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Université Pierre & Marie Curie

Modelling the struggle for existence instructured populations

Vol. 1: Synthesis

Mémoire pour l’obtention de l’Habilitation à diriger des recherchesSoutenance : 10 janvier 2012 à Paris

David Claessen

Laboratoire ”Ecologie & Evolution”,UMR 7625 CNRS-UPMC-ENS

andCentre d’Enseignement et des Recherches sur l’Environnement et la Société -

Environmental Research and Teaching Institute (CERES-ERTI),Ecole Normale Supérieure, 24 rue Lhomond, 75005 Paris

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Jury

• Mikko Heino (University of Bergen, Norway), rapporteur

• Ophelie Ronce (Université Montpellier 2), rapporteur

• Jean Clobert (Station d’Ecologie Experimentale CNRS, Moulis), rapporteur

• Jon Pitchford (University of York), examinateur

• Amaury Lambert (Université Pierre & Marie Curie), examinateur

• Jean-Christophe Poggiale (Université de la Méditerranée, Marseille), examinateur

Illustration on tittle page: ”Cirkellimiet-III” by M.C. Escher. All M.C. Escher works c© 2011 The M.C. EscherCompany - the Netherlands. All rights reserved. Used by permission. www.mcescher.com

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Preface

What is a Habilitation à diriger des recherches? Having grown up outside the french academicworld, as some of the members of my jury, I decided it could be useful to look it up in Wikipedia:

En France, l’habilitation à diriger des recherches est un diplôme national de l’enseignementsupérieur qu’il est possible d’obtenir après un doctorat. Il a été créé en 1984 suite à laloi Savary. Ce diplôme permet de postuler à un poste de professeur des universités(après inscription sur la liste de qualification par le Conseil national des universités),d’être directeur de thèse ou choisi comme rapporteur de thèse.

Elle est définie réglementairement par l’arrêté du 23 novembre 1988 (modifié en 1992,1995 et 2002) : ”L’habilitation à diriger des recherches sanctionne la reconnaissancedu haut niveau scientifique du candidat, du caractère original de sa démarche dansun domaine de la science, de son aptitude à matriser une stratégie de recherche dansun domaine scientifique ou technologique suffisamment large et de sa capacité à en-cadrer de jeunes chercheurs. Elle permet notamment d’être candidat à l’accès au corpsdes professeurs des universités.”

D’après la réglementation en vigueur, le dossier de candidature à l’habilitation àdiriger des recherches comprend soit un ou plusieurs ouvrages publiés ou dactylo-graphiés, soit un dossier de travaux, accompagnés d’une synthèse de l’activité scien-tifique du candidat permettant de faire apparaı̂tre son expérience dans l’animationd’une recherche.

Freely and briefly translated, this means that the HDR diploma is needed for applying tofull professor positions in french universities; for being the main advisor of a PhD student; andfor being examiner of a PhD thesis. It also states that the HDR application consists of a file ofpublications accompanied by a synthesis of scientific activities which should demonstrate thecandidate’s experience in organising and coordinating research.

So preparing the HDR thesis is a moment to take a few steps back and to reflect on what I’vebeen up to the last years. Since my PhD in Amsterdam and Umea, I’ve worked at Rothamsted(UK), in Amsterdam and in Paris. I’ve been working on two postdoc projects before obtaininga permanent teaching and research position at the Ecole Normale Supérieure in Paris. Thesedifferent places and projects have led me to work on a number of different subjects and researchquestions. My current position in Paris has led me to work more as a co-author than a first author(including the indispensable last authorship!). Throughout these years the common theme of mywork has been the modelling of structured populations, in ecological and evolutionary time.In particular I’ve been doing so from the viewpoint of the ”environmental feedback loop”, anapproach that I have learned and developed during my work and discussions with André deRoos, Odo Diekmann, Hans Metz and Régis Ferrière. This mémoire sums up this work, andespecially the role of the latter idea in these modelling efforts, which I see as a ”red thread”running through my work.

I am a biologist, since I’ve studied biology. I am a modeller since that is what I mostly do. I tryto do theoretical ecology, meaning the development of ecological (and evolutionary) ideas usingconceptual tools, i.e., mainly mathematical models. By no means am I a mathematician. I haveinvited a number of mathematicians to take place in my HDR jury since the field of theoreticalecology is, in my opinion, interfacing between mathematics and ”real” (experimental) ecology.Although I try to answer to my research questions using mathematical and computer models,and not by doing experiments myself, I have seen actual animals, plants and micro-organisms; Ihave been in labs, on lakes, in mesoscale enclosures, in fields. I have indeed touched and markedfish and voles (no lemmings, though), inspected cultivated and wild oilseed rape, and estimatedlesion coverage on wheat leaves. And even though I do not do any experiments myself, I usuallytry to work in close collaboration with experimentalists that do. Theoretical ecology exists in thetension field between abstract theory and concrete case studies, and frequently we have to askourselves at what level of generality/specificity we want to work. This is obviously a subjective

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choice, and the ”optimal” choice often depends on the problem at hand. (The same is true for thetype of model we use to tackle a question.) Personally, I think it is useful to keep moving up anddown between the abstract and the concrete. I my work I’ve worked on quite specific models forparticular systems, but also on general ”out of the blue sky” toy models that have no reality tothem (well, perhaps...). I hope this synthesis will make some general sense out of a number ofparticular case studies, combined with a number of more abstract modelling exercises.

In addition to being a synthesis of my research, the text below is also a point of view on evo-lutionary ecology that I have developed to a large extent during my teaching activities at the ENSand for the Master EBE (co-hosted by UPMC, ENS, PSUD, AgroParisTech and MNHN). In partic-ular, I talk a lot about the ideas in the following sections during my courses called ”Evolutionaryecology” (E2, Master 1 ENS), ”Structured populations” (STRU, Master 2 EBE) and ”Adaptive dy-namics” (DYAD, Master 2 EBE). My teaching on these subjects has been largely inspired by otherteachers, in particular André de Roos, Odo Diekmann, Hans Metz and Régis Ferrière, who willno doubt recognize their influence on the following pages.

I want to thank many people that have made it possible for me to do science (and enjoy it).First, I would like to acknowledge my own ”origins”, and thank my family for their support,and especially my father for his contagious curiosity for the (meta-) physical world around us.Next, I thank André de Roos and Lennart Persson, who during my PhD (and afterwards) havebeen very inspiring and taught me all about ecology and how to do it. I thank my colleaguesfrom the Ecology lab in Paris, in particular Minus van Baalen, who introduced me into the lab,and Regis Ferriere, who welcomed me in the Eco-Evolutionary Mathematics team, where I’vehad many interesting and helpful discussions. I want to thank my colleagues at the CERES-ERTIfor a stimulating interdisciplinary and generally intellectual environment (providing lots generaldiscussions about mathemetics, climate, culture, language, history and geography). In particular,I want to thank Denis Rousseau for supporting me in my writing of this mémoire, by giving meall the freedom I needed. In advance, I want to thank the members of my jury for accepting tocritically read this document, and discussing it in January. Not last but not least, I also want tothank my colleagues with whom I have worked together. Without their collaboration, this workwould not have been possible. To name a few: Jens, Amaury, Robin, Vincent, Thomas, Francois,Loic, Boris, Mick, Chris, Jan, Oystein, Eric, Manuela, Jean-Francois, David, Tim, Tobias, Karen,Pär, Ulf, Michael, Andreas, Bernard, Frank.

This mémoire is dedicated to my beloved Corinne, with whom I have discovered not onlythe French academic world, but also this great and beautiful city, Paris, and, most importantly ofcourse, the miracles being in love and having a family, miracles that include our lovely daughtersVéra and Naomi.

