the distribution will become more skewed towards the limits of the

distribution.

We fitted six kinds of occupancy models. Each represented a hypothesis of the

underlying mechanism governing colonization and extinctions in the L. rupestris

metapopulation. Traditionally, patch occupancy models in a metapopulation context

originate from the area-isolation paradigm. Hence, what we named the (1) “area/isolation

model”, includes patch connectivity as a site covariate for colonization and patch area

as a site covariate for extinction (i.e. γ(S), ϵ(A); where S represents a measure of

connectivity either with a symmetric or asymmetric dispersal kernel, and A describes

patch area). Connectivity may also decrease extinction probability by the rescue

effect of nearby patches (e.g., Hanski, 1998). The (2) “rescue effects” model includes

connectivity as a site covariate for colonization and both connectivity and patch area are

incorporated as interacting site covariates for extinction i.e. γ(S), ϵ(A ∗ S). Given the

wind-dispersed nature of Lepanthes rupestris, we expect that larger patches may have

19

a greater likelihood of receiving seeds because they are larger targets. Hence, we also

fitted a (3) “target effects” model in which patch connectivity together with patch area are

incorporated as a covariate for colonization and patch area as a covariate for extinction

i.e. γ(S ∗ A), ϵ(A). We fitted a (4) “target / rescue effects” model, which included both

target and rescue effects i.e. γ(S ∗ A), ϵ(S ∗ A). For each of these models (1–4) we fitted

a model with Ssym and another with Sasym as a site covariate. Because connectivity may

not necessarily be an important factor predicting colonization and extinction dynamics

(Pellet et al., 2007), we also fitted a (5) model that had moss area as site covariates for

colonization and extinction, but no connectivity metric were included. Previous analyses

for this species have found that colonization and extinction dynamics may be different

depending on phorophyte type (tree or rocky boulder; Tremblay et al., 2006). Moreover,

a preliminary analysis showed that models that included phorophyte type as a site

covariate for colonization, extinction and detectability had a better fit than models that

did not incorporated it. Hence, for all of these models (1–5) we include phorophyte

type as an interacting site covariate for colonization, extinction and detectability. We

also included patch area as a site covariate for detectability in all models, because

this preliminary analysis also showed that model fit increased. Finally, we also fitted

a (6) null (intercept-only) model with no covariates to compare with more complex

models. We fitted each model using maximum likelihood with covariates scaled and

centered, and ranked each occupancy model based on Akaike Information Criterion

(AIC; Burnham and Anderson, 2002). Models with the lowest AIC were considered most

parsimonious. Occupancy models were fitted using the package unmarked (Fiske and

Chandler, 2011) in R (Team et al., 2011).

1.3 Results

The calculated values for δ ranged between 1.11 × 10−6 and 9.9 × 10−1. The

distribution of the δij parameter showed a bimodal distribution, skewed towards the limits

20

of the distribution. This distribution was calculated based on an average wind direction

of 2.02 rad (Figure 1-3).

Figure 1-3. Distribution of δij in the Lepanthes system

The “target/rescue” and “target” effects models had better fit than the “no connectivity”

model suggesting that patch connectivity is an important predictor of patch dynamics

in this system. The most parsimonious model was the “target/rescue effects” model

that included patch area and patch connectivity with an asymmetric dispersal kernel

(Sasym) as site covariates for both colonization and extinction (“asymmetric model”

hereafter). The asymmetrical model had considerably more support than the similar

model (“target/rescue effects”) that included the patch connectivity measure with a

symmetric dispersal kernel as a site covariate (Ssym; “symmetric model” hereafter; Table

21

1-1). In all five model-types, the “asymmetric” form of the model had a relative better

than the corresponding “symmetric” model.

Table 1-1. Model selection of occupancy models predicting colonization and extinction ofLepanthes rupestris. Notation: ψ, occupancy;γ, colonization; ϵ, extinction; p,detectability. Covariates: Ssym, connectivity with symmetric kernel; Sasym,connectivity with asymmetric kernel; A, moss area; Ph, phorophyte type

Model �AIC AIC weight1 ψ(.) γ(Sasym ∗ A ∗ Ph) ϵ(Sasym ∗ A ∗ Ph) p(A ∗ Ph) 0 9.70E-012 ψ(.) γ(Sasym ∗ A ∗ Ph) ϵ(A ∗ Ph) p(A ∗ Ph) 7.57 2.20E-023 ψ(.) γ(Ssym ∗ A ∗ Ph) ϵ(Ssym ∗ A ∗ Ph) p(A ∗ Ph) 10.86 4.30E-034 ψ(.) γ(Ssym ∗ A ∗ Ph) ϵ(A ∗ Ph) p(A ∗ Ph) 20.64 3.20E-055 ψ(.) γ(A ∗ Ph) ϵ(A ∗ Ph) p(A ∗ Ph) 24.46 4.80E-066 ψ(.) γ(Sasym ∗ Ph) ϵ(Sasym ∗ A ∗ Ph) p(A ∗ Ph) 30.02 2.90E-077 ψ(.) γ(Ssym ∗ Ph) ϵ(Ssym ∗ A ∗ Ph) p(A ∗ Ph) 31.32 1.50E-078 ψ(.) γ(Sasym ∗ Ph) ϵ(A ∗ Ph) p(A ∗ Ph) 38.93 3.40E-099 ψ(.) γ(Ph) ϵ(A ∗ Ph) p(A ∗ Ph) 41.89 7.80E-1010 ψ(.) γ(S ∗ Ph) ϵ(A ∗ Ph) p(A ∗ Ph) 42.63 5.40E-1011 ψ(.) γ(.) ϵ(.) p(.) 106.57 7.00E-24

Both models (asymmetric and symmetric) predicted similar partial relationships

between colonization, extinction and connectivity (Figure 1-4). Both models predicted

a positive relationship between increasing connectivity and colonization probability

for both phorophytes. Also, both models predict that extinction probability decreases

with increasing connectivity for the rock phorophyte. However, models differed in the

prediction of extinction probabilities for tree phorophyte. The asymmetric model predicts

that extinction probability decreased with increasing connectivity, while the symmetric

model predicts a fixed relationship (Figure 1-4).

In both models, moss area was positively related to colonization and detection

(Figure A-2, A-3), but it was negatively related to extinction. Detectability was high

(p > 0.80) and positively related with patch area in both phorophytes (Figure A-3).

1.4 Discussion

Asymmetric dispersal may be the rule more than the exception in nature (Gustafson

and Gardner, 1996). Nevertheless, most parameterizations of metapopulation models

22

Figure 1-4. Partial relationships between extinction, colonization and connectivitymeasures (Ssym and Sasym). Panels (a) and (b) show the partial relationshipsfor the rock phorophyte, while (c) and (d) show partial relationships for treephorophytes. Shade areas represent 95% confidence interval of theestimates. Note that the range of the connectivity axis represent the valuesof Ssym and Sasym scaled and centered.

23

assume symmetric effective distances between patches. Here we developed a novel

modification of the connectivity formulation traditionally used in incidence function

models that allows for asymmetric effective distances between patches when dispersal

is directed by an advection source. Our results on the wind-dispersed orchid Lepanthes

rupestris suggests that asymmetric dispersal was prevalent in this system and that its

colonization-extinction dynamics were better described by a metapopulation model that

incorporate this modified asymmetric connectivity formulation.

1.4.1 Detectability

Even though Sasym was a relative better descriptor of connectivity than Ssym (models

that incorporated Sasym had always better fit than models that incorporated Ssym),

this model potentially over estimates connectivity. The connectivity measures used

(Equations 1–1 and 1–2) do not incorporate the occupancy state of patch j , which is

commonly used to restrict potential (re)colonization only from occupied patches. This

incorporation implicitly assumes perfect detection, which we found is not the case in this

system.

Imperfect detection is often regarded as an important factor in the study and

management of animal populations (e.g., MacKenzie et al., 2003). Nevertheless, there

is increasing evidence that the probability of detection in plant occupancy studies is

high but often p < 1 (Kery and Gregg, 2003; Kery et al., 2006). Our study showed that

detectability increased with increasing moss area. L. rupestris is a particularly small

orchid; hence the chances of detecting a single individual (particularly if it is missing

leaves) are expected to increase with increasing moss area. Also, generally in epiphytic

species, larger phorophytes will hold more individuals, which will also increase the

probability of detection (Fedrowitz et al., 2012; Snall et al., 2005; Tremblay et al., 2006).

1.4.2 Colonization and Extinction

Our results suggest the potential for both target and rescue effects as important

mechanisms predicting colonization and extinctions (Brown and Kodric-Brown, 1977;

24

Gilpin and Diamond, 1976; Lomolino, 1990). As patches become larger and more

connected they have an increased likelihood of being colonized because they become

bigger targets. Uncertainty also increased with increasing connectivity and area (Figure

1-3 and Figure A-3). In this system, patches are spatially clustered in space with very

few really isolated patches, and there are also very few really large patches, which

may account for the lack of variation at the end of the connectivity and area spectrum.

Even though wind dispersed plant species are characterized for having high gene flow

estimates (Hamrick et al., 1995), L. rupestris has a relatively short mean dispersal

distance and most of the seeds may fall below the maternal plant (Ackerman et al.,

1996). This may be due to the relatively low wind speed in the forest understory and

the relative low frequency of high winds. The importance of the interaction between

connectivity and patch size for colonization has been also reported for other epiphytic

species. For instance, tree diameter and connectivity have been positively related

to colonization of the epiphytic lichen Lobaria pulmonaria (Gustafsson et al., 1992;

Snall et al., 2005) and the epiphytic bryophytes Nyholmiella obtusifolia, Orthortichum

speciosum, Pylaisia polyantha, and Radula complanata (Hazell et al., 1998). The

importance of target effects to predict colonization has also been described in active

dispersers, such as the butterfly Maculinea nausithous (Hovestadt et al., 2011).

The partial relationships between connectivity and extinction for the symmetric

model showed a relatively constant relationship while the best-ranked model (asymmetric)

showed a negative relationship for tree phorophyte. Tree phorophytes are located

relatively higher in elevation than rock phorophytes (Figure A-1) and their occupancy

status may depend more on wind-dispersed seeds.

Metapopulation models often make the plausible assumption that population size

increases with increasing patch area (Hanski and Hanski, 1999). Lower extinction rates

in large patch areas may be the result of positive density-dependent regulation, which

have been previously reported for L. rupestris (Rivera Gomez et al., 2006). Previous

25

research on other epiphytic plants identifies loss of substrate (mostly tree mortality) as

one of the most important driver of local extinction (Snall et al., 2003; Tremblay, 2008).

