postprint - inra

A note on semi-discrete modelling in the

life sciences

By Ludovic Mailleret1† and Valerie Lemesle2‡1. INRA, UR 880, URIH, 06903 Sophia Antipolis, France

2. ENS Lyon, UMPA, 69364 Lyon, France

Semi-discrete models are a particular class of hybrid dynamical systems thatundergo continuous dynamics most of the time but repeatedly experience discretechanges at some given moments. In the life sciences, since the first semi-discretemodel was derived to describe population dynamics by Beverton & Holt (1957), alarge body of literature has been concerned with such modelling approaches. Theaim of the present contribution is twofold. On the one hand, it provides a compre-hensive introduction to semi-discrete modelling through two illustrative examples:the classical work by Beverton and Holt is recalled and an original example onimmigration in a population model affected by a strong Allee effect is worked out.On the other hand, a short overview of the different applications of semi-discretemodels in the life sciences is proposed.

Keywords: impulsive differential equations, epidemiology, medicine,population dynamics, Allee effect model, Beverton Holt model.

1. Introduction

The two most classical modelling techniques of biological phenomena are continuoustime and discrete time models, respectively. Continuous time models in ordinarydifferential equations on the one hand are used to describe the interactions betweencompartments (cells, animals, etc...) that can, from a macroscopic point of view, beconsidered as continuous because the involved processes happen randomly in time(prey-predator encounters, cell division, etc...). The Lotka Volterra predator-preymodel, the Kermack McKendrick SIR epidemic model or Monod’s model of cell(bacteria) growth are typical examples of such systems. On the other hand, somenatural phenomena can not be considered as continuous since they occur at certainmoments of time only: this is for instance the case for animals that reproduceseasonally or animals that are vulnerable to attacks during a certain period of theirlife cycle only. These characteristics gave rise to discrete time models (differenceequations) that were particularly developed by the consumer-resource modellingcommunity with respect to host-parasitoid interactions (Murdoch et al. 2003). Boththese modelling approaches have a long history in the biological sciences: back tothe end of the XVIIIth century for continuous time modelling (Malthus 1798) andto the beginning of the XXth century for discrete time modelling (Nicholson &Bailey 1935).

† Author for correspondence ([email protected])‡ Present address: CIRAD, UR SCA, 34398 Montpellier, France

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2 L. Mailleret & V. Lemesle

There are however a large part of biological systems that do not fit either formal-ism, more precisely that involves some phenomena that are of a continuous natureand some others that are of a discrete one. The state variables of such systemsexperience smooth/continuous dynamics most of the time but at some momentsface abrupt/discrete changes (also termed pulses): at these moments the variables“jump” from some state value to another. One can think for instance to predator-prey or epidemiological systems with seasonal reproduction, emigration processesstarting once a density threshold of a population is reached, intermittent drug ad-ministration in medical programs, etc... The modelling of this kind of systems wouldthen naturally involve two different parts, a continuous one in ordinary differen-tial equations and a discrete one in difference equations, yielding what is called a“hybrid” system. The discrete part may occur as the model variables verify someconditions (recall the emigration process described above) what is called an “im-pact model” (see (Maggi & Rinaldi 2006) for an example on forest fires); otherwisethe discrete part occurs at some given instants (e.g. seasonal processes) and arereferred to as “impulsive” or “pulsed” systems Bainov & Simenov (1989). In thiscontribution we will concentrate on this latter class of models only, and, followingGhosh & Pugliese (2004), Singh & Nisbet (2007), Pachepsky et al. (2008), we willcall them “semi-discrete models”.

The first semi-discrete approach was probably put forward by Beverton & Holt(1957) in their construction of a discrete-time model analogous to the continuoustime logistic model (Verhulst 1838) on the basis of a semi-discrete model. Since then,a large body of literature has proposed semi-discrete models in almost every field ofthe life sciences: population dynamics and ecology, plant pathology, epidemiology,medicine etc... The explicit reference to the use of models that have both continuousand discrete characteristics is usually explicit, but for a non-negligible part it is not.The semi-discrete modelling may even be difficult to identify since the discrete partsof the models may sometimes be described without equations at all (see e.g. (Shaw1994; Swinton et al. 1997; Jansen & Jabelis 1995; Fenton et al.2001). It actuallyappears that different bio-modelling communities work on similar mathematicaltools with little knowledge of what the others do. The aim of this contribution istwofold: providing an introduction to semi discrete modelling as well as a shortreview of the different fields of application in which semi-discrete modelling is usedin the life sciences.

This paper is organised as follows. We first present the general mathematicalformalism of semi-discrete modelling. We then study two illustrative examples ofsemi-discrete models, the classical Beverton-Holt model and an invasion model fora population subjected to a strong Allee effect. It is shown in particular that takinginto account the discrete nature of some phenomena may have an important influ-ence on the behaviour of the system. An overview of the literature is then proposed.The reviewed results are classified with respect to their fields of application: epi-demiology (vaccination strategies and plant diseases), medicine (drug therapy) andpopulation dynamics (harvesting and pest managements programs, seasonal phe-nomena and chemostat modelling). When applicable, we highlight the added valueof semi-discrete models with respect to their continuous (or discrete) counterpart.Finally, we briefly discuss the advantages and drawbacks of semi discrete modelsand some of the actual directions of work in this field.