CONTENTS 5

Contents

1 Introduction 61.1 The struggle for existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 The environmental feedback loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Why study structured populations? . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Size-structured populations 102.1 Physiologically structured population models . . . . . . . . . . . . . . . . . . . . . 102.2 Equilibrium analysis of a PSP model . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 The effect of temperature on generation cycles (fish) . . . . . . . . . . . . . . . . . . 152.4 Stochastic dynamics of small populations (lizards) . . . . . . . . . . . . . . . . . . . 202.5 Size-structure: conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Adaptive dynamics 273.1 Again: the struggle for existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Sympatric speciation in (structured) fish populations . . . . . . . . . . . . . . . . . 293.3 Speciation in dynamic landscapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4 Adaptive dynamics: conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Current research 454.1 Local adaptation and species range shifts, in a size-structured population . . . . . 464.2 Eco-evolutionary feedbacks between climate and phytoplankton . . . . . . . . . . 49

5 Other work 52

6 Discussion 526.1 The struggle for existence in structured populations . . . . . . . . . . . . . . . . . . 536.2 What next? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

7 References 59

A Curriculum Vitae 64A.1 Life history (education and positions) . . . . . . . . . . . . . . . . . . . . . . . . . . 64A.2 Supervision of students and postdocs . . . . . . . . . . . . . . . . . . . . . . . . . . 64A.3 Scientific responsibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65A.4 Teaching responsabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66A.5 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

1 INTRODUCTION 6

1 Introduction

A struggle for existence inevitable follows from the high rate at which all beings tend to increase. (...)As more individuals are produced than can possibly survive, there must in every case be a strugglefor existence, either one individual with another of the same species, or with the individuals of distinctspecies, or with the physical conditions of life. Darwin (1859), page 901

1.1 The struggle for existence

Darwin (1859)’s theory of evolution by way of natural selection is, in retrospect, a very ecologicalvision of evolution. As the driving force of evolutionary change, Darwin identifies the ”strug-gle for existence”, which he considers to be the natural extension of the doctrine of Malthus tothe natural world. He considers this struggle for existence to be the logical result of two basicobservations: the tendency of populations to grow exponentially (”the principle of geometricalincrease”); and the tendency of exponentially growing populations to be kept in check by eco-logical interactions such as food depletion, predation or epidemics. Without heritable variation,the struggle for existence simply results in a regulated population, either at a stable equilibriumor fluctuating. Evolution kicks in as soon as heritable variation between individuals of the samepopulation exists. Allowing for heritable variation, some lineages will be more efficient at ex-ploiting the resources or escaping their natural enemies, and are likely to outcompete the lessefficient types in the population. I think that Darwin’s term for the latter process, ”survival offittest”, is less revealing of the ecological nature of the evolutionary process than the ”strugglefor existence”.

The idea of the environmental feedback loop, the central theme of this synthesis and describedin the next paragraph, is fully analogous to the Darwin (1859)’s description of the struggle forexistence.

1.2 The environmental feedback loop

The idea of the environmental feedback loop (Fig. 1) is the following. Individuals within a pop-ulation (or a community) interact with each other in direct and indirect ways. Usually, in popu-lation biology, this is modelled by direct and indirect density dependence. Density-dependencecan be introduced in many different (ad hoc) ways; a catalogue would be impossible and unuse-ful. Yet for certain (theoretical) purposes, it may be useful to have a generic way to represent andmodel such density dependence; which could pave the way to the development of more generalconcepts, theories and numerical methods. For example in the theories of adaptive dynamicsand of physiologically structured populations, a general concept of density dependence is veryuseful (as illustrated further on in this document).

The idea proposed by Odo Diekmann and his co-authors (Diekmann et al., 1998, 2001, 2010),is to make an explicit distinction between the population dynamic behaviour of individuals (forgiven environmental conditions) and the dynamics of those environmental conditions. Connect-ing these two parts of the system results in a feedback loop since the behaviour of individuals(and their dynamics) depends on the environmental conditions (e.g., food abundance), whereasthose conditions depend on the individuals that use the environment (e.g., through consumption)(Fig. 1).

The idea is to define the ”environment” in such a way, that, knowing the environment, the in-dividuals can be considered independently from each other. That is, if the environment is known,the population dynamically relevant behaviour of an individual (rates of reproduction, growth,survival, migration) can be known as well. All information about density dependence is hencecaptured in the definition of the environment and how the environment and the individuals mu-tually influence each other. This means that in addition to ”individuals” and ”environment”, twoprocesses need to be specified: how do the behaviour and life history of individuals depend on

1Darwin (1859) page numbers refer to the Modern Library Paperback edition, 1998

1 INTRODUCTION 7

Figure 1: The environmental feedback loop. Arrow 1: Population dynamics are governed by the demo-graphic properties of individuals, such as fecundity, survival, individual growth, migration rate. Arrow 2:These individual properties depend, in part, on the state of the environment (food density, abundance ofpredators, temperature, etc). The influence of the environment on individual vital rates and life history iscalled a reaction norm. Arrow 3: Collectively, individuals influence the environment, e.g., though exploita-tion of resources (whereas single individual has a limited impact on its environment).

the environment; and how does the environment change in response to the (collective) behaviourof individuals. The first process can be seen as a reaction norm, whereas the second process, re-ferred to as the population impact on the environment, depends not only on the quality of theindividuals (in terms of their age, size, location, trait values, etc) but also on the quantity of indi-viduals. Whereas a single individual often has negligible impact on its environment, a populationor community of individuals generally has a strong impact (possibly resulting in a struggle forexistence).

Mathematically, applying this idea means that we obtain a linear model for the (short-term)population dynamics. I will illustrate this with a very simple example: logistic growth. Assumean unstructured population in which individuals reproduce with rate ρ(E) and die with rateµ(E), where E refers to the condition of the environment. In continuous time, the populationdynamics can then be written down as

dN

dt= r(E)N (1)

where r(E) = ρ(E)−µ(E). In other words, for any given constant environmentE, the populationwill grow (or decline) exponentially. This corresponds to Darwin’s observation that all popula-tions tend to grow with a (high) ”geometrical ratio of increase”. To give a concrete example forsake of illustration, we can assume that the environment affects reproduction only, in a linearway: ρ(E) = ρ0E and µ(E) = µ0. This trivial choice of vital rate functions may reflect that thereproduction rate depends linearly on food density, here captured by the environmental variableE. Then to complete the model, we need an equation for the dynamics of the environment. Inparticular, we need to know how E depends on the state of the population. Assuming that thefood dynamics are very fast, and hence that food availability is always in steady state with thecurrent consumer population sizeN , we simply specify a direct relation between population sizeand food availability. For example:

E = 1− Nκ

(2)

1 INTRODUCTION 8

The assumption that E decreases as N increases reflects the struggle for existence, that is, thegeometrical growth of populations leads to a deterioration of environmental conditions (resourcedepletion). In this case, the reduction of E with N is a model of intraspecific competition, butsimilar examples can be given for other ecological interactions such as interspecific competitionor predation.

Of course in this particular case, the population dynamic model is more easily presented inthe traditional form of the logistic growth,

dN

dt= rN

(1− N

K

)(3)

with K = κ(1 − µ0/ρ0) and r = ρ0 − µ0. However, making the environmental feedback loopexplicit has the advantage of obtaining a general, linear, equation for the population dynamics.For example, when studying the model above in the context of adaptive dynamics, equation (1)can be used to find the expression for the invasion fitness, which is an extension of the functionr(E). This is true also for more complex models: writing the ecological model in the form of theenvironmental feedback loop (representing all relevant density-dependent interactions), resultsin a model formulation that is naturally designed for extending it in the direction of adaptivedynamics (Diekmann, 2004).

In the context of the more complex physiologically structured populations, this representationhas already paved the way for the development of general numerical techniques for the continu-ation and stability analysis of such models (Kirkilionis et al., 2001; de Roos et al., 2010; Diekmannet al., 2010; Diekmann and Metz, 2010). An example (not the general mathematical theory) of thelatter is given below (section 2.2).

1.3 Why study structured populations?

A great deal of ecological theory is based on simple Lotka-Volterra type models, which usuallyassume that all individuals within a population are identical. This is obviously a very useful sim-plification, which has led to important results on the dynamics of populations and communities(see almost any textbook on ecology). Yet we all know that most populations are structured in dif-ferent ways: spatially, sexually, by age, size, reproductive status, epidemiological status, etc. HereI focus on two kinds of population structure: spatial structure and size structure. Spatial struc-ture is, intuitively, likely to have important implications, which are quite different for ecologicaland evolutionary dynamics, respectively. Ecologically, spatial structure means that individualsinteract with resources and other individuals in a small, local part of the environment. In otherwords, the environmental feedback loop depends on the spatial location. This has been shown toaffect ecological dynamics in different ways: pattern formation, dampening of population cycles,invasion waves, etc (Dieckmann et al., 2000). Evolutionarily, one particular relevance of spatialstructure is that it may lead to reproductive isolation of spatially separated sub-populations. Forexample, the classical distinction between allopatric and sympatric speciation refers to the spa-tial structure of populations. Some examples of the relevance of spatial structure, in particularfor evolutionary dynamics, are given in section 3.3. Spatial structure also directly influences thefitness of different types, albeit mediated through the environmental feedback loop (Ferriere andLe Galliard, 2001; Lion et al., 2011).

Body size is an important structuring variable, since many ecological properties of individ-uals depend strongly on their body size, according to so-called allometric scaling rules (Peters,1983). An example of the ecological relevance of body size is provided by a typical life history ofa Eurasian perch (Perca fluviatilis). At birth an individual has a body mass of about 2 mg and alength of 6-7 mm. After finishing its yolk sac, it starts feeding on zooplankton. When reachingseveral cm in length, it starts including macroinvertebrates in its diet. It matures at around 10-11cm (around 10 g). If lucky, the individual can grow further, and soon will no longer feed on zoo-plankton. Instead, it continues feeding on macroinvertebrates, and will also include small fish inits diet. In fact, it can include any fish in its diet that is between about 5 and 45 % of its own body

1 INTRODUCTION 9

length (which means it can start becoming cannibalistic from about 10 cm in length). Perch canreach lengths up to 80 cm (in Lake Windermere for example), and weigh up to several kilograms.Clearly, the position in the food web of an individual may change significantly throughout itslife time, while body mass may increase by up to a factor of 106. From trophic and metabolicpoints of view, a newborn and an old individuals are hence quite different ”organisms”. This il-lustration should make it intuitive that size structure is likely to have implications for populationdynamics.