In our system flooding is the most important driver of extinction. Even though there has

been substrate loss by flooding which always resulted in local orchid extinction, these

were relatively infrequent (15 trees and 2 boulders; (Tremblay et al., 2006). Trees had,

in general, higher extinction rates than boulders. This is consistent with a previous study

on L. rupestris that found that trees had almost double extinction rates as boulders

(Tremblay et al., 2006). Substrate loss due to flash flooding was higher for trees, which

may suggest that flooding affects extinction in tree phorophytes greater than boulders.

There may be other factors (covariates) that may affect the colonization and

extinction dynamics of L. rupestris, in addition to connectivity, area and phorophyte

type. For instance, seed establishment in orchids is limited by their association with

mycorrhizae (Dressler, 1993; Rasmussen and Whigham, 1993). Also, the amount

of moss moisture may also affect seed establishment (Tremblay et al., 2006). These

variables were not included in this model due to the difficulty of incorporating these into

a long-term metapopulation monitoring program. Nevertheless, their potential interaction

with asymmetric connectivity to predict colonization and extinction dynamics remains

unexplored.

1.4.3 Asymmetric Dispersal and Metapopulation Modeling

There are several examples of organisms, including both active and passive

dispersers that occur as metapopulations with asymmetric connectivity. In the closely

related species L. eltoroensis, individuals more frequently colonized boulders and

trees between 270◦ and 90◦ (Tremblay and Castro, 2009). In aquatic systems, ocean

and river currents are important drivers of asymmetric dispersal for both vertebrates

and invertebrates (Lutscher et al., 2007; Treml et al., 2008; Watson et al., 2011a).

Asymmetric dispersal pattern have also been found in active dispersers such as

Everglades Snail Kites (Rostrhamus sociabilis plumbeus), cactus bugs (Chelinidea

26

vittiger Fletcher et al., 2011) and the endangered Iberan lynx (Lynx pardinus; Ferreras,

2001). Moreover, an individual based model developed by Gustafson and Gardner

(1996) showed that altering landscape heterogeneity resulted in asymmetric rates of

immigration and emigration among resource patches. Hence, asymmetric connectivity

may be the rule more than the exception, given that symmetric connectivity may only be

applicable for organisms which dispersal is not affected by advection sources, spatial

variation in resources or that live in homogeneous landscapes. Such examples are

uncommon in nature.

We found that connectivity was important to predict colonizations and extinctions,

and that asymmetric dispersal is prevalent in the system. This is shown by the skewed

distribution of the δ parameter towards the limits of the distribution and because

the best-ranked model included Sasym. A previous analysis on 10 species of birds,

amphibians and butterflies showed that adding connectivity as a site covariate did not

improved model fit compared to constant colonization parameters (Pellet et al., 2007).

This study argues that using the symmetric connectivity measure traditionally used

in incidence function models may not be adequate to model most species because it

does not account for density-independent movements among patches (e.g., response

to an advection source or taxis) which may significantly affect dispersal rates. The

Sasym connectivity measure accounts for some of these density-idependent movements.

It significantly improved model fit, even when it only took into consideration average

wind direction, which is a broad way to describe advection effects. We expect that if

we incorporate patch-specific measures of wind direction, model fit would improve

even more. In this system which is composed of 920 patches, it is unfeasible to collect

patch-specific wind direction (or other advection); however, in other applications in

smaller metapopulations, this may be feasible.

The lack of recognition of asymmetric connectivity in empirical metapopulation

studies may be due in part to the absence of an estimation framework that can

27

incorporate asymmetric dispersal. Here we provide a general framework using

multi-season occupancy modeling using connectivity as a covariate. The asymmetric

connectivity measure that we applied (Equation 1–2) can be generally applied to any

other simple advection sources (e.g., water current direction) or it can be modified

to adapt it to other systems. For example, wind direction can be replaced by the

angle of riverine or marine currents. The incorporation of asymmetric connectivity

into metapopulation modeling will potentially result in an increase in the accuracy of

conservation and management decisions (Beger et al., 2010).

28

CHAPTER 2THE PROXIMATE CAUSES OF ASYMMETRIC MOVEMENT

2.1 Introduction

Variation in movement has broad implications for many areas of ecology and

evolution, including altering local adaptation via gene flow (Kawecki and Holt, 2002),

community structure (Liebhold et al., 2004; Salomon et al., 2010; Tilman et al., 1994),

and population dynamics (Armsworth and Roughgarden, 2005; Revilla et al., 2004;

Wiegand et al., 1999). In metapopulation theory, movement is particularly critical

because provides the means for (meta)population persistence over time (Hanski, 1998).

Most metapopulation modeling approaches describe inter-site connectivity using

pair-wise Euclidean distance between patches (Dias, 1996; Hanski and Hanski, 1999).

Using these distances typically results in the simplifying assumption of a symmetric

movement pattern in which the probability of moving, p, from patch i to patch j is

the same as moving in the opposite direction (i.e. pij = pji ). A symmetric pattern of

movement may be expected when all the factors that affect dispersal are identical in all

directions (Bode et al., 2008; Gustafson and Gardner, 1996); however, such symmetry

might be rarely observed in nature. Instead, an asymmetric pattern of movement

(e.g. pij = pji ) may more likely emerge from organisms moving across spatially

heterogeneous environments (Ferreras, 2001; Prevedello and Vieira, 2010). Recent

models that incorporate this complexity suggest that movement asymmetries can have

substantial effects on metapopulation dynamics (Armsworth and Roughgarden, 2005;

Vuilleumier et al., 2010; Vuilleumier and Possingham, 2006).

A variety of mechanisms may give rise to asymmetrical pattern of movement. For

instance, asymmetric movement may be the result of environmental heterogeneity such

as variation in habitat quality or patch size (Ferreras, 2001; Gustafson and Gardner,

1996; Holt, 1996; Pulliam, 1988). Individuals may have adapted to seek better habitats

by actively dispersing from low- to high-quality patches (often larger in size; e.g.,

29

Ferreras, 2001; Gustafson and Gardner, 1996), or high quality patches may be net

producers of emigrants to low-quality patches due to density dependence (e.g., in

source-sink dynamics; Holt, 1985; Pulliam, 1988).

Other mechanisms may likely differ between passive and active dispersers. In

passive dispersers, asymmetric movement may be the result of advection through a

directional medium (Armsworth and Bode, 1999; Keddy, 1981; Schick and Lindley,

2007; Treml et al., 2008). In active dispersers, asymmetries may be the result of

changes in the organisms movement behavior due to environmental cues (e.g.

taxis; Compton, 2002). For instance, many species of insects are attracted to visual

or chemical cues that direct their movement (Prokopy and Owens, 1983). Some

Homoptera, Hymenoptera and Hemiptera direct their movement upwind while tracking

volatile chemical signals from vegetation (anemotaxis; e.g. Compton, 2002; Moser et al.,

2009; Williams et al., 2007). The likelihood of an individual to move to a particular patch

may also depend on the patch occupancy state or the abundance conspecifics (Serrano

and Tella, 2003; Serrano et al., 2001; Smith and Peacock, 1990).

Here we use a combination of an observational and experimental approach to

test for mechanisms of asymmetric movement in the cactus-feeding insect, Chelinidea

vittiger. This species is entirely dependent on patchy prickly pear cactus (Opuntia

spp.), where it feeds, breeds and aggregates (Miller et al., 2012). Recently, Fletcher

et al. (2011) found that movement in this species across a patch network was highly

directional, which altered assessments of landscape connectivity. We hypothesized that

three primary mechanisms may cause directed movements: (1) positive anemotaxis

(movement upwind), (2) movement toward larger patches, and (3) conspecific attraction.

Wind may help C. vittiger detect volatile olfactory cues of Opuntia, potentially resulting

in individuals directing their movements towards patches that are located downwind

(Schooley and Wiens, 2004). Similarly, C. vittiger may actively search for larger patches

of Opuntia, because of enhanced resource availability (see Schooley and Wiens, 2005).

30

Because of the aggregative behaviors of this species (Fletcher and Miller, 2008; Miller

et al., 2012), conspecific attraction may occur and also cause directed movements.

To address these mechanisms, we first re-analyzed data from a mark-recapture study

across a 56 patch network (Fletcher et al., 2011) to test for the magnitude of movement

asymmetry and if wind direction or patch area may explain observed variation in

movement across the patch network. Second, we conducted a translocation experiment

in which we manipulated patch area, wind advection and the presence of conspecifics to

determine if, and the extent to which, C. vittiger biases movements in response to these

factors.

2.2 Methods

2.2.1 Study Area and Focal Species

The study was conducted at the Ordway-Swisher Biological Station (29.4◦N,

82.0◦W), in Putnam County, Florida, USA. In this area, C. vittiger uses Opuntia hu-

mifusa, primarily occurring in old-field habitats, which are common at the station. O.

humifusa is native to the eastern U.S. and typically grows in dry, sandy soils in sandhills,

old fields, prairies, or scrub.

We focused our research on adult movements of C. vittiger. Adults of C. vittiger are

winged, but rarely fly; instead, adults typically walk between cactus patches through

an unsuitable matrix (De Vol and Goeden, 1973; Schooley and Wiens, 2004). Median

dispersal ranges from 1–2.5 m/day (Schooley and Wiens, 2004, 2005), which makes

movement tractable and an ideal system to study short-term dispersal (Figure 2-1b).

2.2.2 Study Design

2.2.2.1 Mark-recapture

We re-analyzed data published by Fletcher et al. (2011) in the context of connectivity

modeling. The data consisted of individual movements of marked C. vittiger in all

patches (n = 56) inside a 30 × 30 m plot (Figure 2-1a). Patches were defined following

Schooley and Wiens (2004), where cladodes (i.e., cactus pads) < 25 cm apart were

31

Figure 2-1. Observed inter-patch transitions of C. vittiger in a 30× 30 m plot. (a) Greencircles represent patches that included movement while black circlespatches that did not. Note how most of the connections go in only onedirection showing a pattern of high asymmetric movement. (b) Distribution ofmovement distances divided by the number of patches available formovement (availability). Circle size is proportional to patch size.

considered the same patch. Censuses were conducted from September 2008 to

November 2009 every 2–3 weeks (except during the winter) for a total of 21 censuses.

During each census, all individuals on each patch were counted (both nymphs and

adults), and all adults were sexed and individually marked on the pronotum with nontoxic

permanent marker. Even though this plot was not a closed system, additional recapture

rates in the surrounding area were low.

To address the mechanisms for asymmetric movement, we estimated patch area

using the equation for an ellipse (area = major radius × minor radius × π; Schooley

and Wiens, 2005). The major and minor radii were measured using a ruler. We used

data from the nearest weather station to the site (< 1km), to estimate wind direction. We

averaged daily wind direction from January 1, 2009 to November 27, 2009 to produce

a single estimate of average wind direction relative to each pair-wise combination of

patches in the network.