Article submitted to Royal Society

Semi-discrete models in the life sciences 3

2. Semi-discrete modelling: a general formalism and twoillustrative examples

(a) General formalism

As stated in the introduction, we refer to semi-discrete models as the particularclass of hybrid dynamical systems that undergoes continuous dynamics in ordinarydifferential equations most of the time and that experiences discrete dynamics atsome given time instants. A schematic representation of the dynamics followed bya semi-discrete dynamical system is illustrated on figure 1.

τ τ τ ττkk k+1 k+1

+

k+2k−1

+ +τ

Discrete

DiscreteContinuousContinuous Continuous

Figure 1. Graphical illustration of semi-discrete modelling. The horizontal axis representsthe time; as time t is between two instants τk the system evolves in a continuous way; itexperiences abrupt (discrete) changes at the instants t = τk.

Although most mathematically oriented contributions on semi-discrete (or pulsed,impulsive) systems share the same classical common mathematical formalism (see(Agur et al. 1993; Funasaki & Kot 1993; Panetta 1996)), some others differ (e.g.(Shaw 1994; Ghosh & Pugliese 2004; Singh & Nisbet 2007)). It is then important torecall the standard formalism used to describe semi-discrete models. Let x be thevector of state variables at time t and τk be the instants when the discrete changesoccur. A semi-discrete model reads

dx

dt= f(x, t), t 6= τk,

x(τ+k ) = F (x(τk), τk),

(2.1)

with τ+k denoting the instant just after t = τk. f(.) is the continuous, possibly time

varying, ordinary differential equation followed by the system and F (.) the discretecomponent (also termed pulse, impulse) that may also depend on time. In most ofthe reviewed literature f(.) and F (.) are time independent but this is not necessarilyso (e.g. (Gubbins & Gulligan 1997b; Choisy et al. 2006; Xiao et al. 2006; Braverman& Mamdani 2008)). Moreover in most of the studied cases (τk+1− τk) is a constantfor all k, but once again there are examples where it is not (e.g. (Lakmeche & Arino2001; Liu et al. 2005; Nundloll et al. 2008)).

There are an important part of semi-discrete models dealing with seasonal pro-cesses (like seasonal reproduction) in which time t does not refer to the absolutetime like in (2.1), but rather to the time elapsed between two seasonal processes, i.e.the time within the year. In these approaches, (τk+1 − τk) is constant equal to theyear’s (or season) length; at each instant τk, time t is reset to 0 (e.g. (Gyllenberg etal. 1997; Andreasen & Frommelt 2005; Pachepsky et al. 2008)) and the state vectorcorresponding to the kth year may be denoted with a subscript k on x.



A last remark should be given on what models the discrete part of the system.In most approaches, it represents an instantaneous process: increase of some drugconcentration in blood following a treatment, death of a proportion of an insectpopulation when a pesticide is applied, the introduction of individuals to increasethe size of a population, etc... However, the discrete part of the model may alsosum up what happens within a non-empty period of time. Insect populations forinstance are active and interact with other species during the summer but aremostly dormant in the winter. A semi-discrete model would then represent whathappens during the summer with its continuous part and what happens during thewinter (survival of a proportion of a population only, transition from a juvenileto an adult stage, etc...) with its discrete part (Iwasa & Cohen 1989; Ghosh andPugliese 2004). Hence, strictly speaking, τ+

k is actually not always the instant justafter t = τk.

(b) Illustrative examples

We show on two simple examples the consequences of the explicit modellingof discrete phenomena on the predictions of classical continuous time models. Thefirst example deals with the formulation of the classical discrete-time Beverton-Holt model of population dynamics on the basis of a semi-discrete model. In thesecond example, the effects of pulsed and continuous immigration strategies on theinvasion success of a population undergoing strong Allee effects are compared andcontrasted.

(i) Back to basics: the discrete time Beverton-Holt model

Discrete time models are appealing to biologists and ecologists as they canproduce very complicated dynamical behaviors (cycles, chaos) even with a singlevariable. However, the principles that underlie the model equations are not alwayseasy to interpret from a biological point of view. Hence, the “mechanistic under-pinning” of discrete time population growth models is of interest (Geritz & Kisdi2004).

Beverton & Holt (1957) were already concerned with this need for biologicalinterpretation of discrete-time equations. To obtain their population growth mod-els for fish species from mechanistic assumptions, they postulated that the mainprocesses involved in the population’s growth are, on the one hand, seasonal repro-duction within a very short period of time and, on the other hand, death processesthat occur continuously throughout the season (see also (Gyllenberg et al. 1997)).We recall here how they derived their simplest and most famous model. Let n(t)be the population density at time t, and assume that the per capita mortality rateis proportional to its density. We have the semi-discrete model with discrete repro-duction at time t = kT and continuous death between reproduction instants

dn

dt= −µn2, t 6= kT,

n(kT+) = n(kT ) + αn(kT ),(2.2)

with k ∈ N∗ and where µ is the mortality rate, α the clutch size and T the seasonlength.