So it is as with mountain climbing: Why do we want to model population structure? Becauseit is there. The existence of population structure in natural populations merits the question ofwhat are the consequences of this structure. The consequences of size structure and, more gen-erally, physiological structure, on the dynamics of populations and communities has receivedquite some attention since the 1980s, in particular since the seminal work by Gurney and Nisbet(1985), Metz and Diekmann (1986), de Roos (1997) and Kooijman (2000). Since, it has been shownthat size structure may have a wide range of dynamical consequences, the most basic of whichis the emergence of generation cycles, also known as cohort cycles (Gurney and Nisbet, 1985;de Roos, 1997; Persson et al., 1998), but also alternative stable states (Persson et al., 2007), catas-trophic behaviour (De Roos and Persson, 2002), ecological suicide (Van de Wolfshaar et al., 2008),population dynamics-induced size bimodality (Claessen et al., 2000), and others. This work hasshown that the range of dynamical behaviour of size-structured populations is richer than thatof unstructured models such as the Lotka-Volterra type models. This is no surprise, of course,as size-structured models are more complex and, in particular, they feature more complex andhigher-dimensional environmental feedback loops than unstructured models. While this com-plexity explains the interesting, new types of population dynamics observed in these models, thecomplexity also poses some limitations. Studying models of multiple, interacting size-structuredpopulations quickly becomes exceedingly difficult. Yet the qualitatively different types of dy-namics predicted by size-structured models leads to new research questions as to its occurrenceand importance in natural populations. Physiologically structured population models, describedbelow, allow us to pose a new kind of questions in population ecology. Turning this around, forcertain questions relevant to ecology and evolution, the use of structured populations is neces-sary. In section 2 I argue for this point of view with some examples of research on size-structuredpopulations.

The case of genetic population structure (meaning heritable variation between individuals), isof course a special case, as suggested above (section 1.1). Without genetic structure the strugglefor existence results in population regulation only. With such structure, it will naturally induceevolutionary dynamics in conjunction with the ecological dynamics. Genetic structure, in thesense of genotypical differences between individuals, is also important for a range of dynam-ical phenomena such as speciation, hybridisation, reproductive isolation, and ecological poly-morphism, including ecotypes. Genetical structure may depend to a large extent on the type ofinheritance (sexual reproduction, haploid/diploid, single/multi-locus traits, dominance, etc). Itmay also interact with spatial structure, such as for example local adaptation and maladapta-tion in spatial gradients. Genetical structure, and the ensuing evolutionary dynamics, will bediscussed in section 3.

In my work I have used the concept of the environmental feedback loop to study ecologi-cal and evolutionary dynamics of structured populations. Below is an overview of this work,structured as follows. Section 2 presents research on the ecology of size-structured populations.Section 3 presents work on the adaptive dynamics of structured populations, including size-structured populations (section 3.2) and spatially structured populations (section 3.3). Section4 presents two lines of ongoing research, both including ecological and evolutionary dynamics,and both including physiological and spatial population structure. Throughout the text, the ideaof the struggle for existence and in particular its formalisation as the environmental feedbackform a recurrent theme. The idea is to illustrate how this concept can be used to help formulatingmodels of eco-evolutionary dynamics. In particular, the text should demonstrate how even incomplex situations, this concept can help obtain a straightforward definition of fitness.

2 SIZE-STRUCTURED POPULATIONS 10

2 Size-structured populations

Les petits poissons dans l’eau, nagent, nagent, nagent, nagent, nagentLes petits poissons dans l’eau, nagent aussi bien que les grosLes petits, les gros, nagent comme il fautLes gros, les petits, nagent bien aussi

In this section I describe research on the ecological consequences of size structure. First, I intro-duce the modelling framework that I have used to formulate the models, referred to as physio-logically structured population models (PSP models). Next, I show how the idea of the environ-mental feedback loop helps in developing a method for the equilibrium analysis of such models(section 2.2). Then, two examples of specific questions related to size structure are discussed(sections 2.3 and 2.4).

2.1 Physiologically structured population models

The theory of physiologically structured population models takes into account that individualsmay differ from each other by their physiological state (e.g. age, size, body condition) (Metz andDiekmann, 1986; de Roos, 1997; de Roos and Persson, 2001; Diekmann et al., 2001). In this theory,physiological development (e.g. growth, maturation) is assumed to depend on the current stateof the environment in terms of food availability, abundance of competors and predators, etc.By specifying how, in turn, the environment is affected by the action of the entire population(Fig. 1) a model of population dynamics with plastic, density-dependent life history is obtained.The theory of PSP models is particularly well-suited to study the interaction between populationdynamics and life history (Persson et al., 1998; Claessen et al., 2000, 2002).

I define ”life history” as the history of the physiological state of an individual. Variation inlife history which is caused by variation in environmental conditions I refer to as life historyplasticity. Many environmental factors that influence life history vary in both space and time.First, some of these factors (e.g. food density) interact with the population such that populationfluctuations result in environmental fluctuations which feed back onto life history. In fluctuatingpopulations, life histories of individuals born in different years can therefore be entirely different(e.g. size-dimorphism in fish: Claessen et al., 2000; Persson et al., 2003). Second, the environmen-tal factors are often distributed heterogeneously in space (Hanski and Gilpin, 1997). Life historiesof individuals living in different regions may differ even in the absence of genetic variability (e.g.lizards: Adolph and Porter, 1993; Sorci et al., 1996).

PSP models make an explicit distinction between state variable at different levels: the indi-vidual level (i-state variables); the population level (p-state variables) and the environment (theE variables as introduced above). To formulate a PSP model that takes into account both plasticlife history and the population feedback loop, it is first necessary to decide:

• which i-state variables define an individual (e.g., age, size, sex, ...)

• which state variables define the E environment (e.g., resource abundance; abundance andpossibly size distribution of competitor and/or predators; temperature, ...)

And then to define the environmental feedback loop (Fig. 1) by specifying:

• how individuals develop (in terms of the i-states) given the current state of the environ-ment.

• how the state of the environment changes under the collective influence of all the individ-uals (the population)

The most frequently used formulation consists of a set of ordinary differential equations for thecontinuous dynamics of the i-state variables. A widely used example, the Kooijman-Metz model,is described below.


2.2 Equilibrium analysis of a PSP model

This section illustrates how the environmental feedback loop formulation can help doing an equi-librium analysis of a physiologically structured population model. The general theory for thismethod can be found in Diekmann et al. (1998); Kirkilionis et al. (2001); de Roos et al. (2010);Diekmann et al. (2010). Here I will just outline how using the environment E can be used to re-duce an infinite-dimensional problem to a two-dimensional problem. Although this section mayseem a bit tedious with a certain amount of mathematical detail, I find an explicit introduction ofat least one PSP model a prerequisite for this HDR report.

An example: the Kooijman-Metz model Although nothing new, and certainly not my inven-tion, I will give a fairly detailed description of the Kooijman-Metz (KM) model since it has be-come something of a standard model in size-structured population modelling, and has been afrequent starting point in my teaching and modelling work (Claessen et al., 2000; Claessen andDieckmann, 2002; Claessen and de Roos, 2003, and section 4.1). The KM model comes in variousdegrees of detail, and the one below is quite detailed. The reason is that this version is more ”bi-ological”, that is, the functions describing individual-level properties (maintenance, attack rate,digestion rate, lenght-weight relation) can be easily interpreted and even measured in lab exper-iments. The presentation of the KM model also illustrates the role of the environmental feedbackloop in PSP model formulation. Finally, a cannibalistic extension of this model has been used todemonstrate how the environmental feedback loop approach can be used to arrive at a numericalcontinuation method for PSP models.

The KM-model (Kooijman and Metz, 1984) describes the dynamics of a size-structured popu-lation and its unstructured resource (food) population. The model description below is borrowedfrom Claessen and de Roos (2003), but to keep the presentation simple here, the model below doesnot include cannibalism. The full (cannibalistic) model can be found in the original publication,including all parameter values, based on piscivorous fish, in particular Eurasian perch (Perca flu-viatilis), and zooplankton (Daphnia spp.). Assume that the physiological state of an individualis completely determined by its body length x. Vital rates such as food ingestion, metabolism,reproduction and mortality are assumed to depend entirely on body length and the conditionof the environment. The population size distribution is denoted by n(x) and the density of thealternative resource by R. All individuals are born with the same length xb, and are assumed tomature upon reaching the size xf . Reproduction is assumed to be continuous (in time) whichimplies that the size distribution n(x) is continuous over x (although population cycles may leadto discontinuities in n(x)).