32

2.2.2.2 Field experiment

We used a random block design in which each block contained four plots, one for

each treatment (wind, patch area, and conspecifics) and a control (Figure 2-2). Each

plot included four patches of Opuntia humifusa cladodes located in the four cardinal

directions. The release patch was located in the center of the plot and consisted of

a single cladode (Figure 2-2). We used a small release patch to catalyze movement

(Castellon and Sieving, 2006) because a single cladode is below the minimum patch

area requirements for C. vittiger (Schooley and Wiens, 2005). All experimental patches

were located 1-m away from the release patch, which is within the perceptual range of

this species (Fletcher et al., 2013).

All plots consisted of 32 cladodes that were distributed among four patches,

arranged in different proportions between the patches depending on the treatment. In

the wind and conspecific treatments, and in the control plot, each experimental patch

was composed of eight cladodes per patch (Figure 2-2a,b,c). In the area treatment,

one patch consisted of 16 cladodes (large patch); another consisted of eight cladodes

(medium patch) and two of four cladodes (small patches; Figure 2-2d). These cladodes

were fresh, with no visible feeding marks and were collected in nearby old-fields (< 100

m).

The wind treatment consisted of placing a battery operated Caframo Kona black

color (7cm diameter) fan located 5–10cm away from a randomly selected experimental

patch and 10cm from the soil with wind directed towards the release patch (Figure

2-2b). The fan was operated by a 12V battery and continuously blowing during the

sampling period (48 hrs). Wind speed at the wind treatment experimental patch was

14.8 km/h and 4.7 km/h at the release patch. These wind variation included speeds

previously reported in the literature in a study that found a strong relationship between

wind direction and C. vittiger movement (0.5–7.75 km/h; Schooley and Wiens, 2005).

33

Figure 2-2. Diagram showing the study design. Experimental blocks were composed offour plots, one for each treatment (a) control, (b) wind, (c) conspecifics, and(d) variations in patch area. The study included 3 blocks of this design andall treatments were assigned randomly.

34

The conspecifics treatment consisted of placing one individual male in one patch

and one individual female in another patch in the same plot (Figure 2-2c). We included

males and females separately because previous studies have found differences

in aggregative behavior for males and females (Miller et al., 2012). Treatments

were assigned to the patches randomly. The individual cladodes that contained the

conspecifics were enclosed with fine mesh to prevent the individual from leaving the

patch (Miller et al., 2012). The mesh allowed for visual and/or pheromonal cues to be

potentially detected by the released individuals.

We considered control patches to be included in the treatment plots that were not

subject to the treatment. For example, in the wind treatment plots, only one out of the

four patches had a fan, such that the other three patches in the plot were considered

control plots. Similarly, in the conspecific treatments the two patches that did not have

conspecifics were considered control patches. In the area treatment, patches with eight

cladodes were considered control also. In addition to the control patches, we included

in the block design a control plot in which each patch was composed of 8 cladodes and

releases and censuses were done in the same way as in the treatment plots.

We also performed two types of procedural controls, one for the conspecifics

and another for the wind treatment, to ensure the observed rates of immigration for

each treatment were driven by the treatments and not an artifact of incorporating a

fan or mesh in the plot. We conducted the procedural control experiment for the wind

treatment in the same plot plots (i.e. same location and same cactus patches in each

block) where the wind experiment took place; the only difference was that the fan was

placed blowing in the direction away from the release patch. By doing this, we keep the

potential effects of having a non-natural object in the landscape (i.e. black fan producing

noise), but removed the effect of wind. Similarly, we conducted the procedural control

experiment for the conspecifics treatment in the same plots where the conspecific

treatment experiments took place (i.e. different plots than the wind procedural control

35

experiments). We enclosed the same two patches at used for the conspecific treatments

with mesh, with the only difference being that a conspecific individual was not included.

Sampling was done in the same was as in the other experimental treatments (see

below).

At the beginning of the study, plots were depleted of any other cactus and

individuals of C. vittiger within 10 m of the release patch. Plots were located at least

35 m apart from the nearest experimental plot. Locations of the treatments at both the

plot and patch level were assigned at random.

Individuals were captured in nearby cactus patches (< 100 m). These were

bred and reared in deli containers in a screen house. Nymphs were raised in groups

with fresh and healthy O. humifasa cladodes. As soon as they became adults, they

were marked on their pronotum using non-toxic permanent marker and transferred to

individual containers until they were released in the experimental plots. A greenhouse

experiment found that marking individuals in this way does not alter survival rates

(P > 0.5; Fletcher, unpublished).

In each trial, two males and two females were released at the released patch. The

location of each of these individuals in the plot was censused after 24 and then after 48

hours of being released. Each observed individual was removed from the plot as soon

as its location was noted. We conducted 7 (5 for experimental treatments and 2 for the

procedural control) trials in each of the three blocks between July and November 2012.

Previous studies have found that matrix resistance may be an important determinant

in the movement of C. vittiger (Schooley and Wiens, 2005). To control for this effect, we

measured the structure of the matrix from the release patch to each experimental

patch. We measured matrix structure at 15 points within a 100 × 50cm strip transect

(5 measures in each edge of the transect and 5 in the center spaced every 20 cm)

between the release patch and the experimental patch. For each point, we quantified the

number of times vegetation touched a graded pole in each 10 cm category (vertically)

36

to a maximum of 90 cm. We averaged the number of layers and the maximum height of

each set of 15 measures to characterize the matrix between the release patch and each

experimental patch. Both measures (number of layers and maximum height) were highly

correlated (rp = 0.85); therefore, here we present only the data for maximum height to be

consistent with other investigations on this species (Schooley and Wiens, 2005).

2.2.3 Analyses

2.2.3.1 Mark-recapture

We represented the movements of C. vittiger as a matrix W were wij represents

the total number of movements from patch i to patch j . We calculated the proportion

of symmetric links as the number of links in W in which wij − wji = 0 (i.e. the number

of movements was the same in both directions) divided by the total number of links

that had movement. While Fletcher et al. (2011) noted that movement appeared highly

directional, they did not assess if this putative pattern was greater than what would be

expected based on chance. To test if this proportion was different from the proportion

of symmetric links expected by chance, we compared the proportion of symmetric links

in this matrix of movements W, to the proportion of symmetric links in 1000 randomly

generated movement matrices. These random matrices had the same number of

movements as W, but the positions of the links were randomly assigned among the

links that had movement (i.e. shuffling the weights of the matrix while keeping the same

topology).

We used a linear regression model to test for the relationship between a pattern of

asymmetric movement and variations in patch size and the angle of wind direction. The

response variables for both of these models was the difference in movement between

i and j , �wij = |wij − wji |. Similarly, we calculated the difference in patch area as

�Aij = |Ai − Aj | where Ai is the area of patch i . We also calculated the difference (δij ),

between the angle of wind direction and the angle between patches i and j , in radians

37

with respect to the horizontal axis. For simplicity, δij is scaled by π to constrain δij ∈ [0, 1].

See Chapter 1.

2.2.3.2 Field experiment

To test for the effect of the treatments (variations in patch area, wind and conspecific

presence) driving the directed movements of C. vittiger, we modeled the rate of

movement from the release to each of the experimental patches. This rate of movement

described the proportion of the released individuals that moved from the release patch

to an experimental patch in a period of 48 hrs. This is an appropriate response given

that experimental patches were located 1-m away from the target patch and the average

individual movement rate is < 1 m/day (Figure 2-1b).

Each treatment was analyzed independently because variation within plots for

different treatments made pooling data across treatments impractical. In the conspecific

treatment, we fitted separate models to describe the rate of movement of males, females

and another for both sexes together, because previous research has shown different

aggregative behaviors between males and females C. vittiger (Miller et al., 2012). For

each treatment consideration (area, wind, and conspecifics male, females and both),

we fitted five types of models that represented different hypotheses explaining the

rate of movement. An (1) intercept-only model representing the null hypothesis that

the experimental treatments are not driving the movements of C. vittiger. We, also

fitted a model representing the (2) treatment. Because each treatment was model

independently, the variable used was different for each treatment. In the area treatment,

variation in patch area was represented by the number of cladodes in each patch (4,

8 or 16 cladodes). In the wind treatment, the presence of wind was represented by a

binary variable indicating the presence or not of the fan in the experimental patch. In

the conspecific treatment (response of males, females and both), the presence of a

conspecific was treated as categorical variable with three factors: male, female and

control. The effect of the matrix, represented by the maximum average vegetation height

38

between the release patch and the experimental patch, was incorporated into models

both as a (3) single main effect, and also as an (4) additive and (5) interacting covariate

with the treatment.

We used a negative binomial regression to model the rate of movement toward

experimental patches with an offset representing the total number of individuals

released (4 individuals, except for the response of males and female conspecifics

in which two individuals were released). The negative binomial distribution was an

appropriate choice because the count data showed over-dispersion. In addition, we

compared model fit to a Poisson distribution and the negative binomial resulted in better

fit (all negative binomial models resulted in AIC values < 2 units than the corresponding

Poisson model). In a preliminary analysis, we incorporated the date of the census,

block and plot as random effects, but their variance resulted < 10−8 and thus their

incorporation did not significantly increase model fit.

Negative binomial regressions were fitted by maximum likelihood using the MASS

package in R. We used Akaikes Information Criterion (AIC) to compare model fit and

choose the most parsimonious model.

2.3 Results

We observed a total of 70 movements in the 30 × 30m patch network (see Fletcher

et al., 2011, for more details). The proportion of symmetric connections in this network

(0.04) was lower than the average in the randomly generated movement networks

(0.10 ± 0.04; z = −1.77,P = 0.04). Patch area ranged from 1.25–252 cm2. Average

wind direction was 131 ± 99◦, with average prevailing wind speeds of 4.07 ± 3.03 km/hr.

We found a weakly significant, positive relationship between the difference in movement

(�wij ) and difference in patch area (�Aij ) between patches (R2 = 0.09,P = 0.05;

Figure 2-3). In contrast, there was no significant relationship between the difference in

movement and increasing deviation from the angle of wind direction (R2 = 0.03,P =

0.25; Figure 2-3).

39

Figure 2-3. Linear regression analysis. Figure shows the relationship between thedifference in movement (�wij ), and (a) the difference in patch area (�Aij ) and(b) deviation from average wind direction (δij ) in the mark-recapture study

In the field experiment, a total of 240 individuals were marked and released of which

86 were recaptured. We recaptured 19 individuals in the area treatment, 16 individuals

in the wind treatment, 25 (14 females and 11 males) in the conspecifics treatment and

26 in the control plot. The most parsimonious model explaining the rate of movement

in the area treatment included the number of cladodes (patch size) interacting with the

maximum height of the vegetation in the matrix as covariates (Table 2-1). This model

predicted a relative small and constant rate of movement to all size patches when the

matrix was low, but a higher rate of movement towards bigger patches when the matrix

vegetation was higher than the median (Figure 2-4). The null model (intercept only) was

the most parsimonious model explaining the rate of movement in the wind treatment

and control plots. A null model was also the most parsimonious explaining the rate of

movement in the conspecific treatments when modeling the response of both males and

females together and when modeling the movement of females. When modeling the

40

rate of movement of males in the conspecifics treatment, the most parsimonious model

incorporated the presence of conspecifics; however, this model had no better fit than the

null (intercept-only) model (�AIC < 2; Table 2-1). The null (intercept-only) model was the

most parsimonious model explaining the rate of movement in both procedural controls

(Table 2-1).