Integrating the continuous part of (2.2) for t ∈ (kT, (k + 1)T ) gives

n(t) =n(kT+)

1 + µTn(kT+),

so that the post reproduction map of the population density between seasons is

n((k + 1)T+) =(1 + α)n(kT+)1 + µTn(kT+)

,

which is the classical discrete-time “Beverton-Holt” model. This model is a discrete-time analogous to Verhulst (1838) continuous time logistic equation: it has a singleglobally stable equilibrium, similar transient dynamics, and sound biological bases.

(ii) Pulsed immigration in a model with strong Allee effects

In population models, the Allee effect is the reduction of the population growthrate at low density that can for instance result from failure in mate searching. Thepopulation growth rate may remain positive for small population densities (weakAllee effect) or even drop below zero at some threshold, under which the populationwill go extinct (strong Allee effect, see e.g. (Taylor & Hastings 2005)). According toCourchamp et al. (1999), one-population growth models with strong Allee effectscan be written as

dx

dt= rx

(x

Ka− 1)(

1− x

K

), (2.3)

with x the population density, r the Malthusian parameter, Ka the Allee thresholdand K the carrying capacity of the environment. Model (2.3) is a very simpleexample of what is called bi-stability: if at the initial moment the population densityis below the Allee threshold it will decline to extinction, if it is above, it will growto the carrying capacity of the environment. This bi-stability is typical of a strongAllee effect.

In what follows we investigate the outcome of different immigration strategies,namely continuous and pulsed immigration, for a population subjected to a strongAllee effect. We say that an invasion succeeds if the immigration strategy can drivethe population density from x = 0 to some value above the Allee threshold Ka, andthat it fails otherwise. The interaction between strong Allee effects and (continuous)immigration was previously studied by Keitt et al. (2001).

Since we are interested in the invasion process only, i.e. as x stays beyondthe Allee threshold Ka, we assume for mathematical simplicity that the carryingcapacity K is large with respect to Ka, so that model (2.3) can, as long as x ∈(0,Ka), be approximated by

dx

dt= rx

(x

Ka− 1).

Actually, the computations we show next may also be worked out with the originalAllee effect model (2.3), but they would be unnecessarily complicated and we prefer,for the sake of clarity, to consider this approximation.

In the continuous immigration case, the model reads

dx

dt= rx

(x

Ka− 1)

+ σ, (2.4)



with σ the (constant) immigration rate of the population. Let T be a reference timeperiod (say e.g. one year) then, according to (2.4), σT individuals migrate into thesystem during this period.

Consider now the pulsed immigration problem: immigrating individuals are nomore evenly spread over the time period T but immigrate at one time (or two,etc...). Assume that

(σTn

)individuals immigrate into the system every

(Tn

)period

of time (with n some positive integer). Then, for the pulsed immigration problem,we have the semi-discrete model

dx

dt= rx

(x

Ka− 1), t 6= kT

n ,

x

(kT

n

+)= x

(kT

n

)+σT

n,

(2.5)

for all k ∈ N∗. Notice that during a time period T , σT individuals migrate intothe system modelled by (2.5). Hence the immigration rate (i.e. the number ofimmigrants per unit time) is the same in the continuous model (2.4) and in thesemi-discrete model (2.5): it is simply σ. This property is of prime importancesince we want to compare the continuous immigration strategy to the pulsed one.As a consequence, as n approaches +∞, i.e. as immigration numbers are infinitelysmall but immigration occur at an infinite frequency, model (2.5) reduces to (2.4).Of course it is not realistic to imagine an immigration of an “infinitely small numberof individuals”: the minimal number is one individual. This reveals two things. First,although widely used in e.g. metapopulation models, the continuous immigrationmetaphor actually lacks biological realism. Second, the pulsed immigration model(2.5), that corrects some of the flaws of the continuous immigration model, is onlyrealistic up to some final n.

In the continuous immigration case (2.4), it is fairly easy to show that theinvasion will be successful, i.e. x(t, x0 = 0) overshoots Ka for some t > 0, if andonly if

σ >rKa

4. (2.6)

In the pulsed immigration case (2.5), it is proved in Appendix A-(a) that theinvasion succeeds if and only if

σ >nKa

Ttanh

(rT

4n

). (2.7)

Again, as n approaches +∞, condition (2.7) reduces to (2.6) because tanh(z) isequivalent to z at z = 0.

Let us now compare conditions (2.7) with (2.6) for some finite z. Since tanh(z) <z for all positive z it is easy to show that invasibility condition (2.6) is more restric-tive than (2.7). Both strategies will of course succeed if σ is large enough, but thereare some intermediate values of the immigration rate with which an invasion basedon continuous immigration would fail, but its pulsed counterpart would succeed. Inother words, given an immigration rate, an invasion may more easily succeed if theimmigration strategy is pulsed rather than continuous.

This result does not come from the fact that with a single immigration pulse,the population would overstep the Allee threshold Ka (although it is possible if σ



is large enough). Indeed, one can show that there always exist immigration ratesthat would not succeed in a single migration pulse, but do so in several pulses (seeAppendix A-(b)).