The assimilation rate follows a size-dependent, type II functional response

F (x) = caA(x)R

1 +H(x)A(x)R(4)

where ca is the assimilation efficiency, A(x) is the attack rate, and H(x) is the size-dependentdigestion time per gram of prey mass (Table 1).

We assume that a fraction κ of assimilated energy is allocated to growth and maintenance(Kooijman and Metz, 1984), and the remainder to reproduction. The growth rate in mass is ob-tained by subtracting the metabolic rate from the energy intake rate. Assuming that the bodyweight and the metabolic rate scales both scale with the cube of body length, λx3 and ρx3, re-spectively, then the growth rate in length becomes

dx

dt= g(x) =

1

3λx2(κF (x)− ρx3

)(5)

The length for which the metabolic rate equals the intake rate allocated to growth (κF (x) = ρx3)is referred to as the maximum length, denoted by xmax.

For adults, the per capita birth rate is calculated by dividing the investment in reproduction


Table 1: The Kooijman-Metz model: individual level functions. The weight-length relation, attackrate, digestion time and maintenance rate are basic (”empirical”) functions; the other functionsare derived expressions based on the assumptions.

Weight-length w(x) = λx3

Attack rate A(x) =

{αx2 (x− xp)2 if x ≤ xp0 otherwise

Digestion time1 H(x) = ξx−2

Maintenance M(x) = ρx3

Holling type II F (x) = caA(x)R

1+H(x)A(x)R

Growth rate g(x) =

{0 if κF (x) < M(x)

13λx2 [κF (x)−M(x)] otherwise

Birth rate b(x) =

{cr(1− κ)F (x) 1λxb3 if x > xf0 otherwise

Total mortality µ(x) = µ0 + µs(x)

Starvation µs(x) =

{s [M(x)− κF (x)] if κF (x) < M(x)0 otherwise

1 The published version of this table (Claessen and de Roos, 2003), incorrectly states H(x) = ξx−3

by the energy cost of producing a single newborn:

b(x) =

{cr(1− κ)F (x) 1λxb3 if x ≥ xf ,0 otherwise,

(6)

with the conversion efficiency cr.The mortality rate is assumed to be the sum of a constant background mortality rate µ0, and

a starvation mortality:µ(x) = µ0 + µs(x) (7)

In equilibrium individuals cannot grow beyond the maximum sustainable size, so for an equi-librium analysis we do not have to consider starvation mortality. However, in population cyclesindividuals may go through periods of food shortage and starvation. For such cases we assumethat starvation mortality rate increases linearly with the difference between the metabolic rateand the food assimilation rate (Table 1).

We assume that the alternative resource population is unstructured. In our model it followssemi-chemostat dynamics extended with a term to account for the effect of consumption by thestructured population,

dR

dt= r(K −R)−

∫ ∞xb

A(x)R

1 +H(x)A(x)Rn(x)dx (8)

with A(x) and H(x) as defined in Table 1.The individual-level model is summarized in Table 1 and the PDE formulation for the population-


Table 2: The Kooijman-Metz model: specification of the dynamics of p-state variables. Theindividual-level functions are listed in Table 1.

PDE∂n

∂t+∂gn

∂x= −µ(x) n(x)

Boundary condition g(xb)n(xb) =∫ xmaxxf

b(x)n(x) dx

Resource dynamicsdR

dt= r(K −R)−

∫ xmaxxb

A(x)R

1 +H(x)A(x)Rn(x)dx

level model is presented in Table 2. The list of parameters and their values can be found inClaessen and de Roos (2003).

PSP models such as this one are often studied by numerical intergration (simulation). Anefficient method for this is the Escalator Boxcar Train (EBT) (de Roos, 1997). In order to simulatethe model, the population size distribution n(x) needs to be discretised, and the EBT method doesso in a natural way by keeping track of a (variable) number of cohorts; for each cohort, the EBTintegrates ordinary differential equations for the cohort abundance and the i-state variables (bodylength x in this case). There are two typical population dynamic behaviours of the KM model: astable equilibrium and generation cycles. Generation cycles are discussed in more detail below(section 2.3).

The environmental feedback loop How does all this illustrates the principle of the environ-mental feedback loop? The interaction environment, denoted by E in the Introduction, is heredefined as the resource density (R). On the one hand, knowing R, the life history of an indi-vidual is entirely specified: its growth trajectory is obtained by integration of the growth rate;its reproductive output is obtained by computing its per-capita, size-dependent birth rate; itssurvival curve can be obtained by integrating the size-dependent per capita mortality rate. Thisillustrates the statement that onceE is known, individuals can be considered in isolation (despitethe presence of density dependent interactions). On the other hand, the impact of an individualon its environment can also be computed, by integrating its feeding rate along its life history (seebelow for equations). Since this represents the impact on the environment by a single individualonly, we need a measure of the total population size in order to complete the description of theenvironmental feedback loop (Fig. 1). A convenient measure is the total population birth rateP (that is, P is the product of the number of individuals and their total expected reproductiveoutput). Then, multiplying the cumulative consumption rate with P gives the consumption rateof the whole population. In other words: the impact of the population on its environment (Fig.1).

Life history as an input-output map In more mathematical terms, the above paragraph can bemade explicit as follows. Elements from the individual-level model outlined above can be usedto construct a life history if the appropriate input is given. We subdivide the life history into threeaspects; survival, growth and reproduction. The probability to survive to age a is denoted S(a)and is the solution of the ODE

dS

da= −µ(x(a))S(a) , S(0) = 1, (9)


where the function µ(x) is the size-dependent mortality rate. The growth trajectory, denoted x(a),is the solution of

dx

da= g(x(a)) , x(0) = xb, (10)

with g(x) the growth rate in length. The expected, cumulative reproduction up to age a, denotedB(a), is the solution of

dB

da= b(x(a))S(a) , B(0) = 0, (11)

in which b(x) is the size-dependent, per capita birth rate. The expected, life-time reproductiveoutput, denoted R0, is then given by:

R0 = B(∞) (12)

Due to the occurrence of x(a) in (9) and (11), (10) has to be solved first, then (9), and finally (11).Alternatively, the ODEs (9-11) can be solved simultaneously. Together, S(a), x(a) and R0 definea life history. In other words, the recipe for translating a given environmental condition into thecorresponding life history (arrow from ”Environment” to ”Individual” in Fig. 1) is by solvingequations (9-11).

The next thing we need to do, is to find the return map, that is from a given life history backto the impact on the environment. For that we need two ingredients: the life history and thenumber of individuals (since the impact is determined by the whole population collectively). Fora single individuals with a given life history, the expected, cumulative consumption up to age a,is denoted with θ(a,R). It can be calculated in parallel with (9-10) by integrating

dθ

da=

A(x(a))R

1 +A(x(a))RH(x(a))S(a) , θ(0, R) = 0 (13)

(cf. (4)). The total population consumption rate of alternative resource is then the product of Pand θ(∞, R). Note that for this computation P is required as an input variable, since it cannot bederived from the life history. In other words, the population impact on the environment given acertain life history (arrow from ”Individual” passing through ”Population” to ”Environment” inFig. 1) equals P θ(∞, R).

A merit of this way to characterise the environmental feedback loop, is that it provides us witha low-dimensional definition of the population dynamical equilibrium, and a tool to computethis equilibrium using continuation techniques. It should be noted that the population-levelmodel (Table 2) is an infinite-dimensional object (i.e., a continuous function). Characterisingthe population equilibrium with the function n(x) therefore does not lend itself to numericalequilibrium analysis. By contrast, the environmental feedback loop is characterised by only two(unkown) variables: the food density R and the population birth rate P . Now observe that thepopulation dynamic equilibrium of the KM model can be charcterised by two criteria, being therequirements that each individuals replaces itself (R0 = 1) and that the resource is at equilibrium(r(K −R) = P θ(∞, R)). These are two equations in the two unknowns R and P .

Following Diekmann et al. (1998, 2010), we refer to the unkown variables R := I1 and P := I2as the input variables I (where I is the vector of the input variables), and to the equilibriumconditions R0 − 1 := O1 and r(K − R) − P θ(∞, R) := O2 as output variables O (the vector ofourput variables). Then the environmental feedback loop yields the map

f : Rk → Rk; I 7→ O (14)

which is referred to as the input-output map. (In this particular example, k = 2 since we have twoinput and output variables. See Claessen and de Roos (2003) for another example with k ≈ ∞).An equilibrium can be found via an input I∗ ∈ Rk for which the equilibrium and feedbackconditions

f(I∗) = 0 (15)

hold. The condition f(I∗) = 0 can now be used in numerical continuation. The continuation


method of Kirkilionis et al. (2001) can trace the equilibrium as a function of one free parameter.More recent development of the mathematical theory has enabled stability analysis of the

equilibrium along the traced equilibrium curve, and in particular of the detection of a Hopf bi-furcation and the two-parameter continuation of the Hopf bifurcation (de Roos et al., 2010; Diek-mann et al., 2010). These are the first steps towards more general numerical continuation toolsfor PSP models. Given the enormous contribution for the study of models based on ODEs withcontinuation tools such as AUTO and Content/Matcont, this is a very promising development.