Figure 2-4. Partial relationships of the best fit model explaining the rate of movement inthe area treatments of the field experiment. This model included patch area(number of cladodes) interacting with vegetation height. Lines represent 1stquartile (low matrix height), median (moderate matrix height) and 3rdquartile (high matrix height) of maximum vegetation height. Shaded areasrepresent the 95% confidence intervals

2.4 Discussion

Asymmetric dispersal may be the rule more than the exception in nature. While

recent metapopulation models suggest that movement asymmetries can have

substantial effects on metapopulation dynamics, empirical tests of dispersal asymmetries

across landscapes remain rare with even fewer tests of the mechanisms driving such

patterns. We found that movements of C. vittiger were generally asymmetric and better

explained by movements from small-to-large patches.

41

Table 2-1. Model selection for each experimental treatment. The number of cladodes isrepresented by n.cladodes, matrix refers to the maximum height of vegetationin the matrix

Treatment Sex Response Model k �AIC DevianceArea Both N.cladodes*Matrix 4 0 36.1

N.cladodes+Matrix 3 1.4 38.8N.cladodes 2 3.5 37.4Matrix 2 12 36Intercept only 1 12.1 37.3

Wind Both Intercept only 1 0 33.5Wind 2 1.2 33.7Matrix 2 1.9 33.7Wind+Matrix 3 3.4 33.8Wind*Matrix 4 5.6 33.6

Conspecifics Both Intercept only 1 0 38.3Matrix 2 1.4 38.6Conspecifics 3 2.7 38.7Conspecifics+Matrix 2 4.6 38.3Conspecifics*Matrix 3 9.2 38.1

Males Conspecifics 3 0 35Intercept only 1 1.7 36.1Conspecifics+Matrix 4 2.2 35Matrix 2 3.8 36.1Conspecifics*Matrix 6 4.7 33.9

Females Intercept only 1 0 60.2Conspecifics 3 1.6 57.5Matrix 2 2 60.1Conspecifics+Matrix 4 3.6 57.3Conspecifics*Matrix 6 5.6 54.9

Control Both Intercept only 1 0 54.59Matrix 2 2.1 54.6

Procedural Cont. Both Intercept only 1 0 20.8Fan 2 0.68 21.78

Intercept only 1 0 20.35Mesh 2 1.5 20.35

42

Both the observational and experimental results showed no support for prevailing

or artificial wind driving movement decisions. Even though previous studies have

shown that C. vittiger directs its movements towards prevailing winds (Schooley and

Wiens, 2003) in open-short grass prairies in Colorado, Fletcher et al. (2013) also

found little support for this wind hypothesis while studying the perceptual range of C.

vittiger in our study area. They suggested that wind speeds at old-field habitats in our

study area are relatively lower than in the open short-grass prairie habitat than where

(Schooley and Wiens, 2003) conducted their study. Nevertheless, in our experiments,

artificial wind speeds at ∼ 10cm from the ground were within the range of those in

the open short-grass prairie (Schooley and Wiens, 2003). This discrepancy may be

explained by at least two alternative hypotheses related to habitat differences between

open short-grass prairie in Colorado and old-field habitats in Florida. First, C. vittiger

populations at OSBS may have developed alternative ways to detect patches that

do not include relying on wind because wind is not a prevalent factor in this system.

Alternatively, the type or amount of olfactory cues may differ between O. humifusa and

O. polycantha (the Opuntia species in Schooley and Wiens, 2003).

We found no strong support for the presence of adult conspecifics driving

the directed movements of C. vittiger. Nevertheless, multiple studies have found

that this species uses social information to guide its behavior (e.g., Fletcher and

Miller, 2008; Miller et al., 2012), suggesting that there may still potential for other

(non-tested) conspecific interactions driving directed movements. Given that conspecific

interactions have been documented for both aggregative and non-aggregative species

(e.g., Fletcher, 2009; Serrano and Tella, 2003), and can affect colonization rates

(Hahn and Silverman, 2006), functional connectivity estimates (Zeigler et al., 2011)

and metapopulation dynamics (Alonso et al., 2004; Ray et al., 1991), its potential

implications as a driver of asymmetric dispersal remains an alternative hypothesis to be

further tested.

43

Patch size has been found to be an important determinant of patch immigration

rates (Bowler and Benton, 2005). Both observational and experimental results showed

support for the hypothesis that variation in patch area is a driving mechanism of directed

movements that result in an asymmetric pattern of movement. In the field experiments,

the rate of immigration increased with increasing patch size. Two potential mechanisms

may account for this pattern. First, organisms may be dispersing by a passive-diffusive

process (Kindvall and Petersson, 2000; Travis and French, 2000), which makes large

patches (i.e. patches with greater circumference) more prone to be detected by a

randomly moving individual. On the other hand, there is ample evidence showing that

animals tend to move toward larger patches (e.g., Diffendorfer et al., 1995), suggesting

that other factors may be covarying with patch size and individuals may have strong

preference for larger patches (Bowler and Benton, 2005). Our results suggest that C.

vittiger prefer large patches. If individuals of C. vittiger were randomly finding patches

in the area treatment, we would have expected the immigration rate to be proportional

to the linear dimension of the patch, because all patches were located at the same

distance from the release patch (Bowman et al., 2002). Given that in our experimental

design the linear dimension of the patches was proportional to the number of cladodes,

we would have expected the average movement rate towards the 16 cladode patches to

be 4 times that of the 4-cladode patches. Instead, we found that the rate of movement

towards the 16 cladodes patch was 11 times that of the 4 cladodes patches (Figure

2-5c). Moreover, the average movement rate towards the 16 cladode patches was 2.5

times that of the 8 cladode patches suggesting a non-linear relationship between patch

size and immigration rate (Figure 2-5c). C. vittiger may direct their movements towards

large cactus patches simply because larger patches provide a greater amount of food

resources. Alternatively, large patches may provide more spines for females to lay eggs

or may include complex vertical structures that may serve as refugia from predators,

thereby decreasing predation risk.

44

Figure 2-5. Movement rate (± SE) at the patch level in the experimental treatments. Thesolid gray line represents the mean immigration rate for patches in thecontrol treatment and the dashed line the associated standard errors

A combination of matrix characteristics and other landscape features such as patch

size and isolation are the most important drivers of inter-patch movements (Bender and

Fahrig, 2005). Moreover, variations in matrix permeability and landscape characteristics

may result in an asymmetric pattern of dispersal

The cost of dispersal is an important determinant of whether dispersal leads to

fitness benefits (Bowler and Benton, 2005). Our study system could be an example in

which, even though increased matrix structure may provide resistance to movement

(Schooley and Wiens, 2005), it may increase survival by decreasing predation risk.

Predation estimates for C. vittiger are scarce, but spiders have been shown to predate

them and other cactus-feeding insects (De Vol and Goeden, 1973; Miller, 2008).

Regardless of treatment, most plots in which we observed no recaptures had little or no

matrix structure. These individuals either were predated or permanently emigrated from

the experimental plot, suggesting that the risk of longer distance dispersal may be less

than the risk of being exposed moving through the matrix without cover. On the contrary,

45

moving through a more complex matrix may provide cover in which individuals could

move more safely and assess their surroundings for a patch to settle. A similar pattern

has been found in other taxa. For instance, predation risk for forest birds is higher while

trying to cross an agricultural matrix than secondary forests with relatively more vertical

structure

Inter-patch matrix heterogeneity and movement behavior has strong implications

for population dynamics and biodiversity patterns (Armsworth and Roughgarden, 2005;

Salomon et al., 2010). We found that patch size and matrix structure were the most

important predictors of inter-patch movements and immigration rate in C. vittiger, leading

to directed movements that resulted in an asymmetric pattern of dispersal. Given that a

relationship between patch size and matrix heterogeneity is common in many taxa, our

results support the idea that asymmetric dispersal may be common in nature (Gustafson

and Gardner, 1996). Variation in landscape features such as matrix structure and

patch size may be important drivers of asymmetric dispersal and hence have broad

implications for (meta)population dynamics.

46

CHAPTER 3OPTIMAL SITE SELECTION STRATEGIES: PROTECTING AGAINST WORST-CASE

DISTURBANCE SCENARIOS

3.1 Introduction

Human-induced habitat loss and global climate change have been identified as

some of the most important large-scale drivers of biodiversity declines (Fahrig, 2003;

Lindenmayer and Fischer, 2006; Mawdsley et al., 2009; Sala et al., 2000); however,

there is much uncertainty about their future ecological consequences (Elith et al., 2002;

Kujala et al., 2012; Moilanen et al., 2006a; Thomas et al., 2004). There is an increasing

interest in identifying conservation strategies that can ameliorate some of the expected

future impacts of these drivers, while at the same time accounting for uncertainties

(Moilanen et al., 2006b), because failing to account for these uncertainties may result in

inadequate conservation and management actions (Regan et al., 2005).

Managing networks of protected areas is one of the most common biodiversity

protection strategies (Heller and Zavaleta, 2009; Kukkala and Moilanen, 2012; Mawdsley

et al., 2009). Often, these area networks are designed under a systematic conservation

planning framework that employs quantitative tools for strategically selecting reserves.

These quantitative tools identify areas for the representation and/or persistence of

biodiversity given socioeconomic or biological constraints. These problems can be

described as linear or integer optimization programs (that maximize or minimize a linear

function over a set of linear constraints) and benefit from a wide array of models and

algorithms that have been developed in the operations research field (Cabeza and

Moilanen, 2001; Kukkala and Moilanen, 2012; Margules and Pressey, 2000; Moilanen,

2008; Moilanen et al., 2009). The end goal of these models is to determine the optimal

set of resource patches to protect, given a conservation objective. A wide array of

objective functions have been employed in these optimization models, such as the

maximization of species representation, the maximization of species persistence or the

minimization of costs (Cabeza and Moilanen, 2001). Recently, the focus of these models

47

have shifted from these pattern-based objectives to process-based objectives that drive

ecosystem functioning and species persistence (Possingham et al., 2005).