Keitt et al (2001), based on an analysis very similar to the one presented abovein the continuous case, show that the interaction between (continuous) migrationand strong Allee effects gives rise to what they call “invasion pinning”: the spatialfront of an invasion may be stopped by the strong Allee effect if the migration rateis too low. The study that was reported above show that their results still holdtrue in the case of pulsed immigration, although the upper bound on the migrationrate required to exhibit invasion pinning would be smaller than with continuousinvasion. This result may have important consequences in the field of biologicalinvasions management. Species can indeed exhibit pulsed migrations through, forinstance, accidental transportation by humans: this is what happens for the gypsymoth, one of the major insect pest in the U.S.A (Sharov & Liebhold 1998). Forsuch organisms, disregarding the fact that the migration is pulsed, rather thancontinuous, leads to overestimate the “pinning” induced by strong Allee effects.As a consequence it results in the underestimation of the species ability to spreadout in the environment (see also (Johnson et al. 2006) for a related but somewhatdifferent discussion on this issue).

Conversely, in the field of reintroduction biology, repeated pulsed introductions(also termed re-stocking (Armstrong & Seddon 2008)) are worthwhile since they arelikely to enhance reintroduction programmes success in comparison to single and tocontinuous ones (see also the discussion on re-stocking by Deredec & Courchamp(2007)).

With this rather simple example that compared pulsed and continuous immi-gration strategies with the same overall immigration pressure, we have shown oneof the important consequences of semi-discrete modelling: although in some casessemi-discrete models and their purely continuous (or discrete) equivalent give sim-ilar results, this is not true in general. Taking explicitly into account the discretenature of some phenomena and the continuous nature of some others may insteadexhibit emergent properties that yield to new conclusions.

3. Literature overview

Although it is not possible to provide an exhaustive review of the use of semi-discrete(pulsed, impulsive) models in the life sciences, we try to propose a representativeoverview. The modelling approaches are classified with respect to their application:epidemiology (vaccination strategies and epidemiological control, plant epidemiol-ogy), medicine (mainly drug therapy) and population dynamics (harvesting andpest management programs, population and chemostat models).

(a) Epidemiology

One of the major field in which semi-discrete models are used is epidemiology,may it relate to humans, animals or plants. The most important application isthe comparison between classical (continuous) vaccination and pulsed vaccinationin epidemiological models (Agur et al. 1993). Some other applications deal withthe effect of seasonal demographic processes (birth, death). Another important



field is that of plant epidemiology in which pathogens face seasonal demographicbottlenecks each winter.

(i) Pulse vaccination strategy

In a seminal paper, (Agur et al. 1993) proposed the idea of “pulse vaccinationstrategy” to improve the control of measles epidemics. Instead of a constant vacci-nation effort spread throughout the year, they proposed to vaccinate a proportionof the population within a very short period of time: vaccination was then a dis-crete process while epidemic evolution was assumed continuous. Their study, basedon the numerical analysis of an aged-structured epidemiological model and on themathematical analysis of a simple susceptible-infective (SI) model, yielded to theconclusion that pulse vaccination should give better results than the classical con-stant vaccination programs. Pulse vaccination campaigns performed in 1994 in theUK against measles confirmed the theory that this strategy has a dramatic impacton the development of the epidemic.

Subsequently, numerous authors compared both vaccination strategies (contin-uous versus discrete) for various epidemiological systems. Shulgin et al. (1998) com-pleted the study done by (Agur et al.1993) for the SIR (R for recovered or removed)model and proved how mixed strategies can affect the level of infected population.D’Onofrio (2002) considered an SEIR (E for exposed) model and other more specificones were examined in (D’Onofrio 2005). A complete mathematical study based onthe classical mathematical tools of dynamical systems was proposed in these papers.Gakkhar & Negi (2008) explored a SIRS model with a non monotonic incidence rateand proposed a bifurcation analysis showing that pulse vaccination can lead to verycomplex dynamics. Liu et al. (2008b) added a vaccinated class to the population(denoted V) in the SIR model and a mathematical analysis based on the value ofthe threshold R0 was done. Meng & Chen (2008) studied through a bifurcationanalysis a new SIR model where the offspring of infected parents can be infectedor susceptible. Again the complicated dynamics that may arise in such systems,and not with continuous vaccination, were highlighted. In all these contributions,pulse vaccination was always shown to be capable of eradicating the diseases, doingusually better than continuous vaccination.

Other authors followed different approaches in order to accurately describe epi-demic dynamics. Wagner & Earn (2008) focused on a particular disease (Polio) forwhich the vaccine can sometimes mutate back and actually provoke the disease.The influence of different vaccination programs were studied and this time pulsevaccination appeared less effective than more classical strategies. In other papers,seasonality had been added to the epidemiological models to obtain a more precisedescription of the epidemic (Liu & Jin 2006). Choisy et al. (2006) studied numer-ically a time varying SEIR model with pulse vaccinations and warned that such astrategy can actually lead to complex dynamics and an increase of the infective pop-ulation. Galvani et al. (2007) proposed a model of influenza in which the populationwas structured in elderly and younger people; seasons were accounted for as wellas a pulse vaccination occurring each fall. It was shown through a numerical anal-ysis that the classical vaccination program (elderly vaccination) was actually notthe better way to control influenza epidemics. The vaccination of younger peopleinstead would give more acceptable results.