Applications An application of this method is illutrated in Fig. 11 (page 32). The continuationmethod enables us to trace the equilibrium curve of a PSP model while varying a model param-eter. Even if the equilibrium is unstable, and even if the curve folds backwards (as in Fig. 11),the equilibrium can be traced. It is clear that such an unstable equilibrium curve, in between twoalternative stable states, would be impossible to find by simulation only. In Claessen and de Roos(2003) we use this method to study the influence of the size-dependent nature of a cannibalisticinteraction on the equilibrium. The method allows us to detect two fold bifurcations (similar tothe folded curve in Fig. 11) in the equilibrium curve. The found equilibrium curve helps inter-preting the population dynamics observed in simulations (which are limited to stable equilibriaand other attractors). In particular, we are able to identify the ”biological” process that causesbistability in the cannibalistic model: only if the cannibals spare their smallest victims, i.e., if thevictims are invulnerable to cannibalism up to a critical size, then cannibals are able to reach giantsizes. If, by contrast, cannibals are able to include even the smallest indviduals in their diet, thenthey are bound to reach a maximum body size not much bigger than their maturation size. Wecalled this the ”Hansel and Gretel effect”.

2.3 The effect of temperature on generation cycles (fish)

This section describes the work done in a collaboration with Jan Ohlberger (postdoc at CEES,Oslo), Eric Edeline (UPMC and Bioemco Lab), Oystein Langangen (postdoc at CEES, Oslo), andco-workers at CEES, Oslo and the CEH, Lake Windermere. The modelling work has led to twopublications: Ohlberger et al. (2011a) and Ohlberger et al. (2011b) and is still ongoing. The firstobjective, described here, is to adapt the PSP model that I developed during my thesis for can-nibalistic freshwater fish (Claessen et al., 2000) in order to answer to a number of questions con-cerning the dynamics of Lake Windermere fish in particular, and some questions on the effect oftemperature on cohort cycles in general. This work was done in the wider context of a projectproposed by Eric Edeline to the Norwegian Science Council to study the effect of climate changeon lake ecosystems, using both population dynamic modelling and time series analysis. Theempirical part of the work is based on the long term observations of the fish populations inLake Windermere (UK). The general aim of the project is to try to disentangle the effects on thefish community dynamics of climate change, ecological interactions, fisheries, short-term evolu-tion, and nutrient loading; processes that are all known or likely to influence the perch and pikepopulations of Lake Windermere. In addition to the work on a PSP model, described in somedetail below, we also studied a more simple stage-structured population model (Ohlberger et al.,2011b), using the biomass modelling approach of De Roos et al. (2007, 2008).

A first study of the effect of temperature on population cycles was published by Vasseur andMcCann (2005). They used a simple, unstructured bioenergetics model to determine the influ-ence of temperature on a consumer-resource interaction in order to predict the consequencesof temperature changes on the dynamics and persistence of consumer populations. Their re-sults indicate that warming is likely to destabilize consumer-resource interactions and that thequalitative response of the population dynamics depends on whether individual metabolic rateincreases faster or slower with temperature than ingestion rate. Their model is a first step towarda bioenergetics theory of the impact of climate change on food web dynamics. The simplicity oftheir model allows them to analyse the model in quite some detail. The drawback is that they can-not address the size-dependent influence of temperature on organisms, and its consequences. In


particular, empirical evidence suggests that small and large individuals do not respond equallyto temperature. For example, in cold environments, large individuals have a higher metabolicefficiency compared to small individuals (Kozlowski et al., 2004). Van de Wolfshaar et al. (2008)present the first size-structured population model that accounts explicitly for seasonal temper-ature effects on vital rates. They show that the combined temperature- and size-dependence ofvital rates may have fatal consequences for winter survival of both individuals and the popula-tion as a whole. However, they did not study the effect of changing temperature on populationdynamics, the objective of the modelling exercise described here (Ohlberger et al., 2011a).

Life history and population dynamics of a number of freshwater fish species including roach(Rutilus rutilus), Eurasian perch (Perca fluviatilis), yellow perch (P. flavescens) and Northern pike(Esox lucius), have been modelled with a model in which the i-state of a fish is defined by two vari-ables: the amount of irreversible mass (x) and reversible mass (y) (Persson et al., 1998; Claessenet al., 2000, 2002; de Roos and Persson, 2001; Persson et al., 2004; Persson and De Roos, 2006). Themodel assumes that if x and y are known, all ecological functions can be derived from these twoquantities: x and y hence completely define the state of an individual. For example, total bodymass equals w = x + y, gonad mass equals y − qx (where q is a constant), body length dependson x only, the search rates for different types of food are functions of x only. The state of thepopulation is defined as the distribution of the number of individuals over the individual state(e.g., n(x, y)). Inspired by the biology of temperate freshwater fish, these models assume thatreproduction occurs in a pulsed way during spring. At this moment, the gonad mass of adultindividuals is converted into newborns. Together, the newborns form a new ”cohort” of iden-tical individuals. Thus the population consists of a variable number of cohorts, each describedby a set of differential equations for the dynamics of x, y and Ni, where the latter is the abun-dance of cohort i. Cohorts disappear from the population when their abundance drops below atrivial threshold (e.g., a single individuals per lake). In these models, the environment is char-acterised by the population densities of prey (e.g., zooplankton) and, possibly, predators (e.g.,cannibalistic conspecifics). Population feedback arises from the assumption that the dynamics ofthe environment depend on the state of the fish population: consumption by the fish depletesthe zooplanton population and possibly causes mortality of small fish. Here I will not give anydetailed description of these models, which can be found in the original publications cited aboveand in Ohlberger et al. (2011a) .

Generation cycles A very general result from PSP modelling is that intra-specific competitiontends to cause ”generation cycles”, also known as ”cohort cycles” (Gurney and Nisbet, 1985;de Roos, 1997; Persson et al., 1998; Claessen et al., 2000; de Roos and Persson, 2003). This kindof population cycles is distinct from the better-known predator-prey cycles, also referred to as”consumer-resource cycles” or ”delayed-feedback cycles”. The two types of population cyclescan theoretically be distinguished by the cycle periodicity (Murdoch et al., 2002). Predator-preycycles are expected to have a periodicity of at least 4TC + 2TR, where TC and TR are the mat-uration times of the consumer and resources, respectively. Intuitively, this can be understoodby considering the classical Lotka-Volterra model and its predator-prey cycles. Each cycle con-sists a phase of exponential predator growth, depleting the prey population; followed by a phaseof predator exponential decline; followed by exponential prey growth. Each bit of exponentialgrowth or decline requires at least a few generations to complete. By contrast, generation cy-cles have a periodicity of one generation (exceptions include 0.5 generations or 2 generations percycle).

The basic mechanism of generation cycles is an inter-generational conflict: each generationreplaces its parental generation through intra-specific competition. The mechanism can be il-lustrated by considering an example. In Ohlberger et al. (2011a) we present a model of a size-structured fish population. Under certain conditions (e.g., 17◦C in Fig. 2) the populations exhibitscycles of which the periodicity corresponds to the maturation time. The figure shows that thebirth of each new generation (black dots, lower panel) is followed by a sudden depletion of thezooplankton population (resource, upper panel). The adult portion of the population goes extinct


Figure 2: Population dynamics of themodel of Ohlberger et al. (2011a) at temper-atures 15◦C and 17◦C (for the case with-out cannibalism). Top plots: predicted re-source density (solid lines) and critical re-source densities (defined as zero-growth re-source level) for newborns (dashed lines)and mature fish (dotted lines). Bottomplots: predicted consumer density of new-borns (circles), juveniles (grey lines) andadults (black lines). The left panel showsfixed point (FP) dynamics, whereas theright panel shows recruit-driven generationcycles (GC), here also referred to as single-cohort cycles. From: Ohlberger et al. (2011a)

rapidly, starved to death by the low resource abundance. The new generation remains juvenilefor several years, during which it declines exponentially due to a constant mortality rate, andthe individuals grow in size. The resource gradually increases (due to the decline of the juvenilecohort). The juveniles mature in their 5th year, and the following spring they reproduce the nextgeneration.

The underlying mechanism of this type of cycles depends on the size scaling of the ecologicalproperties of individuals. The basic, size-dependent functional relations in the model are similarto that of the Kooijman-Metz model (Table 1), i.e., a type II functional response, a hump-shapedattack rate on zooplankton, an increasing maintenance rate with body size, decreasing digestiontime with body size. From these basic ingredients we can derive a dependent relationship, whichis referred to as the ”critical resource density” R∗(x). As its notation suggests, this quantity isanalogous to Tilman (1982)’s R∗, except that in our case it is a function of body size rather thana single measure for the whole population. As in Tilman’s theory, an individual’s competitiveability is measured by R∗(x): the lower its value, the better the individual can deal with severelycompetitive situations. In the case of the fish model, R∗(x) is an increasing function of bodysize. This results mainly from the fact that maintenance requirements increase faster with bodysize than the feeding rate. This result is general for PSP models that have been parametrisedfor particular fish species (Persson et al., 1998; de Roos and Persson, 2003; Persson and De Roos,2006). More theoretically, we can note that the feeding rate is generally a surface-limited processand hence likely to scale with the body mass to the power 2/3, whereas maintenance is a mass-limited process and hence likely to scale linearly with body mass (Kooijman, 2000) (even thoughan alternative theory postulates a 3/4 scaling rule, Brown et al., 2004). Thus, also based on thesesimple theoretical considerations, R∗(x) is expected to increase with body size (even under the3/4 scaling rule!).