Connectivity is an ecological process that promotes species persistence in

fragmented landscapes, the maintenance of biodiversity, and other ecosystem

processes (Crooks and Sanjayan, 2006; Heller and Zavaleta, 2009; Lindenmayer

et al., 2008; Possingham et al., 2005; Taylor et al., 1993; Tischendorf and Fahrig,

2000). Improving connectivity is one of the most frequent recommendations when

managing landscapes for future environmental changes because highly connected

sites are expected to have greater persistence and population densities (Gillson et al.,

2013; Hanski, 1998; Heller and Zavaleta, 2009). The rationale behind this idea is that

increasing connectivity will allow individuals to respond to future changes by moving into

new regions (Krosby et al., 2010). Even though connectivity is commonly considered

in a variety of management actions, recently its relevance for systematic conservation

planning has been acknowledged (Fagan and Lutscher, 2006; Moilanen et al., 2007,

2009; Wilson et al., 2009) by incorporating a penalizing factor in the objective function to

promote the selection of sites that are closer to each other (e.g., Ball et al., 2009; Beger

et al., 2010).

Decisions based on information from reserve selection tools are, however, subject

to different types of uncertainties (Pressey, 2004). These uncertainties can be generally

classified into epistemic (related to lack of knowledge or data) or linguistic (due to

misunderstanding of concepts or terms) (Regan et al., 2002). Statistical epistemic

uncertainties include those arising from insufficient data. Non-statistical epistemic

uncertainties arise from plausible assumptions made in the planning process such

as static environmental or biological conditions, such as those related to static

species distributions or static species interactions (Moilanen et al., 2006a). Yet

future anthropogenic actions, climate change or natural disasters may change both

environmental and biological conditions in uncertain ways (Elith et al., 2002; Thomas

48

et al., 2004). Uncertanity analysis is rarely incorporated into systematic conservation

planning. Some uncertainty analysis methods such as sensitivity analysis, scenario

building and information gap approaches have been proposed as feasible ways to

address statistical uncertainties (Moilanen et al., 2006b; Regan et al., 2005), but we

currently lack a formal approach to account for non-statistical epistemic uncertainty.

Here we describe a novel application of network flow interdiction models, that

deals with non-statistical epistemic uncertainty by optimally determining which patches

to protect based on a worst-case disturbance scenario for connectivity. We describe

the modeling framework and apply it to two empirical systems to show its application,

assumptions, sensitivity, and potential modifications.

3.2 Methods

3.2.1 Network Flow Interdiction Models

Ecological spatial networks consists of nodes that represent spatial locations such

as forest patches, grasslands, ponds, lakes, or animal colonies and links (“arcs” from

now on) that describe some flow between node pairs. In ecological spatial networks,

arcs typically describe the flow of individuals (proportion or the probability of movement)

between nodes.

Arcs and nodes may be vulnerable to disturbances that impair flow in the network.

The purpose of network flow interdiction models is to assess vulnerabilities in networks

and plan protective measures (Brown et al., 2006; Smith, 2010). These models analyze

the worst-case scenario for network infrastructure by assuming an intelligent adversary

that has limited resources (i.e. a budget on how many arcs or nodes can be disturbed).

The adversary’s objective is to maximize damage to the network. The end goal of

network flow interdiction models is to determine the value of protecting a given set

nodes or arcs that will best maintain network flow after this worst-case disturbance.

Even though interdiction models are widely applied in military and anti-terrorism

scenarios, they can be generally applied to any kind of scenario when a decision maker

49

is concerned with designing a robust system that withstands any possible combination

of disruptions. This viewpoint is often taken when analyzing critical systems (e.g., power

systems and national defense infrastructure), where resilience to worst-case disruptions,

accidental or intentional, is vital.

In ecological spatial networks, large-scale disturbances, such as land use or natural

disasters, may decrease connectivity. This decrease may have important ecological

consequences. For instance, the loss of an important patch in a metapopulation may

cause local extinctions or a decrease in metapopulation persistence (Hanski, 1998).

Given the uncertainty releated to future disturbances in natural systems, network flow

interdiciton models are an appropriate modeling tool to assess the vulnerability of an

ecological spatial network to a potential worst-case scenario.

Here we applied a defender-attacker model (i.e., Stackelberg game Von Stackelberg,

1952), which involves two players: a leader and a follower. The leader picks a subset

of patches to protect, which become immune to disruption. The follower then acts

to optimally disrupt a subset of unprotected patches with the goal of minimizing

connectivity. We emphasize that the follower need not be a malicious or sentient agent,

but rather simply a representation of a worst-case scenario given the leaders protection

strategy.

3.2.2 Model Description

In the context of connectivity conservation, a common goal is to identify a land

protection strategy given a budget, under the assumption that connectivity may be

reduced in the future. Disturbances may reduce connectivity by limiting the ability of

individuals to move within the landscape as a result of decreased matrix permeability or

patch quality (Kindlmann and Burel, 2008). Here we consider the case where protection

and disturbance occurs at the patch (node) level; however, the modeling approach can

be easily modified to incorporate disturbances to the matrix (arcs), see Section 3.4.1.

50

Decreasing patch-quality reduces connectivity by increasing patch level mortality and

decreasing movement.

The modeling approach consists of three interrelated stages: assessment,

disturbance, and protection (Figure 3-1). In the assessment stage we use patch

transition probabilities (including probabilities of staying in a given patch) and individual

survival associated with each patch (Figure 3-2) to quantify connectivity. These

transition probabilities and survival are often estimated from mark-recapture data in

a multi-state modeling framework (Nichols and Kendall, 1995; White et al., 2006). We

describe these transitions as a reducible Markov chain (see Section 3.2.2.2) to quantify

total individual life expectancy. This connectivity measure describes the amount of time

an individual remains alive and moving through the network.

The usefulness of various connectivity measures for management and conservation

has been debated in the literature (Calabrese and Fagan, 2004; Kindlmann and

Burel, 2008). Generally, measures that incorporate realistic estimates of movement

in combination with integrative fitness currencies (e.g., survival) are preferred over

measures that make plausible assumptions about movement (Belisle, 2005). Hence,

quantifying connectivity as total individual life expectancy is an appropriate measure

because it incorporates precise between-patch movement probability estimates (from

mark-recapture), site fidelity, and patch-specific survival.

A protection strategy is attained by optimizing disturbance and protection actions.

First, a worst-case scenario for connectivity is assessed by determining an optimal

disturbance action (a set of patches to disturb) that will minimize connectivity (total

individual life expectancy). These disturbance actions may include factors that decrease

patch quality such as deforestation, fire, hurricanes, flooding or changes in abiotic

factors. Second, we simulate an optimal protection action (an optimal set of patches

to protect) that will mitigate the worst-case scenario by maximizing the amount of

connectivity loss prevented (i.e. maxmin bilevel problem). These protection measures

51

include actions that prevent patch deterioration such as protecting against future

development, reduce human use (e.g., reduce fishing or hunting), managing for fire

or control predator populations. In this context, we are uncertain about how future

disturbances will affect connectivity, hence, we protect against the worst-possible

scenario (Figure 3-1).

Figure 3-1. Schematic diagram of the three stages of the modeling process:assessment, disturbance and protection. The protection and disturbancemodels interact recursively to assess the optimal set of patches to protectgiven a worst-case scenario for connectivity.

3.2.2.1 Case studies

We applied the network flow interdiction modeling framework to two published

datasets: adult Roseate Terns (Sterna dougallii) living in northeastern USA (Spendelow

et al., 1995) and Skipjack Tuna (Euthynnus pelamis) living the western Pacific (Hilborn,

1990). These datasets included survival, site fidelity and between-patch movement

estimates.

52

Roseate Tern colonies were located in Bird Island (BI) in Marion Massachusetts,

Great Gull Island in New York (GG), Falkner Island in Connecticut (FI) and Cedar Beach

(CB) in Long Island, New York (Figure 3-2a). Individual adults were captured yearly from

1988–1990 in these sites using treadle traps and individually marked with color bands.

See Spendelow et al. (1995) for more details on potential bias due to colorband loss and

fading, and how they controlled for it. We used the arithmetic mean for 1988–1990 of

their survival, transition and colony size estimates (see below).

Skipjack Tuna were tagged and recovered (>150,000 individuals tagged), monthly

from 1977–1981, in seven fisheries sites that included waters near Palau (PAL), Yap

(YAP), Truk (TRK), Ponape (PON), Papua New Guinea (PNG), Solomon Islands (SOL)

and the international waters between YAP, TRK, PON and Indonesia (Figure 3-2b).

The data analyzed were collected as part of the Skipjack Tuna survey and assessment

program of the South Pacific Commission. Fish tagged and/or recovered from Fiji were

excluded from their analyses. For more details on the data or the estimates see Hilborn

(1990).

Both datasets varied in the number of patches, scale of movement, survival, and

level of phylopatry. Survival was more variable for the Skipjack Tuna (0.664–0.99) than

the Roseate Terns (0.74–84). Phylopatry was generally lower in the Skipjack Tuna

(0.60–0.87) than in the Roseate Terns (0.88–0.98). Both species had similar ranges of

between-patch movement rates (Skipjack Tuna: 0–0.11; Roseate Terns: 0–0.09).

Survival and transition probabilities for both species were estimated using variations

of the Arnason-Schwarz multi-state model. In this multi-state modeling framework,

movement is modeled as a first-order Markov chain, which models the stochastic

movement patterns of individuals among patches, accounting for the possibility that an

individual dies at each time step. This multi-state approach allows the decomposition

of transition probabilities into survival (Si ) and movement (�qij ). Note that in this case

mortality happens before an individual moves. See Spendelow et al. (1995) for

53

comparison with alternate modeling approaches to estimate survival and transition

probabilities.

Figure 3-2. Spatial network representation of Roseate Tern colonies in northeast USAand Skipjack Tuna in the Western Pacific. The color of the arcs correspondto the probability of movement from patch i to j . Note that self arcs had thelargest probabilities showing high site fidelity.

3.2.2.2 Connectivity assessment

We calculate functional connectivity as the total individual life expectancy, which

describes the number of time-steps an individual remains alive in the network. To

add more biological reality to this measure, we weight total life expectancy by patch

abundance to account for the number of individuals subject to conditions in patch i . At

each time-step an individual has some probability of moving to another patch, staying in

the same patch, or dying. The units of this measure are the same as the input data. For

example, if the estimated transition probabilities correspond to a period of two weeks,

then the life expectancy describe how many periods of two weeks an individual will

survive.

We calculate individual life expectancy by modeling transition probabilities using a

reducible Markov chain. Accordingly, we consider a landscape composed of discrete

patches denoted by P = {1, ..., |P|}. The Markov chain contains |P| + 1 states, where

the additional state is an absorbing state d that corresponds to mortality (Figure 3-2).

54

The Markov chain states are given by M = P∪{d}. We initially represent this Markov

chain as a matrix �Q, comprised of elements �qij for i ∈ P and j ∈ P, and �qdi = 0,

�qid = 1 − Si for each i ∈ P, and �qdd = 1. Each column of �Q represents a patch in the

network, except for the last column, which represents mortality. By assumption we have

that �qij ≥ 0 for all i , j ∈ M and∑

j∈M �qij = 1 fro all i ∈ M.