(ii) Discrete demographic processes

Other studies were concerned with the influence of discrete demographic pro-cesses (seasonal birth or death, discrete culling of a proportion of a population,etc..) on the dynamics of epidemiological systems. For example Roberts & Kao(1998) considered a SI model with discrete birth processes and applied it to a tu-berculosis in possums system. In this particular system, the authors showed thatthe discrete birth process was well approximated by an easier to study continuousbirth model. Swinton et al. (1997) also studied an SI model with a discrete birthprocess. They analysed the influence of fertility control (removal of a proportion ofthe newborns at each birth) or lethal control (continuous removal of individuals) onthe epidemic dynamics and applied it to the problem of bovine tuberculosis trans-mission by badgers. Their work argued that lethal control of badgers is better thanfertility control. Fuhrman et al. (2004) examined a SI model with discrete culling ofa proportion of the infective population and concluded that such a strategy madethe eradication of a disease possible. Gao et al. (2005) also studied a SI modelwith birth pulses and seasonal parameters. They performed a bifurcation analy-sis, showing once more that very complicated dynamics may arise in such systems.Finally, Andreasen & Frommelt (2005) proposed an interesting “school-oriented”epidemic model. The population was structured in age classes corresponding to thestudent level at school. Each year, the entire cohort of students was instantaneouslytranslated from one school level to the next. A complete mathematical analysis hasbeen done and the model was shown to be able to convincingly mimic recurrentepidemics like influenza or measles, but also to produce complex dynamics thatwere difficult to interpret.

(iii) Plant epidemiology

Another important domain is that of plant epidemiology which also has a stronglink with seasonality. An important proportion of plants are deciduous or cultivatedin annual cropping systems so that they mostly disappear from the environment atthe end of each year or season. This leads pathogens that are responsible for plantdiseases to regularly face demographic bottlenecks at each disappearance of theirhost population, a situation particularly well described with semi-discrete models.Shaw (1994) first proposed a plant disease model accounting for these demographicbottlenecks. He numerically showed the very complex dynamics that were inducedby these seasonal processes, and warned that “observed” chaotic patterns were notnecessarily due to some random external forcing. Truscott et al. (1997) examineda fungal root disease model in an annual cropping system. A local mathematicalanalysis and some simulations allowed to determine parameter values under whichthe disease was eradicated or persisted. Gubbins & Gilligan (1997a, b) studied thepersistence of continuous time host plant - parasite system in an environment sub-mitted to discrete perturbations due to cropping. They were particularly interestedin the influence of the transmission functions and the type of perturbation on thepersistence of the infection. Madden & van der Bosch (2002) presented a SEIRmodel of a plant-pathogen introduced in an annual cropping system with a semi-discrete formalism. They investigated to what extent plant pathogens may be usedas biological weapons and the conditions under which the pathogen may persistfrom season to season were given.



Zhang & Holt (2001) investigated a plant - two virus model and compared con-tinuous host plant availability (perennial systems) to semi-discrete host plant avail-ability (annual systems). They (numerically) showed that the competitive exclusionprinciple applies for the continuous system (the virus with the higher reproductivenumber always outcompeted the other one) while this was not necessarily the casein the semi-discrete model. Conversely, Pietravalle et al. (2006) numerically stud-ied an annual cropping system composed of two cultivars and the evolution of itsresistance against a single strain of a pathogen.

The performance of chemical treatments in the control of plant diseases werealso studied with semi-discrete models. Vendite & Ghini (1999) modelled the ap-plication of fungicide as a discrete process and the evolution of fungicide resistancein the fungal population as a continuous process. They studied the influence of thefungicide dose and show that small doses are less likely to lead to fungal resistancethan large ones. With a similar approach, (Hall et al. 2007) studied numerically theinfluence of the dose and frequency of pesticide use on the evolution of pathogen re-sistance; they actually recommended the use of frequent and small doses of pesticiderather than infrequent and large ones.

(b) Medicine

Semi-discrete models have also been used in the medical sciences, mainly in themodelling of drug administration. To our knowledge, one of the first authors whoinvestigated this theme was Panetta (1996, 1998). In these contributions, he pro-posed two semi-discrete models of chemotherapeutic cancer treatment in which theevolution of the tumours were described continuously while the chemotherapeutictreatments were modelled as discrete processes. These models were quite simple sothat a complete mathematical analysis was performed, investigating different thera-peutic strategies. Lakmeche & Arino (2000, 2001) extended these results consideringmore general models. In (Lakmeche & Arino 2000) in particular, an important bi-furcation theorem for two-dimensional systems was proved: under some conditionsit guaranteed the existence of a stable non trivial periodic solution emerging froma “trivial” one. This contribution was major in the field of semi-discrete modellingand used in almost all references that perform bifurcation analyses. Optimal pro-tocols of drug administration in cancer immunotherapy were investigated througha numerical optimisation algorithm in Cappuccio et al. (2007).