Generation cycles result from the asymmetric competition between differently sized individ-uals, if R∗(x) is sufficiently steeply increasing (or decreasing) with body size (Persson et al., 1998;de Roos and Persson, 2003). In our example, R∗(x) increases with body size and hence newbornsare competitively superior to adults. A sufficiently high fecundity of adults then automaticallyresults in generation cycles.

Generation cycles occur even in more complex models that include multiple ecological inter-actions. Cannibalism has the potential to dampen generation cycles, because it allows adults toreduce the competition with newborns in two ways: killing newborns reduces the abundance ofcompetitors, and eating newborns provides extra energy (Claessen et al., 2000). Yet even with


Figure 3: Temperature dependence terms (A) forperch consumption (solid), perch metabolism (dashed)and zooplankton growth rate (dotted), and the indi-vidual net energy gain (B) as a function of tempera-ture for perch of body weight 0.1g (solid), 1g (dashed)and 10g (dotted). Thin dotted lines (A) indicate cal-ibration to a value of 1 at 20◦C (see text). The bodysize of perch was set to 8.2g, the optimal size for preyattack. The net energy gain for differently sized perch(B) was calculated at a zooplankton density of 2 Ind/L,which resembles rather low resource levels where ex-ploitative competition in perch is expected to be high.From: Ohlberger et al. (2011a)

strong cannibalism generation cycles may be present, even though they are modified by the can-nibalistic interaction; for example, the period length may be longer, and the population maycontain very big, cannibalistic individuals in addition to the cohort of juveniles that drive theperiodicity (Claessen et al., 2000; Persson et al., 2004).

The effect of temperature What are the consequences of climate change on population dynam-ics? Or more generally, what is the relation between abiotic factors and ecological dynamics?Answering these questions may provide clues for testing theoretical predictions of our ecologicaltheories with empirical data. And it may provide clues for how ecosystems respond to environ-mental change. Modelling dynamical consequences of the physiological response to temperaturechange requires the ability to translate individual-level physiological processes into populationlevel dynamics. The framework of PSP models allow us to do so. The model formulation focusseson the description of individual-level properties of individuals such as the maintenance require-ments, feeding rate, etc. Parameters are often numerous in such models (see the long tables inmany of the above cited articles). Yet most of these parameters are fairly ”easy” to measure inlab settings. The model then provides independent population-level predictions of the emergentdynamics. This contrast with more simplistic modelling frameworks, such as the Lotka-Volterraand derived models, which form the basis of theoretical ecology. Finding the temperature depen-dence of parameters of such simpler models may be more difficult than finding all parameters ofa PSP model. For instance, the parameters r and K of the logistic growth equation (a frequent in-gredient of Lotka-Volterra type models) are essentially population-level quantities, and are henceinherently impossible to measure without actually measuring the population level dynamics inthe lab or field.

The goal of the modelling exercise in Ohlberger et al. (2011a) is to make the basic eco-physiologicalfunctions in the above described ”fish” model (Claessen et al., 2000, 2002) temperature depen-dent; the consumption rate (parametrised by the attack rate and the digestion time), the metabolicrate, and the zooplankton renewal rate. To do so, we multiplied each function by a temperature-dependent scaling factor (Fig. 3). We us empirically-based scaling relations rather than the more


Figure 4: The regime diagram of the PSP modelin Ohlberger et al. (2011a). GC=generation cycles.FP=fixed point dynamics. CD=cannibal-driven dy-namics. CD-GC=alternation of CD and GC dynam-ics (both types of dynamics are unstable). FP orGC=coexistence of two attractors (corresponding toFP and GC).

commonly used theoretical model for temperature dependence of chemical reactions, referred toas the Boltzmann factor or the Van ’t Hoff-Arrhenius equation (Brown et al., 2004). The latterequation is one of the elements of the metabolic theory of ecology. However, looking into the lit-erature on fish ecology, we found that the actual measurements of the temperature dependenceof these relations deviates significantly from the theoretical (chemical) model. In particular, allthree empirical relations (Fig. 3) display a maximum (at different optimum temperatures) ratherthan a monotonically increasing shape, as is the case of the Arrhenius equation.

Note that in the model of Ohlberger et al. (2011a), these functions are size and temperature-dependent. Despite this complexity, we obtained fairly simple and seemingly general results:increasing temperature tends to increase the level of intra-specific competition and tends to re-sult in the onset of generation cycles. That is, on a temperature gradient we expect generationcycles at high temperatures and stable populations (”fixed point dynamics”) at low tempera-tures. This result is illustrated with the example of the predicted population dynamics at twodifferent temperatures (Fig. 2). At 15◦C, the population displays so-called fixed point dynamics(FP), which means that the within-year year dynamics are more or less the same from year toyear. (Note that reproduction is pulsed at the beginning of the growing season). Such FP dynam-ics are characterised by the coexistence of a large number of cohorts (age classes). Intra-specificcompetition is too weak to result in the exclusion of some cohorts by other ones. By contrast, at17◦C, the model displays typical generation cycles, in this case characterised by the existence ofa single cohort during most of the cycle (except during a short period following reproduction).

While this provides only two examples, a more general result can be found in Fig. 4. In factthe model we studied includes cannibalism as well as competition (in Fig. 2 cannibalism is as-sumed to be absent). Cannibalism was included in the analysis to assess the generality of theeffect of temperature on generation cycles, since we know generation cycles occur without andwith cannibalism, although in modified form (see above). The figure clearly shows that increas-ing temperature tends to result in generation cycles irrespective of the level of cannibalism. Theamplitude of population fluctuations is predicted to increase with temperature, in both cannibal-istic and non-cannibalistic populations Ohlberger et al. (2011a).

Throughout the studied temperature range (12-22◦C), the model assumes that the resourcepopulation growth rate increases with temperature. Yet the model predicts an overall decreaseof the mean resource density (i.e., the long-term average given a fixed temperature), caused byincreased food intake by consumers, and increased consumer total reproduction rate (Ohlbergeret al., 2011a). The lower resource level at high temperatures reflects the increased level of intra-specific competition. A direct consequence of the different temperature dependencies of the vitalrates at the individual level is that energy gain increases faster with temperature for small indi-viduals (Fig. 3). Therefore, cool conditions favour big individuals, while warm conditions favoursmall ones. Increasing temperature hence reinforces the competitive advantage of small overlarge individuals, which enhances the mechanism that causes generation cycles (Persson et al.,


1998).Above we have seen that generation cycles are often the result of the fact that the critical re-

source density R(x) increases with body size (which is the case for all fish for which the dataexist, Persson and De Roos, 2006). The size dependence of intraspecific competition, however,changes with temperature. The net energy gain of an individual increases faster with tempera-ture at smaller sizes (Fig. 3), thereby magnifying the competitive advantage of small over largeindividuals. This effect is reflected in the general observation that optimum growth temperatures(which for immature fish can be assumed to be equal to those of the net energy gain) decreasewith increasing body size (Kozlowski et al., 2004). This has been reported for several fish species(Karas and Thoresson, 1992; Bjornsson and Steinarsson, 2002; Imsland et al., 2006) and otherectotherms such as amphipods (Panov and McQueen, 1998). Thus, this functional form of thetemperature-size relationships may be valid for other fish species and possibly ectotherms ingeneral.

Our conclusion on the effect of temperature on intra-specific competition and the resultinggeneration cycles are in line with those of Vasseur and McCann (2005): higher temperatures arepredicted to destabilise population dynamics. Recall that that the model of Vasseur and Mc-Cann (2005) concerned consumer-resource (predator-prey) cycles. Interestingly, these two typesof population cycles (generation cycles vs predator-prey cycles) are caused by ecologically anddynamically very different processes (size-asymmetric competition vs delayed feedback throughthe predator-prey interaction), and characterised by very different cycle periodicities (Murdochet al., 2002). Whether these predictions hold for more complex food webs than the ones modelledin these studies remains to be seen. Yet these predictions can be put to the test relatively easily,by comparing population dynamics across temperature gradients, albeit natural ones or in thelaboratory (Ohlberger et al., 2011a).