In the absence of any disturbances, the total individual life expectancy (z) can be

calculated as follows. Let p be a |P|-dimensional column vector whose i th element, pi ,

describes the abundance in patch i ∈ P, and let �A be the submatrix of �Q obtained by

removing the last row and column of �Q (Ross, 2006). See Appendix B for additional

mathematical details of the connectivity assessment.

z = pT (I− �A)−11, (3–1)

where I is the |P| × |P| identitiy matrix and 1 is the |P|-dimensional column vector of

all ones. In the unweighted case p = 1, and hence life expectancy is based solely on the

transition probabilities.

3.2.2.3 Disturbance: simulating a worst-case scenario

The disturbance model identifies a worst-case scenario for connectivity by

selecting a limited number of unprotected patches to disturb that will minimize total

life expectancy. Here we define disturbance as any action that may decrease patch

quality. To describe the disturbance decision we use the binary variable yi , which equals

1 if patch i ∈ P is disturbed, and equals 0, otherwise. Disturbance to any given patch will

affect the individual life expectancy by changing the transition probabilities in the Markov

chain. We define qij as the updated transition probability of going from state i ∈ M to

state j ∈ M. The magnitude of this change depends on the disturbance regime, which

is described by the parameters δij , γij , and ηij . Biologically, these parameters describe

how a disturbance action in a patch (decreasing habitat quality) affects patch-specific

55

survival and the between-patch movements associated with the disturbed patch. We

assume that disturbance actions will decrease connectivity; hence, δij and γij are both

nonnegative. For instance, if patch j is disturbed (yj = 1), then the probability of moving

from patch i to patch j (qij ) decreases by δij . Similarly, if patch i is disturbed (yi = 1),

qij decreases by γij . The parameter ηij is used as a correction term when both yi = 1,

and yj = 1. Also we set δii = γii = 0, meaning that if patch i is disturbed the updated

probability of staying in the same patch (�qii ) is affected only by ηii (see Appendix C for

more details).

qij =

�qij − δijyj − γijyi + ηijyiyj if i , j ∈ P

�qid +∑k∈P

(δikyk + γikyi − ηikyiyk) if i ∈ P and j = d

�qij if i = d and j ∈ M

(3–2)

where P describes the set of all patches in the network and M the set of all states

in the Markov chain. We estimate δij , γij , and ηij as a percentage of the initial transition

probability �qij . To this end, we define δij = αij�qij , γij = βij�qij , and ηij = ωij�qij , where

αij , βij ∈ [0, 1] and ωij ∈ [αij + βij − 1,αij + βij ], ∀i , j ∈ P.

We select the optimal set of patches (H) to disturb by means of the mixed-integer

linear program described in Appendix C.

3.2.2.4 Protection

The end goal of the protection model is to determine the optimal subset of patches

to protect. We define binary variables wi , ∀i ∈ P , which equal 1 if and only if patch i is

protected. The w variables affect the y -variables in the disturbance problem because

yi = 0 whenever wi = 1.

The protection model maximizes the potential loss in the total life expectancy that

can be prevented by protecting a particular set of patches. We use the integer linear

56

program described in Appendix D to determine an optimal set of patches to protect,

given a set of possible disturbance actions.

3.2.2.5 General model assumptions

We assume that individuals follow a random walk between patches. That is, an

individual attempt to move from one patch to another is independent of the sequence of

patches previously visited. This assumption is common to other connectivity modeling

approaches (McRae et al., 2008). We also assume that �qid > 0, for all i = 1, ... , |P|,

which means that no matter the patch where the individual is currently living, there is

always a positive probability of death. We consider the survival rates (Si ) estimated from

mark-recapture as true survival; however, these survival estimates may be negatively

biased due to the difficulty of discerning between mortality and permanent emigration

(Gilroy et al., 2012). We consider the network to be a closed system, hence, the only

available patches to move are included in P. We assume individual homogeneity in the

factors that affect movement and survival. We also assume that movement decisions

and mortality are density independent.

3.2.3 Scenarios

We illustrate the modeling framework by building different protection scenarios for

each species. We developed scenarios in which the protection budget (u) was roughly

half the landscape (Roseate Tern: u = 2, b = 2; Skipjack Tuna: u = 3, b = 4). We

optimized both the weighted and unweighted total life expectancy for the Roseate Tern,

because the data included patch abundances. We optimized only the unweighted total

life expectancy for the Skipjack Tuna, because patch abundances were not included in

the dataset. We also show how total individual life expectancy varies for all combinations

of protection budgets and number of patches allowed to be disturbed.

Because there was no data available to estimate δij and γij , we generated δij =

αij�qij and γij = βij�qij as described in Section 3.2.2.3. Parameters αij and βij were

randomly generated on the continuous uniform distributions with bounds 0 and 0.35,

57

which represents disturbance actions of low to moderate magnitude. Furthermore, we

generate parameters ωij on the continuous uniform interval [αij + βij − 1,αij + βij ]. All the

code was written in C++, using CPLEX 12.3 optimization software.

3.2.4 Sensitivity Analysis

We performed two types of sensitivity analysis. First, we assess the robustness

of the solution (set of optimal patches to protect) given changes in the input data (�Q).

Second, we study how the optimal individual life expectancy changes with increasing

protection budget (amount of sites allowed to be protected) or number of patches

allowed to be disturbed. We observed high site fidelity in both species (high probability

of individuals staying in the same patch). We wanted to know how much change in

these probabilities will take to change the optimal solution. Recall that ωii describes the

proportional decrease in the probability of staying in patch i given that the patch was

disturbed. We vary ωii(1+ ϵ), where ϵ ∈ [−1, 0], to assess how much variation is required

to change the optimal solution.

From Section 3.2.2.3 we have that qii = �qii(1+ωiiyi) ∀i ∈ P, showing that a source of

change in qii is ωii . We define ϵ as the rate of change in ωii , so that qii = �qii(1+(1+ϵ)ωiiyi)

∀i ∈ P. Since qii ∈ [0, �qii ], we know that ϵ ∈ [−1, 0]. In addition, we have that qii = �qii

∀i ∈ T , so we only allow ϵ to affect qii ∀i /∈ T . We define z(ϵ) as the optimal life

expectancy obtained after solving the protection problem using (1 + ϵ)ωii (instead of ωii )

∀i ∈ T among the input data.

Define z(ϵ) as the life expectancy calculated by assuming that set of patches T

is protected, as input data. The ratio ρ(ϵ) = z(ϵ)/z(ϵ) measures the change in the

optimal total life expectancy as a function of ϵ. If ρ(ϵ) = 1, then T is still optimal after

the perturbation given by ϵ, whereas if ρ(ϵ) < 1 then T provides a suboptimal solution

with a gap from the optimal life expectancy given by 1− ρ(ϵ). Given that abundance data

was not available for the Skipjack Tuna, for consistency we perform this analysis on the

unweighted individual total life expectancy.

58

3.3 Results

3.3.1 Roseate Terns: Step-by-Step

Total individual life expectancy (unweighted), in the case that no patch is disturbed

was z = 20.43. If we allowed at least half of the network to be disturbed (b = 2 patches),

with a protection budget of two patches (u = 2). The worst-case for connectivity

happens when the set of patches H = {2, 4} are disturbed, and a corresponding total

life expectancy of z = 15.59 (Figure 3-3b). The corresponding optimal set of patches

to protect that will prevent the most loss of total life expectancy is T = {1, 4} with an

associated life expectancy of z = 15.97 (Figure 3-3j).

When the weighted total life expectancy is optimized using patch abundances

pT = [3773, 2467, 350, 172] (Spendelow et al., 1995), the total life expectancy when

no patch is disturbed was z = 32368.9. The worst-case scenario for connectivity

happens when patches H = {1, 2} are disturbed with an associated life expectancy of

z = 20545.7. The corresponding optimal set of patches to protect is T = {1, 2} with an

associated life expectancy of z = 30638.9 (Appendix E).

3.3.2 Skipjack Tuna

The total individual life expectancy in the case that no patch is disturbed was

z = 46.07. If we allowed at most four patches to be disturbed (i.e. b = 4), and have

a protection budget of at most three patches (i.e. u = 3), the optimal strategy is to

protect the set of patches T = {2, 3, 7}, with an associated optimal total individual life

expectancy of z = 35.99.

3.3.3 Sensitivity Analysis

We found that, in both species, as the protection budget (u) decreases, total life

expectancy decreases and as the number of patches allowed to be disturbed increases,

total life expectancy decreases asymptotically (Figure 3-4). When optimizing the

unweighted total individual life expectancy, the optimal solution for the Skipjack Tuna

was more robust than the optimal solution for Roseate Terns (Figure 3-5). The optimal

59

Figure 3-3. Example of the iterative process between the disturbance and protectionmodels for a protection budget of u = 2 and allowing b = 2 patches to bedisturbed. The figure shows the (a) total individual life expectancy withoutany protection or disturbance, the (b) worst-case scenario for connectivityand (j) the optimal protection against this worst-case scenario.

60

solution for the Skipjack Tuna remained optimal for ϵ ∈ [−0.7, 0], which implies that site

fidelity estimates can decreased by at most 70% and the solution will remain optimal.

On the contrary, the optimal solution for the Roseate Tern network remained optimal

for ϵ ∈ [−0.2, 0], which implies that side fidelity estimates can decrease at most 20%

for the solution to remain optimal. When considering the weighted total individual life

expectancy, the solution remained optimal regardless of the amount of variation (Figure

3-5).

Figure 3-4. Sensitivity of the optimal life expectancy in the protection stage givenchanges (ϵ) in probability of staying in the same patch (�qii ).

3.4 Discussion

Systematic conservation planning tools provide guidance for the proper management

of protected area networks. Uncertainty is present in many stages of this planning

process, and often times not acknowledged, which may result in inadequate management

actions. Recently, methods have been developed that test the robustness of site

selection models given uncertainty in the input data (Moilanen et al., 2006a,b); however,

even when uncertainty in the data is accounted for, there is still uncertainty about

future changes in the system due to human disturbances, or natural disasters. Here

we presented a novel application of network interdiction models that deals with these

61

Figure 3-5. Optimal life expectancy (z) given all possible protection budgets (u) andnumber of patches allowed to be disturbed (b). Life expectancy decreaseswith decreasing protection budget and with increasing number of patchesallowed to be disturbed

unaccounted uncertainties by proposing a protection plan given a worst-case scenario

for connectivity.