With respect to other medical problems than cancer, Smith? & Schwartz (2008)analysed the influence of frequency and intensity of vaccine administration, ap-proximated as a discrete process, in a continuous time within-host HIV model.They theoretically showed that sufficiently large or frequent vaccine administra-tion would keep the infection arbitrarily low. A last medical domain concernedwith semi-discrete models is endocrinology. Kroll (2000) studied the effects of con-tinuous and pulse hormone (parathyroid hormone, PTH) releases in a continuousmodel of bone formation. It appeared that pulse hormone administration was bet-ter than continuous administration, what, according to the author, explained someof the osteoporosis mechanisms. More recently in chronobiology, Vidal et al. (2009)modelled luteinizing hormone (LH) secretion and evolution using a time-variablesemi-discrete model that accounted for variations in the photo-period. The dynam-ical patterns produced by the model were very close to observations made on ewes.



A similar approach was proposed by Shurilov et al. (2009) to model the regulationof non-basal testosterone secretion in males though pulses of gonadotropin-releasinghormone (GnRH).

(c) Population dynamics

With epidemiology, population dynamics modelling is the major (and historical)field in which semi-discrete models have been used. The approaches are even morediverse than for epidemic problems. We report here different studies related tothe modelling of populations by itself, some works considering harvest and pestmanagement problems and finally the application of the technique to chemostatmodelling.

(i) Population models

The mathematical modelling of population dynamics is an important field ofapplication of dynamical systems. We highlight in this section different applicationsof semi-discrete modelling to population biology issues.

Some studies on population dynamics with impulsive birth were developed in(Ballinger & Liu 1997). The permanence of population growth was studied usingclassical mathematical tools. Other interacting population models with birth pulseshave been studied, showing that complex dynamics aroused (e.g. Tang & Chen(2004)). Recently, Wang et al. (2008) also considered the influence of birth pulsesand other impulsive perturbations on evolutionary game dynamics.

In line with the important work by Beverton & Holt (1957) recalled in §2 b (i),the first set of contributions is concerned with the rationale that underlies discretetime models of population dynamics. De Roos et al. (1992) proposed a semi-discretemodelling technique dedicated to structured populations that extends the classicaldiscrete time Leslie matrix models. Such techniques were mode recently used tomodel structured populations in an evolutionary context (Persson & de Roos 2003;van de Wolfshaar 2008). Nedorezov & Nedorezova (1995) provided mechanisticarguments to the formulations of some discrete time models on the basis of semi-discrete ones (continuous death and discrete reproduction at some fixed moments).A complete mathematical analysis was proposed for one-dimensional populationmodels. Gyllenberg et al. (1997) used the same kind of models to compare differentreproduction strategies. The “mechanistic underpinning” discrete time models wasfurther developed by Geritz & Kisdi (2004) with a stage structured consumer -resource semi-discrete model (continuous consumer resource interaction, discreteconsumers death and reproduction). A time-scale separation in the continuous partwas assumed to simplify the model, so that a complete mathematical analysis waspossible. A similar approach was developed by Eskola & Parvinen (2007) to derivediscrete time Allee effect models. Eskola & Geritz (2007) proposed mechanisticarguments to the derivation of other discrete time population models. Followingthe same modelling techniques, Rueffler et al. (2006, 2007) studied the evolutionarydynamics of resource specialization. Kisdi & Utz (2005) put the model by Geritz &Kisdi (2004) in the spatially explicit context of a patchy environment and studiedthe formation of patterns along a chain of patches. Utz (2007) prolonged this work,



adding a more detailed structure of the consumer population to the model andproposed an explanation to the emergence to an Allee effect.

Dynamics of populations interactions with seasonal reproduction was first evokedin Hochberg et al.(1990) who studied by numerical tools the possible exclusion ornot of a mutant population in a host-parasitoid-pathogen system. Later Briggs &Godfray (1996) developed a season dependent model describing insect-pathogeninteractions. A local stability and bifurcation analysis showed the emergence ofcomplex dynamics. Seasonal population dynamics of ticks (ticks are active duringsummer and dormant during winter) were described by Ghosh & Pugliese (2004)and their influence on disease transmission of tick-borne diseases were investigatedmathematically. On the basis of a similar model of ticks dynamics, Ding (2007)proposed a numerical method to find the optimal protocol of pesticide applicationto control ticks population.

More recently, Singh & Nisbet (2007) derived a semi-discrete host-parasitoidmodel with stage structure of the host population depending on their status (par-asitised or not). The influence of the functional response on the durability of thehost-parasitoid interaction was studied and their conclusion contradicted classi-cal results obtained from discrete time models. Pachepsky et al. (2008) analysed aconsumer-resource model with continuous reproduction of the resource and discretetime reproduction of the consumer. The model was shown to produce dynamicsrepresentative of classical consumer-resource models, but showed also a richer dy-namical behavior than these. Following the same modelling idea, Akhmetzhanovet al.(2009) studied a semi-discrete seasonal consumer-resource system. The con-sumers were assumed to be optimal foragers so that the within-season dynamics(continuous part) was formulated as an optimal control problem and analyticallysolved. A numerical bifurcation analysis of the long term dynamics of such a systemwas also proposed and compared with the case of consumers that were non-optimalforagers. This work was actually also a reminiscence of previous works by Schaffer(1983) and Iwasa & Cohen (1989) on the optimal allocation of resources by plantsfor growth in seasonal systems.