The environmental feedback loop In the model of Ohlberger et al. (2011a), the environmentalfeedback loop includes two different ecological interactions: competition via depletion of thezooplankton resource; and cannibalism. Competition for zooplankton is an indirect density-dependent interaction, that operates through the E-variable R. The impact of R on individuallife history is via the size-dependent functional response (similar to equation (4)). The impactof the population on this E-variable is through the population-level, total feeding rate, a verysimilar expression to equation (8). Cannibalism is a direct-dependent interaction, in the sense thatcannibalistic mortality and feeding rates depend directly on the current population abundanceand size distribution. For this interaction, the E-variable hence contains the entire populationsize distribution. The impact of this E-variable on the individual life history is mediated by acomplex function that takes into account the body sizes of any pair of interacting cannibals andvictims (Claessen et al., 2000, 2002, 2004), see equation Ac(c, v, x, y, T ) in Table 2 of Ohlbergeret al. (2011a)), as well as the abundances of victims and cannibals.

2.4 Stochastic dynamics of small populations (lizards)

This section describes the work that has been done in collaboration with Manuela Gonzalez-Suarez and Jean-Francois le Galliard. Manuela has been a postdoc in the Eco-Evo Laboratory(UMR 7625) supervised by Jean-Francois and myself. The work has so far resulted in two publi-cations: Gonzalez-Suarez et al. (2011a) and Gonzalez-Suarez et al. (2011b). The goal of this workis to study the consequences of demographic stochasticity for populations of which life history issubject to plasticity. Demographic stochasticity is randomness that results from the discretenessof individuals and the probabilistic nature of demographic processes such as birth, death, clutchsize, etc. Its influence is expected to be inversely proportional to population abundance: it shouldhence be most important in small population.

We have developed a PSP model for a model system, the common lizard (Lacerta vivipara) (Fig.5), based on experimental data obtained by Jean-Francois Le Galliard and Manuela Gonzalez-Suarez as well as other members of the laboratory, and complemented with literature data. The


Figure 5: The common lizard (Lac-erta vivipara) in its experimen-tal environment: the semi-naturalenclosures at the CEREEP fieldstation.

objective is to obtain a model of population dynamics for this species that can be used to predictor interpret the dynamics of semi-natural enclosures at the CEREEP experimental station2. Themodel is used to infer general aspects of the population dynamics of small, structured popula-tions, subject to demographic stochasticity.

General context Spatio-temporal stochastic variability is ubiquitous in ecological systems. It isparticularly important in extinction and invasion which are inherently stochastic processes dueto low population numbers. To understand extinction and invasion dynamics it is hence crucialto understand the consequences of stochasticity. Heterogeneity in the environment is expectedto result in life history variability between individuals (Van Kooten et al., 2004, 2007). Owingto the mutual dependence of life history and population dynamics, such variability is boundto have consequences for extinction dynamics of small populations and for mutant invasion inevolutionary dynamics.

To date, stochasticity and plastic (density-dependent) life history have rarely been studiedin conjunction. On the one hand there is a large body of theory on the role of stochasticity onstructured-population dynamics which has yielded powerful tools to predict population growthrate, extinction risk, evolutionarily stable strategies, etc (e.g. Tuljapurkar, 1997; Orzack, 1997; Hac-cou et al., 2005). However, most of this theory assumes density-independent population dynam-ics (e.g. Tuljapurkar, 1997) or non-plastic life history (e.g. age-structured populations, Bjornstadet al., 2004; Le Galliard et al., 2005) and therefore cannot account for plastic life histories. Onthe other hand, a dynamic theory about plastic life history has been developed based on physi-ologically structured population models (de Roos et al., 2003). The message from this theory istwofold: population feedback on life history (i.e., density-dependent effects on life history) is crit-ical to the realization of life histories while in turn population dynamics depend on realized lifehistories. However, current theory of PSP models assumes deterministic population dynamics(but see some exceptions: Claessen and Dieckmann, 2002; Van Kooten et al., 2004, 2007; De Rooset al., 2009).

Considering the consequences of the link between plastic life history and stochastic popula-tion dynamics, two interesting questions emerge:

2CEREEP Ecotron-Ile-de-France = Centre de Recherche en Ecologie Expérimentale et Prédictive & Ecotron Ile-de-France, UMS 3194 ENS-CNRS. Based at the Foljuif domain of the ENS, at the southern end of the Fontainebleau forest(Nemours, Seine et Marne).


1. Life history provides the population with a “memory” of past stochastic fluctuations. Whatis the role of such delayed effects on the dynamics of physiologically structured popula-tions?

2. What is the role of plastic life history in the population dynamics of small populations?First, density-dependent compensatory effects are expected to increase growth rate andfecundity at low densities. Second, demographic stochasticity is expected to result in lifehistory variability between individuals. Do these effects alter the predicted extinction riskand expected time to extinction?

Modelling the common lizard With these questions in mind, we started out building a PSPmodel for the common lizard. The full model description can be found in Gonzalez-Suarez et al.(2011a). There are three i-state variables: age, structural mass (i.e., bone, organs), and reservesmass (i.e., adipose and reproductive tissues). We assume that energy acquisition, growth, sur-vival, and reproduction are functions of body mass defined by an energy budget model. Foodintake and metabolism also depend on environmental conditions, that is, sunshine duration, toreflect the importance of weather on lizard life history (Adolph and Porter, 1993). Food intake isalso a function of a density-dependent scaling function D(B) that provides feedback from popu-lation dynamics to the individual process of food consumption. Note thatD(B) represents hencethe environmental feedback loop in this model (Fig. 1). Whereas PSP models usually model thedynamics of the resource population explicitly, we cannot accurately model prey dynamics be-cause the common lizard feeds on a large variety of prey and its functional response is not wellunderstood (Avery, 1971; Gonzalez-Suarez et al., 2011b). In the absence of quantitative empiricaldata to adequately define the consumer-resource interaction in this species (in terms of the func-tional response, prey renewal rates, etc), we model density dependence in a phenomenologicalway, using a simple function D(B) that reflects our general knowledge of the species feedingbiology. An individuals feeding rate is obtained by multiplying its empirical, size-dependentfeeding rate under standard conditions (Gonzalez-Suarez et al., 2011b) by the function D(B),which is a decreasing function of the populations weighted abundance B.

Understanding how assimilated energy is actually channelled in an organism is complicated,and numerous energy allocation rules have been proposed (Kooijman, 2000; Claessen et al.,2009). We assume that individuals follow a ”net-production allocation rule” before first reproduc-tion and a ”gross-production allocation rule” (Kooijman 2000) after the first reproduction event.These two allocation models reflect observed differences in prioritization between reproductivelizards, which prioritize reproduction, and nonreproductive individuals, which prioritize struc-tural growth (Andrews, 1982).

The environmental feedback loop The above model is an example of a PSP model with directdensity dependence, that is, the interaction is not mediated through a resource. (Note that thecannibalistic interaction in Claessen et al. (2000, 2002); Claessen and de Roos (2003); Ohlbergeret al. (2011a) is also a direct density dependent interaction). In more detail, the environmentalfeedback loop is modelled as follows. First, the environment E impacts individual life historythrough the E-dependent feeding rate (in mass per day):

C(x,B) = sunD(B) γ1xγ2 (16)

where x is body size, sun is the number of hours of sunshine per day. The allometric functionγ1x

γ2 is the empirical relation between feeding rate and body size under standardized conditions.Thus, under ”typical” population densities, the feeding rate of an individual depends linearlyon sunshine duration, and depends allometrically on body size. Whenever population densityexceeds the ”typical” density, denoted by B0, the feeding rate should be below the standard rate,


and vice versa. We modelling this in a phenomenological way using the following equation:

D(B) = exp

(δ

(1− B

B0

))(17)

in which δ is parameter tuning the sensitivity of the feeding rate to changes in population densityB. To complete the description of the feedback loop we need to specify the ”population density”B. Even though this is a phenomenological model (i.e., non-process based, non-mechanistic), theidea behind it is that the lizards interact mainly through the depletion of their resources. There-fore, we assume that the contribution of an individual to the collective impact on the environmentdepends on its individual feeding rate. We define B as the total, standardized feeding rate:

B =

n∑i=1

γ1xγ2i (18)

where n is the current number of individuals in the population, and xi is the body size of indi-vidual i.

Thus, in terms of Fig. 1, the arrow from environment to individual is captured by equations(16) and (17); the arrow from population to environment corresponds to equation (18). The ar-row from individual to population represents the contribution of individuals to their collectivedynamics: reproduction, growth and mortality, as described briefly above.