We group the data requirements to parameterized the model into two: movement

data and scenario building variables. Movement data includes transition estimates

(ψ), patch survival (S) and patch abundances. It also includes the estimates on how

disturbance in a patch will affect the transition probabilities (γ, δ and η). Scenario

building variables include the maximum number of patches allowed to be disturbed (b)

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and the protection budget (the maximum number of patches to protect u). The data

requirements to parameterize the interdiction model may be an important limitation of

this approach. Even though between-patch transition and survival estimates are one

of the best data available to quantify landscape connectivity (Belisle, 2005; Kindlmann

and Burel, 2008), these datasets are scarce in the literature because it requires much

field effort and they are complex to fit in a multi-state mark-recapture context (Calabrese

and Fagan, 2004). Nevertheless, these datasets are becoming more common in

conservation efforts. Parameter estimates that describe how movement changes when

patches are disturbed can be taken from the literature when available (e.g., Bechet

et al., 2003; Bowne et al., 2006). If these estimates are not readily available for a

particular species, the model can be parameterized assuming different scenarios for

these parameters.

The sensitivity of the optimal solution to variations in the data will depend on the

structure of the network. We found the model to be very robust to changes in site

fidelity estimates for the Skipjack Tuna, but less robust for the Roseate Terns. The

Roseate Tern network consisted of only 4 nodes with little variation in site fidelity and

mortality probabilities (Figure 3-2), which means that even small changes in some of

the movement estimates may significantly change the optimal solution. In contrast,

the robustness of the solution for the Skipjack Tuna network may be due to increase

variation in site fidelity and mortality probabilities. The optimal protection for Roseate

Tern network, when considering the weighted total individual life expectancy, was very

robust. This is because the abundances in the optimal patches to be protected was an

order of magnitude greater than the other patches.

Scenario building variables are useful to understand various scenarios for

connectivity, depending on how many patches are expected to be disturbed and the

protection budget. Generally, life expectancy is expected to decreases with increasing

number of patches allowed to be disturbed and increase with increasing number of

63

patches allowed to be protected (Figure 3-5). The shape of these relationships will be

specific for each application and will be a useful tool for managers making prioritizing

decisions. In some instances, increasing the protection budget by one patch may result

in a significant increase in life expectancy, suggesting that increasing the protection

budget may be useful. In other instances, decreasing the protection budget may not

necessarily result in a significant life expectancy decrease suggesting that a reduced

protection budget may be appropriate.

3.4.1 Modifications

The network interdiction model presented here is a flexible framework that

can be applied to a wide array of instances in conservation and management. This

approach can be applied to organisms, both active and passive dispersers, for which

between-patch movement and survival data are available. Even though we describe

a model in which the goal is to asses vulnerabilities to patch disturbances, the model

can be modified to assess vulnerabilities in the matrix. In this case the disturbance

and protection models will optimize life expectancy given the protection or disturbance

of arcs in the network instead of nodes. This modification will allow the identification

of places in the matrix to perform restoration activities or where to build corridors to

increase connectivity.

The cost of acquiring or protecting a patch is commonly incorporated in most

prioritization models (Cabeza and Moilanen, 2001; Kukkala and Moilanen, 2012;

Margules and Pressey, 2000; Moilanen, 2008; Sarkar et al., 2006). In this modeling

approach the protection model is given a budget or maximum number of patches to

protect. This inherently assumes that the cost of protecting each patch will be the same.

Variations in cost can be incorporated as constraints in both disturbance and protection

models.

Prioritizing disturbance actions to minimize connectivity for invasive species is a

potential alternate application of network interdiction models. In this case the problem

64

may be simpler because it only requires the disturbance model, in which we identify the

worst-case scenario for connectivity. Hence, the optimal set of patches that minimizes

connectivity will be the best patches to disturbed to control the invasive species.

3.4.2 Conservation Planning for a Worst-case Scenario for Connectivity

The few prioritization models that account for connectivity follow the area/isolation

paradigm from metapopulation theory that states that the probability of recolonization

increases with decreasing isolation (Hanski, 1998) and attempt to increase structural

connectivity by either minimizing the perimeter length of the network or the distance

between patches (Ball et al., 2009; Beger et al., 2010; Nalle et al., 2002; Williams

et al., 2004). Here, when prioritizing areas for the Roseate Terns, patches 1 and 4 were

selected as the optimal patches to be protected, even when the distance between these

two patches is the longest when compared to other between-patch distances in this

network (Figure 3-2a). Similarly, the optimal solution for the Skipjack Tuna includes the

node that represents international waters, which is the most isolated patch in terms of

distance (Figure 3-2b). This patch was part of the optimal protection solution because

it had high site fidelity and very low mortality. These results suggests that minimizing

between-patch distance may not be always an appropriate criteria for prioritization.

The practicality of increasing structural connectivity between patches has been

debated. Decreasing distance between patches may decrease the probability of

local extinction and reduce potential inbreeding, but it may increase the spread

of diseases or the persistence of exotic species (Williams et al., 2005). Hence, it

has been argued that increasing structural connectivity (i.e. distances between

sites without information on how organisms actually move among sites) may not

be an optimal management strategy, when prioritizing areas for an uncertain future

(Hodgson et al., 2011, 2009). Measures of structural connectivity do not incorporate

information about the behavior of the species, or how moving through landscape affects

individual fitness. Here, the optimal protection is assessed optimizing a connectivity

65

measure based on precise movement data (i.e. movement probabilities are quantified

through rigorous mark-recapture methods) and patch-specific survival estimates,

which ameliorates significantly some of the concerns that have been brought about

connectivity conservation for future changes (Doerr et al., 2011).

The current paradigm of metapopulation ecology also determines patch importance

by considering patches individually (Hanski, 1998). Highly connected patches are

considered more important to maintain ecological process in a spatial network (Bunn

et al., 2000; Minor and Urban, 2008). Moreover, the resilience of a network is often

measured by quantifying how many individual nodes can be removed without altering

network connectivity (Bodin and Saura, 2010; Bunn et al., 2000; Pascual-Hortal and

Saura, 2006; Urban and Keitt, 2001; Watson et al., 2011b). Here we presented an

alternative method by considering the importance of a particular subset of nodes instead

nodes individually. Considering a combination of patches instead of the sum contribution

of individual patches, often results in non-intuitive solutions. This is a common result in

the operations research literature (Brown et al., 2006).

Solutions from the network flow interdiction models, like any other optimization

model, are simply a tool for decision support. Its purpose is to guide the management

process and its solutions should not be necessarily considered in isolation (Williams

et al., 2004). Instead, it should be considered in conjunction with other models with

different objectives as part of the conservation planning process. Pressey and Bottrill

(2009) summarized the conservation planning process in 11 stages. These stages,

include the identification of (1) the scope, (2) the stakeholders, (3) the spatial extent

and the (4) goals of the conservation program, (5) the collection of socio-economic

variables and threats, (6) collect data on biodiversity, (7) set conservation targets,

(8) evaluation of the existing area network, (9) identification of important areas, (10)

applying the conservation actions and (11) monitoring. The modeling framework

presented here encompasses stages 6 to 9 of this process. The data collection (stage 6)

66

requires between-patch transition probabilities and patch specific survivals. The network

interdiction framework also allows to set the specific conservation objectives (stage 7)

in terms of the number of patches to be protected (and the number of patches to be

allowed to be disturbed). The connectivity assessment stage (Section 3.2.2.2; stage 8)

which calculates the expected life expectancy given the current conditions. The result of

the model is the set of patches that will minimize future decreases to the life expectancy

of individuals given the scenarios built in in the model (stage 9).

3.4.3 Uncertainty and Worst-case Scenario Planning

Uncertainty may be present in multiple stages of the conservation planning

process, but rarely acknowledged. Information gap decision theory has been proposed

as an efficient framework to incorporate statistical uncertainties in conservation

planning (Moilanen et al., 2006a,b). The aim of information gap decision models is

to identify solutions that are robust to the presence of uncertainty. Even when statistical

uncertainties are incorporated in the planning process by the use of information gap, or

other sensitivity analysis, other non-statistical uncertainties may still affect the system in

the future and result in inadequate management actions. The network flow interdiction

model acknowledges that there are additional uncertainties which are difficult to quantify,

but uses the available data to protect against the worst possible disturbance. Hence,

even when climate change or regional-scale changes in environmental conditions may

affect the network in an unpredictable manner, the system has already been protected

against the worst possible scenario.

67

APPENDIX AMODEL SUMMARIES AND ADDITIONAL FIGURES

Table A-1. Parameter estimates and standard errors (SE) for the multi-seasonoccupancy model that resulted in the best fit; ψ(.), γ(Sasym ∗ A ∗ Ph),ϵ(Sasym ∗ A ∗ Ph), P(A ∗ Ph).

Colonization Extinction DetectionEstimate SE Estimate SE Estimate SE

INTERCEPT -4.897 0.198 -2.724 0.285 2.142 0.113Sasym 0.330 0.174 0.565 0.200A 0.008 0.001 -0.016 0.005 0.004 0.001Ph(TREES) 0.800 0.383 0.575 0.415 -0.410 0.184Sasym : A 0.002 0.001 -0.016 0.005Sasym : Ph(TREES) 0.410 0.255 -0.414 0.394A : Ph(TREES) -0.003 0.006 -0.005 0.010 0.001 0.002Sasym : A : Ph(TREES) -0.015 0.005 0.004 0.012

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Table A-2. Parameter estimates and standard errors (SE) for the model that included asymmetric dispersal kernel (Ssym) as a site covariate; ψ(.), γ(Ssym ∗ A ∗ Ph),ϵ(Ssym ∗ A ∗ Ph), P(A ∗ Ph).

Colonization Extinction DetectionEstimate SE Estimate SE Estimate SE

INTERCEPT -4.867 0.195 -2.60 0.267 2.134 0.112Ssym 0.231 0.192 0.558 0.219A 0.007 0.001 -0.021 0.006 0.004 0.001Ph(TREES) 0.800 0.409 0.400 0.400 -0.413 0.184Ssym : A 0.001 0.001 -0.018 0.005Ssym : Ph(TREES) 0.289 0.284 -0.667 0.380A : Ph(TREES) -0.002 0.007 -0.001 0.010 0.001 0.002Ssym : A : Ph(TREES) -0.008 0.005 0.020 0.009

Figure A-1. Spatial representation of patches (boulders and tree trunks) sampled forLepanthes rupestris occurrence in the Luquillo Experimental Forest.

69

Figure A-2. Theoretical distributions of the parameter δij . A random spatial arrangement(a) results in a relatively uniform distribution (b). If patches are alignedalmost perfectly in the same angle of wind direction (c), it results in abimodal distribution highly skewed to the limits of the distribution (d).

70

Figure A-3. Predicted partial relationships between colonization, extinction and patcharea for both phorophytes for the best-supported model

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Figure A-4. Partial relationship between patch area and detectability for (a) rock and (b)tree phorophyte for the best-supported model

72

APPENDIX BCONNECTIVITY ASSESSMENT: MATHEMATICAL DETAILS

We calculate total individual life expectancy (�gi ) as the number of transitions that

an individual living in patch i survives before dying (going to state d) (Ross, 2006). This

calculation is derived based on the following linear equation. The expected lifetime, �gi ,

starting from patch i ∈ P, is given by one plus

• zero if the individual dies at the next period (with probability 1− Si )

• �gi if the individual successfully reaches patch j ∈ P (with probability �qij ).