(ii) Impulsive harvesting

Impulsive harvesting, the instantaneous removal of a proportion of a continu-ously growing population, has recently been considered in several research articles.Some harvesting problems linked to annual cropping systems have already beenreported in the plant epidemiology §3 a (iii) and will not be repeated here; someproblems related to harvests are also reported in the pest management §3 c (iii).

Ives et al. (2000) studied the effects of recurrent periodic mortality events in-duced by harvests on the dynamics of a predator prey system modelling an aphidparasitoid interaction in an alfalfa field. Alfalfa was harvested several times a year,what induced some periodic mortality events on both predator and prey popula-tions. The authors showed through a numerical analysis that such a system exhib-ited more complicated dynamical patterns that were not possible without account-ing for the harvest events. Following this idea, some papers studied the complexdynamics produced by population models experiencing impulsive harvesting (e.g.(Liu & Chen 2008)). Using a somewhat different approach, Drury & Lodge (2008)identified a kind of “semi-discrete hysteresis” due to impulsive harvesting in an



intraguild predation model. This phenomenon is probably common to a large partof semi-discrete models whose continuous part is bistable.

An important direction of research is the optimal management of renewableresource (Clark 1990). Some authors were interested in this question consideringimpulsive harvesting rather than continuous harvesting. Zhang & Wang (2003) stud-ied the impulsive harvesting policy that maximized the per unit time sustainableyield for a single population with logistic growth. Independently, Xiao et al. (2006)proposed later a similar analysis for a population following logistic growth withperiodic carrying capacity and Malthusian parameter, accounting for seasonal vari-ations in the ecosystem. Dong et al (2007) studied this issue for a population with(periodic time varying) Gompertz growth and not logistic growth. These contribu-tions showed that optimal continuous harvesting was superior to optimal impulsiveharvesting, but argued that the differences were small and that impulsive harvestingwould be easier to apply to real life situations. Braverman & Mamdani (2008) alsoconsidered the same issue but assumed that harvesting has some harmful effects onthe population. They showed that in this situation, (optimal) impulsive harvestinggave better results than (optimal) continuous harvesting.

(iii) Pest management

The control of insect pests in cropping systems is mainly based on the appli-cation of chemical pesticides and/or the release of natural enemies of the pests.Since these two processes are by their very nature discrete phenomena (instanta-neous death after pesticide application, instantaneous increase of the natural enemydensity after a release), pest management modelling has made a significant use ofsemi-discrete models. Jansen & Sabelis (1995) analysed first the consequences ofrecurrent death processes induced by pesticide on a tri-trophic chain (plant - pest- natural enemy). Although there was no mathematical analysis, the evolutionaryconsequences of pesticide use and its consequences for the design of pest manage-ment programs were discussed.

Lu et al. (2003) analysed a predator - prey (pest - natural enemy) model withimpulsive use of a pest-specific pesticide. They showed that pesticide use may leadto both population extinction, to the natural enemy extinction only and to thepersistence of both species. Most modern pest management programs are so-called”integrated” pest managements: chemical pesticides are coupled with biologicalcontrol, i.e. the release of natural enemies of the pests. Liu et al. (2005) studieda predator - prey model with both pulses of pesticide and release of biologicalcontrol agents. A mathematical analysis of the condition under which the pest waseradicated as well as a numerical bifurcation analysis showing various routes to verycomplicated dynamics were proposed. Similar studies were performed with variousforms of predator-prey models (e.g. Li et al. (2006)), more complicated food chains(Georgescu & Morosanu 2008; Xiang et al. 2008) and stage structured populationsShi & Chen (2008), to cite a few.

Some other contributions related to pest management focused on the effect ofdose and release frequency of biological control agents on the efficiency of biologicalcontrol. Mailleret & Grognard (2006, 2009) showed that frequent small releasesshould be preferred to large and infrequent ones, a recommendation being even moreimportant when biological control agents interfere with each other (Nundloll et al.



2009a, 2009b). In a similar context, Nundloll et al. (2008) analysed the effects ofpulse harvesting on a biological control model with discrete natural enemy releases.It is shown in particular that in a biological control context, biological control agentsshould not be released more frequently than the pulse harvesting operates.

(iv) Chemostat models

The idea of studying the effects of nutrient pulsing in the chemostat is due toEbenhoh (1988) who proposed an algae competition model that was mainly stud-ied numerically. It was shown that multiple species can coexist in a chemostat withpulsed nutrient input. Funasaki & Kot (1993) followed the same modelling andstudied a mass-action tri-trophic chain (nutrient - phytoplankton - zooplankton)in a chemostat with nutrient pulses. Through a mathematical analysis and simula-tions they showed the emergence of very rich dynamics. Conversely, Song & Zhao(2006) considered a two nutrients - one microorganism system in a chemostat withsynchronised pulses of the two nutrients. The extinction and permanence of thepopulation is studied. Wang et al. (2007) followed the same idea with a Monodtype tri-trophic food chain and pulse input and removal. A numerical bifurcationanalysis enlightened the route to complicated dynamics. The same authors alsostudied in a similar way food chains with other functional responses (Beddingtonand Tessiet type). Most recently, Toth (2008) explored resonance and chaos in asingle species chemostat model with nutrient pulses. Stage structured populationwas used to describe the microorganisms population and bifurcation and simulationanalyses illustrated the produced complex dynamics.