Stochastic population dynamics We studied the effect of demographic stochasticity on lifehistory and population dynamics by comparing different model versions, that are increasinglystochastic. The first version is a fully deterministic model, in which the abundance of each yearclass j is described by a differential equation dNj/dt. All other model versions are stochastic. Inmodel version 2, cohorts are discretized into individuals. This results in a size-structured birth-death process, in which individuals are discrete units that are born (all on the same day in theyear), grow continuously in time, reproduce discretely in time, and then die. Model version 3modifies the second model with a stochastic component to the food intake rate. This reflects thatthe function D(B) describes the average environment, whereas each individual experiences itslocal environment in a more heterogeneous way. One day, some individuals get lucky and finda large prey, whereas others find almost nothing. Each day, an individual’s local environmentis randomly drawn from a normal distribution with a mean of D(B) and a standard deviationreflecting observed variability in consumption rates of a lizard population. An alternative exten-sion of the second model (version 4) relaxes the assumption that all individuals of a year classare born on the same day. Rather, a female may reproduce on any day in a given birthing periodof about a month (drawn from a truncated normal distribution). Finally, version 5 combines ver-sions 3 and 4: discrete individuals with stochasticity in food intake and in birthdays. We chosethese two stochastic processes (food intake rate and birthday) because they are thought to beimportant heterogeneous components of the life history of the studied species.

The model has been thoroughly parametrized to laboratory, field and literature data. A fulllist of model parameters, their values and how the data was obtained is given in Table 1 ofGonzalez-Suarez et al. (2011a). All versions of the model thus parametrized, predict relativelystable population dynamics, in the sense that generation cycles or predator-prey cycles are notobserved. One explanation of the absence of generation cycles is the relatively low population-level reproduction rate, which is insufficient to cause the level of size-dependent competitionrequired for generation cycles. The dynamics of the stochastic model versions are quite simi-lar to the deterministic version, although of course population dynamics and growth trajectoriesdisplay more variability in the latter cases. Yet the stochasticity does not qualitatively affect thepopulation-level population dynamics. It should be noted that up to date we have focused onnon-extinction population dynamics, with fairly high population numbers (around 100 to 200 in-dividuals). It remains an open question whether closer to extinction the stochastic model versionsmay display qualitatively different behaviour.


Figure 6: Comparison of population-level model predictions (A) and individual-level predictions of survival rate (B).Model versions are version 1=Det, version 2=Dis, version 3=Food, version 4=Birth, version 5=F&B. Emp=empiricallyobserved. In both panels longer error bars represent standard deviation (SD) among years, except for the ”Emp” age>1+estimate which represent estimates of SD among cohorts because annual estimates are not available. Shorter error barsillustrate SD among 10 stochastic replicates. From: Gonzalez-Suarez et al. (2011a)

Note that even though the model is heavily parametrized with empirical data, in particular onindividual-level aspects, many aspects of life history and population dynamics remain dynamic(output) variables of the model. In particular, model predictions of life history include growthtrajectories, maximum body size, age at maturation, realized fecundity, the survival function.Population-level predictions include the emerging population structure, for example the propor-tion of age-0, age-1 and older individuals, proportion of juveniles and adults, level of fluctua-tions. The stochastic models also provide predictions of the level of variability in any of these lifehistory and population aspects.

Fig. 6A shows that the difference between model versions is slight in terms of populationage structure. Independent empirical observations are generally consistent with these modelpredictions. All model versions predict multiple coexisting cohorts (age classes) as observed innatural populations (Massot et al., 1992). The predicted age structure (Fig. 6A) agrees well withempirical estimates (Massot et al., 1992).

The annual survival rates of the age classes are very similar and they correspond well toempirically observed survival rates (Fig. 6B). This is most interesting for the youngest age class,for which the survival rate is strongly size-dependent, and hence the model predictions dependon the realised population dynamics through the environmental feedback loop. The survivalrate of older individuals merely reflects our assumption of the size-independent component ofthe mortality rate.

More interesting are the life history comparisons of fecundity and the growth trajectories (Fig.7). Although our model underestimates the level of variability in fecundity, the mean fecundityis well captured (Fig. 7A). The growth trajectories predicted by the model are very similar toempirically observed ones, although the stochastic model versions tend to overestimate the av-erage growth rate and the maximum body size (Fig. 7B), especially the model versions 3 and 5.The increase in mean snout-vent length (SVL) in older individuals in model versions 3 and 5 isaccompanied by a decrease in body condition, so that these adults are characterized by beinglonger and skinnier. Variation in birth date alone (model version 4) results in differences in SVLamong individuals born early in the birthing period (first 10 days) vs. those born late (last 10days), with the former being significantly larger at all ages and reaching larger maximum sizes(Student t tests P < 0.031). Although significant, the actual differences in size are generally small(< 1 mm). Interestingly, variability in food consumption eliminates these differences: there areno differences in SVL between individuals born early or later in the F&B version (Student t testsP > 0.10).


Figure 7: Comparison of individual-level model predictions of life history characteristics and empirical observations.A: Fecundity (number of female offspring per adult female). B: growth trajectories. Model versions are version 1=Det,version 2=Dis, version 3=Food, version 4=Birth, version 5=F&B. Emp=empirically observed. In A, longer error bars areSD among years, and shorter error bars are SD among 10 stochastic replicates. Stars represent extreme values estimateas the mean of yearly maxima and minima. In panel B, round symbols represent mean snout-vent length (SVL) at fixedages (0, 200, 400, 600, and 800 days), error bars are SD among years, and stars represent maximum SVL estimate as themean of yearly maxima. From: Gonzalez-Suarez et al. (2011a)

Figure 8: Illustration of how Jensen’s inequality influences the energy budget for adults (body mass 4.5 g; panel A) andfor juveniles (body mass 0.4 g; panel B). Thick lines: assuming a constant food intake rate, the relationship between thegrowth rates of reserves mass (grey line), or structural mass (black line), with food intake rate is piece-wise linear. Thearrows indicate the transition in growth rates that occurs when the assimilated energy is insufficient to cover metaboliccosts (to the left of the arrow). Dotted lines: the average growth rates assuming daily stochastic variation in food intake,assuming that the daily food intake rate takes the extreme values only (i.e., either 0 or 0.7), with a certain probability.Along the dotted line, the probability of ”lucky days” changes from zero to one. From: Gonzalez-Suarez et al. (2011a)


The most striking result of the above modelling exercise is that stochastic variation in the foodintake rate results in long and skinny individuals even though the mean food intake remainsconstant. Skinny individuals are also less fecund, and thus the number of offspring per femaledecreases, which reduces competition among the newborn class and leads to higher survival ofyoung individuals. This predicted change in individual morphology can be explained by Jensen’sinequality and the non-linear relationship between body growth and daily food intake (Fig. 8).

Jensen’s inequality states that for a set of values xi with mean E(xi), the average resultof a non-linear function f(xi) (denoted by E(f(xi)) need not equal the function of the aver-age f(E(xi)) (Ruel and Ayres, 1999). Here, the non-linearity in body growth is caused by thetransition that occurs in the energy allocation when the assimilated energy is not sufficient tocover metabolic costs. This transition leads to a concave up relationship between structural-massgrowth and resource availability because growth is halted when energy intake is below mainte-nance costs (Fig. 8). As a result the mean growth rate of structural mass when food intake variesdaily is higher than the growth rate predicted for the mean food intake. Conversely, reservesmass growth has a concave down relationship as reserves are converted back to energy used tocover maintenance costs when food intake is insufficient. As a result the mean growth rate ofreserves mass is lower in a stochastic environment.

Transitions in growth are expected whenever individuals are able to survive for some timeusing energy reserves and body growth is reduced or stopped at the time when food intake isnot sufficient to cover maintenance costs. These simple requirements are met by a wide variety oftaxa (Kooijman, 2000); thus, the non-linear relationship between body growth and food availabil-ity should be very widespread. However, morphological changes may not be apparent if foodavailability always remains above or below the transition point. Changes in morphology will be-come apparent only when food intake falls below maintenance costs for some individuals at somepoint in time. This is likely to occur in food-regulated populations when population size is nearcarrying capacity, or in habitats with high intrinsic stochasticity in food availability. Confirm-ing our predictions, laboratory studies have shown that changing the temporal variance in foodavailability, while keeping the mean constant, results in morphological changes in sticklebacksand sea urchin larvae (Ali and Wootton, 1999; Miner and Vonesh, 2004). Whether the observedmorphological changes have demographic consequences in natural populations remains to beclarified. However, our results suggest population sizes may change, even though slightly in ourcase, thereby affecting overall resource levels.

2.5 Size-structure: conclusions

Above I have described and illustrated a theoretical framework (PSP models) that allows us tomodel the dynamics of size-structured populations. Size-dependent interactions may give rise toa number of dynamical behaviours that go beyond the scope of dynamics of unstructured modelssuch as Lotka-Volterra type models. I have given the example of generation cycles: asymmetriccompetition between small and large individuals may give rise to population cycles with a peri-odicity of one generation.

A characteristic of populations with plastic life history is that the environment (and hence theenvironmental feedback loop) determines, to a certain extent, the realised phenotype of individu-als, in terms of their growth trajectory and possibly other physiological traits such as corpulence.If the environment is constant, all individuals will have the same life history. If it is periodic (asin generation cycles), life histories will vary periodically, too. If the environment is stochasticand variable between individuals (as in the lizard model), this will result in between-individualvariability in life histories. In the case of the lizard model, the stochastic food environment in-fluences life history in an unexpected way, shifting life history systematically towards taller andskinnier phenotype

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