The relationship can be written as

�gi = 1 +∑j∈P

�qij�gij , (B–1)

or in matrix from:

I�g = �A�g+ 1. (B–2)

If (I− �A) is invertible (which we assume to be the case, by a a slight perturbation of

�A if necessary), then

�g = (I− �A)−11. (B–3)

The weighted total life expectancy, z , is calculated as:

z = pT (I− �A)−11 = pT�g. (B–4)

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APPENDIX CDISTURBANCE OPTIMIZATION

The disturbance optimization model includes the initial transition probabilities given

by �qij (∀i , j ∈ M) and the estimated impacts on the transition probabilities caused by

the interdictor’s disturbances, which are given by δij , γij , and ηij (∀i , j ∈ M). A formal

statement that describes the calculation of qij , ∀i , j ∈ M is given by Equation 3–2.

To illustrate the use of ηij , consider the case in which disturbing patches i and j

together reduces qij by less than the sum of the individual disturbances δij + γij . In this

case, ηij is a positive correction term. On the other hand, if there is a synergistic effect

attributable to jointly disturbing patches i and j , such that disturbing i and j together

reduces qij by more than the individual disturbances, then ηij is as a negative correction

term. We assume that ηij ∈ [δij + γij − �qij , δij + γij ], so that in case of disturbing both

patches i and j , we have qij ∈ [0, �qij ]. In the case that only one patch is disturbed, we

require the condition �qij − max{δij , γij} ≥ 0 so that qij ∈ [0, �qij ]. Also, we set δii = γii = 0

without loss of generality, meaning that if patch i is disturbed, then qii is updated only

according to ηii ∈ [−�qii , 0].

We define the parameter b ∈ {1, · · · , |P| − 1} as the maximum number of patches

that can be disturbed. Our model also uses the parameter �gi (∀i ∈ P), given by Equation

B–3. We know that �gi is an upper bound on the number of transitions spent in transient

states, because the transition probabilities between transient states are either reduced

when some patches are disturbed or remain unchanged when no patch is disturbed.

The model identifies those patches that should be disturbed to minimize the

individual total life expectancy. The disturbance optimization model uses variables yi ,

∀i ∈ P, and updated transition probability variables qij , ∀i , j ∈ P as defined by the first

case of Equation 3–2.

The disturbance problem objective is given by

min z = pTg, (C–1)

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and g is again calculated as

g = (I− A)−11, (C–2)

where A is obtained by deleting the last row and column from Q. By substituting

qij using Equation 3–2, and writing Equation C–2 in functional form, we can rewrite

Equation C–2 as

gi −∑j∈P

�qijgj +∑j∈P

δijyjgj +∑j∈P

γijyigj −∑j∈P

ηijyiyjgj = 1, ∀i ∈ P. (C–3)

Note that each equation in Equation C–3 is nonlinear in the y and g variables. To

linearize these equations in Equation C–3, we introduce the additional variables wij

(= yigj ) and xij (= yiyjgj ). The resulting system is given by

gi −∑j∈P

�qijgj +∑j∈P

δijwjj +∑j∈P

γijwij −∑j∈P

ηijxij = 1,∀i ∈ P. (C–4)

The system shown in Equation C–4 is the first set of constraints in our model.

This set describes not only the way to update the transition probabilities given the

interdictor’s actions (q variables), but also the calculation of the expected time spent in

transient states (g variables). To guarantee that wij = yigj , we introduce the set of linear

constraints shown in Equations C–5–C–8. Note that if yi = 1, then wij = gij . If yi = 0,

then wij = 0, as desired.

wij ≥ 0, ∀i , j ∈ P (C–5)

wij ≤ gj , ∀i , j ∈ P (C–6)

wij ≥ gj − �gj(1− yi), ∀i , j ∈ P (C–7)

wij ≤ �gjyi , ∀i , j ∈ P (C–8)

Similarly, we define the set of constraints shown in Equations C–9–C–13 to

guarantee that xij = yiyjgj . Since the term yiyjgj is also nonlinear, we introduce the new

75

variable vij and the corresponding set of constraints shown in Equations C–13–C–16

to guarantee that vij = yiyj . Note that if yi = 0, then vij = 0 and xij = 0. Moreover, if

yi = yj = 1, then vij = 1 and xij = gj , as desired.

xij ≥ 0, ∀i , j ∈ P (C–9)

xij ≤ gj , ∀i , j ∈ P (C–10)

xij ≥ gj − �gj(1− vij), ∀i , j ∈ P (C–11)

xij ≤ �gjvij , ∀i , j ∈ P (C–12)

vij ≥ 0, ∀i , j ∈ P (C–13)

vij ≤ yi , ∀i , j ∈ P (C–14)

vij ≤ yj , ∀i , j ∈ P (C–15)

vij ≥ yi + yj − 1, ∀i , j ∈ P. (C–16)

The maximum number of patches that can be disturbed is considered in Equation

C–17, while the constraint shown in Equation C–18 defines the binary nature of the

y -variables.

∑i∈P

yi ≤ b (C–17)

yi ∈ {0, 1}, ∀i ∈ P (C–18)

The mixed-integer linear problem consists in minimizing the objective function shown in

Equation C–1 subject to constraints in Equations C–4–C–18.

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APPENDIX DPROTECTION OPTIMIZATION

The protection optimization model includes the parameter u ∈ {1, · · · , |P| − 1} to

denote the protection budget (the maximum number of patches that can be protected).

We let the binary decision variable wi equal 1 if and only if patch i ∈ P is protected.

First observe that all feasible protection solutions obey the constraints:

∑i∈P

wi ≤ u (D–1)

wi ∈ {0, 1}, ∀i ∈ P. (D–2)

Equation D–1 enforces the maximum number of patches that can be protected,

while constraints in Equation D–2 define the binary nature of the decision variables.

The objective for the protection problem is less straightforward to model. Conceptually,

let H be the set of all cardinality b subsets of P, i.e., the set of all possible disturbance

solutions. For every H ∈ H, let zH be the individual life time given disturbance set H.

We can define binary variables δH,∀H ∈ H, which equal 1 if and only if the protection

action “disables” disturbance set H (i.e., if there exists an i ∈ P such that i ∈ H and

wi = 1). Then we can state constraints:

∑i∈H

wi ≥ δH, ∀H ∈ H. (D–3)

The protection problem objective is denoted by

max

{min

H∈H:δH=0

zH

}. (D–4)

However, such a problem is non linear and requires exponentially many constraints

and variables. As an alternative consider the feasibility problem given by F(H):

Constraints in Equations D–1 and D–2,

77

∑i∈H

wi ≥ 1 ∀H ∈ �H, (D–5)

where �H ⊆ H is a subset of the set H. Note that F( �H) is feasible if and only if all

possible disturbances in the set �H can be disabled by a protection action. Therefore,

consider the following algorithm, in which an upper bound value is initially set to UB= �z

(computed by Equation B–4), �H is initially empty, and which the protection solution is

given by w∗ = 0.

• Step 1. Solve F( �H), and if feasible solution w exists, go to the second step;otherwise, go the third step.

• Step 2. Solve the optimal disturbance problem augmented with the constraints thatyi ≤ wi ,∀i ∈ P, which enforce the condition that no protected patch is disturbed.Let H be an optimal solution to the disturbance problem, given w, noting thatconstraints yi ≤ 1− wi ,∀i ∈ P, are added to the disturbance problem. Add H to �H.If UB > zH, then set UB = zH and w∗ = w. This step returns to step 1.

• Step 3. Terminate with an optimal solution given by w∗ having an objective UB.

Two notes are necessary regarding this algorithm. The first regards the convergence

and correctness of the algorithm. Each time F( �H) is solved, one more subset is added

to �H. A second issue is that solving F( �H) in early iterations of the algorithm yields

protection strategies that are blind to many possible disturbance solutions that are not

yet in �H, but are likely to be added to �H in future iterations. Accordingly ,we can apply a

lower-boundary objective function to F( �H).

To estimate the lowest possible life expectancy we define ~g as a vector whose

entries ~gi represent the mean number of transitions spent in the transient states, given

that the Markov chain starts at state i . To guarantee that ~g is a lower-bound on life

expectancy, we assume that the transition probabilities between transient states are the

minimum possible, which is equivalent to assume that the transition probabilities from

any transient state to the death state are maximum. To this end we define the transition

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matrix ~Q, whose entries ~qij are given by

~qij = (1−max{δij , γij , δij + γij − ηij})�qij . (D–6)

By our construction of the transition probabilities in Equation D–6, no disturbance can

produce a total life expectancy of less than pT~g, and so pT~g is a valid lower bound on

total life expectancy. The objective function for F( �H) in this case can be written as

f (w) =∑i∈P

pi(�gi − ~gi)wi , (D–7)

where pi(�gi − ~gi) is an estimate of the potential loss in the total life expectancy that can

be prevented if patch i is protected.

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APPENDIX EMODEL ITERATIONS FOR WEIGHTED LIFE EXPECTANCY OPTIMIZATION

Table E-1. Model iterations showing the interaction between disturbance and protectionstages when pT = [3773, 2467, 350, 172]

Iteration Stage Optimal solution z �z

1 Protection wT = [ 0 0 0 0 ] 20545.7 −10093.2Disturbance yT = [ 1 1 0 0 ]

2 Protection wT = [ 1 1 0 0 ] 30638.9 0Disturbance yT = [ 0 0 1 1 ]

3 Protection wT = [ 1 0 1 0 ] 27084.8 −3554.1Disturbance yT = [ 0 1 0 1 ]

4 Protection wT = [ 1 0 0 1 ] 26465.4 −4173.5Disturbance yT = [ 0 1 1 0 ]

5 Protection wT = [ 0 1 1 0 ] 24499.1 −6139.8Disturbance yT = [ 1 0 0 1 ]

6 Protection wT = [ 0 1 0 1 ] 23846.9 −6792Disturbance yT = [ 1 0 1 0 ]

80

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BIOGRAPHICAL SKETCH

Miguel A. Acevedo was born in San Juan, Puerto Rico in 1980. He graduated from

the University of Puerto Rico with a B.S. in biology 2002. He later graduated with a M.S.

in the Tropical Biology Program from the University of Puerto Rico-Rıo Piedras in 2006.

Miguel received his Ph.D. from the University of Florida in the spring of 2013 where,

as part of the Quantitative Spatial Ecology, Evolution and Environment (QSE3 IGERT),

developed an interdisciplinary dissertation research that incorporated spatial ecology,

network theory and concepts from industrial and systems engineering. He hopes to

return someday to Puerto Rico and join their professorate conducting quantitative

ecological research, and teaching.

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