4. Conclusion

In this article the different kinds of application of semi-discrete modelling in the lifesciences have been briefly reviewed. Although it was not possible to be exhaustive,it is hoped that no major contribution has been forgotten. The main advantages ofsemi-discrete modelling compared to its continuous or discrete counterpart are:

- it allows an accurate description of some phenomena that can not be ac-counted for using “classical” modelling techniques,

- semi-discrete models can exhibit emergent properties (see e.g. the immigrationmodel in §2 b (ii)),

- as a special case of emergent properties (mostly a consequence of their discretecomponent), semi-discrete models can produce very complicated dynamics(co-existence of multiple complicated attractors, chaos, etc...), even in lowdimensions. As such, they can provide explanations, neglected with othermodelling approaches, to the real-life observation of complex dynamics,

- semi-discrete models are extremely versatile and can be used to describe alarge variety of systems.

There remains however shortcomings, not restricted to semi-discrete modellingbut yet important, that can restrain its wider adoption:



- as a combination of a discrete and a continuous component, such systemsare intrinsically complex and their mathematical analysis is more difficultthan purely continuous or discrete models. A consequence is that numericalcomputing is sometimes the only way to study such systems,

- since semi-discrete models easily produce complex dynamics, there is an im-portant risk that these dynamics are mathematical artifacts and do not cor-respond to real phenomena. This issue is however common to most modellingtechniques (Sherratt et al. 1997).

Notwithstanding these drawbacks, semi-discrete models are mathematical toolsthat can not be ignored when modelling biological phenomena. The formalism thatwe proposed in §2 a is even a little narrow in comparison to what can actually beput in a semi-discrete context. Liu et al. (2008a) recently proposed a HIV infectionmodel with a structured population described by a continuous parameter (age sinceinfection), rather than with different variables, and pulsed drug administration. Inthis contribution, the continuous part of the model is then a system of partial differ-ential equations (PDE), not of ordinary differential equations. Akhmet et al. (2006)also studied a predator-prey semi-discrete PDE model, but related to diffusion ofthe populations in space. Other recent work were also concerned with the effects oftime delay in the continuous part of their models (see e.g. (Gao et al. 2006, 2007;Li & Fan 2007)), or those of stochasticity (Wagner & Earn 2008).

To finish with, one should notice that most of the actual work seems dedicatedto modify the continuous part of the semi-discrete models, while the discrete partremains classical. A future direction of work may be to consider biological systemsthat require to alter the discrete part as well.

L.M. is grateful to the SPE Department of INRA for financial support through the “IA2L”project. V.L. was funded by a post-doctoral grant from the CNRS. Both authors thankFrederic Grognard for insightful discussions on the Allee effect model with pulsed im-migration, especially for the proof that condition (2.7) is necessary to the success of aninvasion (see the end of Appendix A-(a)).

Appendix A.

(a) Computation of the invasibility condition (2.7)

From the form of the RHS of the continuous part of (2.5), it is possible to obtainan analytical expression for the population density as long as it stays below the Alleethreshold Ka. Indeed, through a classical separation of variables technique, we getfor a population initiated at x0 < Ka at time t0 > 0

x(t) =x0Kae

−r(t−t0)

Ka − x0(1− e−r(t−t0)),

as long as x(t) remains below Ka. From this equation, it is possible to deduce theone dimensional map of post immigration population densities

x

((k + 1)T

n

+)

=σT

n+

x(kTn

+)Kae

− rTn

Ka − x(kTn

+)

(1− e− rTn )

, (A 1)



as long as x(kTn

+)

does not overshoot Ka. Then, a necessary and sufficient con-dition for the invasion to succeed is that there exists a positive integer ks suchthat

x

((ks + 1)T

n

+)≥ Ka.

Since the RHS of (A 1) is increasing in x(kTn

+)

and positive at 0, a simpleargumentation based on a cobweb plot drawing shows that the invasion succeeds if

ω =σT

n+

ωKae− rT

n

Ka − ω(1− e− rTn )

(A 2)

does not have a solution ω∗ ∈ (0,Ka). Some algebra show that a sufficient conditionfor this to hold is

1− σT

nKa<

(1 +

σT

nKa

)e−

rT2n ,

which after some manipulations is shown to be the same as (2.7).Now suppose that (2.7) does not hold. Then some calculations show that equa-

tion (A 2) has at least one solution. Moreover, this solution is positive and smallerthan Ka therefore preventing the invasion from success: this shows that the suffi-cient invasibility condition (2.7) is necessary to the success of the invasion.

(b) Invasion after multiple pulses, not after a single one in (2.5)

From model (2.5), in one pulse(σTn

)individuals migrate into the system. Since

the initial population density is zero, a necessary condition for the invasion tosucceed after one single pulse is

σT

n≥ Ka.

Since tanh(.) is smaller than one, it is then easily seen from condition (2.7) thatthere always exist immigration rates that allow invasions to succeed after multiplepulses, though it would fail after a single one.

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