modelling the struggle for existence in structured populations1 introduction 6 1 introduction a...

Universite Pierre & Marie Curie

Modelling the struggle for existence instructured populations

Vol. 1: Synthesis

Memoire pour l’obtention de l’Habilitation a diriger des recherchesSoutenance : 10 janvier 2012 a Paris

David Claessen

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

andCentre d’Enseignement et des Recherches sur l’Environnementet la Societe - Environmental Research and Teaching Institute

(CERES-ERTI),Ecole Normale Superieure,

24 rue Lhomond,75005 Paris

2

Jury

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

• Ophelie Ronce (Universite Montpellier 2), rapporteur

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

• Jon Pitchford (University of York), examinateur

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

• Jean-Christophe Poggiale (Universite de la Mediterranee, Marseille), examinateur

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

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Preface

What is a Habilitation a 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 inWikipedia:

En France, l’habilitation a diriger des recherches est un diplome national del’enseignement superieur qu’il est possible d’obtenir apres un doctorat. Il a etecree en 1984 suite a la loi Savary. Ce diplome permet de postuler a un postede professeur des universites (apres inscription sur la liste de qualification parle Conseil national des universites), d’etre directeur de these ou choisi commerapporteur de these.

Elle est definie reglementairement par l’arrete du 23 novembre 1988 (modifie en1992, 1995 et 2002) : ”L’habilitation a diriger des recherches sanctionne la recon-naissance du haut niveau scientifique du candidat, du caractere original de sademarche dans un domaine de la science, de son aptitude a matriser une strategiede recherche dans un domaine scientifique ou technologique suffisamment largeet de sa capacite a encadrer de jeunes chercheurs. Elle permet notamment d’etrecandidat a l’acces au corps des professeurs des universites.”

D’apres la reglementation en vigueur, le dossier de candidature a l’habilitationa diriger des recherches comprend soit un ou plusieurs ouvrages publies oudactylographies, soit un dossier de travaux, accompagnes d’une synthese del’activite scientifique du candidat permettant de faire apparaıtre son experiencedans l’animation d’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;and for being examiner of a PhD thesis. It also states that the HDR application consists of afile of publications accompanied by a synthesis of scientific activities which should demon-strate the candidate’s experience in organising and coordinating research.

So preparing the HDR thesis is a moment to take a few steps back and to reflect onwhat I’ve been up to the last years. Since my PhD in Amsterdam and Umea, I’ve worked atRothamsted (UK), in Amsterdam and in Paris. I’ve been working on two postdoc projects be-fore obtaining a permanent teaching and research position at the Ecole Normale Superieurein Paris. These different places and projects have led me to work on a number of differentsubjects and research questions. My current position in Paris has led me to work more asa co-author than a first author (including the indispensable last authorship!). Throughoutthese years the common theme of my work has been the modelling of structured popula-tions, in ecological and evolutionary time. In particular I’ve been doing so from the view-point of the ”environmental feedback loop”, an approach that I have learned and developedduring my work and discussions with Andre de Roos, Odo Diekmann, Hans Metz and RegisFerriere. This memoire sums up this work, and especially the role of the latter idea in thesemodelling efforts, which I see as a ”red threat” running through my work.

I am a biologist, since I’ve studied biology. I am a modeller since that is what I mostlydo. I try to do theoretical ecology, meaning the development of ecological (and evolution-ary) ideas using conceptual tools, i.e., mainly mathematical models. By no means am I amathematician. I have invited a number of mathematicians to take place in my HDR jury

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since the field of theoretical ecology is, in my opinion, interfacing between mathematicsand ”real” (experimental) ecology. Although I try to answer to my research questions us-ing mathematical and computer models, and not by doing experiments myself, I have seenactual animals, plants and micro-organisms; I have been in labs, on lakes, in mesoscale en-closures, in fields. I have indeed touched and marked fish and voles (no lemmings, though),inspected cultivated and wild oilseed rape, and estimated lesion coverage on wheat leaves.And even though I do not do any experiments myself, I usually try to work in close collabo-ration with experimentalists that do. Theoretical ecology exists in the tension field betweenabstract theory and concrete case studies, and frequently we have to ask ourselves at whatlevel of generality/specificity we want to work. This is obviously a subjective choice, andthe ”optimal” choice often depends on the problem at hand. (The same is true for the typeof model we use to tackle a question.) Personally, I think it is useful to keep moving upand down between the abstract and the concrete. I my work I’ve worked on quite specificmodels for particular systems, but also on general ”out of the blue sky” toy models that haveno reality to them (well, perhaps...). I hope this synthesis will make some general sense outof a number of particular case studies, combined with a number of more abstract modellingexercises.

In addition to being a synthesis of my research, the text below is also a point of view onevolutionary ecology that I have developed to a large extent during my teaching activitiesat the ENS and for the Master EBE (co-hosted by UPMC, ENS, PSUD, AgroParisTech andMNHN). In particular, I talk a lot about the ideas in the following sections during my coursescalled ”Evolutionary ecology” (E2, Master 1 ENS), ”Structured populations” (STRU, Master2 EBE) and ”Adaptive dynamics” (DYAD, Master 2 EBE). My teaching on these subjects hasbeen largely inspired by other teachers, in particular Andre de Roos, Odo Diekmann, HansMetz and Regis Ferriere, who will no 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 enjoyit). First, I would like to acknowledge my own ”origins”, and thank my family for theirsupport, and especially my father for his contagious curiosity for the (meta-) physical worldaround us. Next, I thank Andre de Roos and Lennart Persson, who during my PhD (andafterwards) have been very inspiring and taught me all about ecology and how to do it. Ithank my colleagues from the Ecology lab in Paris, in particular Minus van Baalen, whointroduced me into the lab, and Regis Ferriere, who welcomed me in the Eco-EvolutionaryMathematics team, where I’ve had many interesting and helpful discussions. I want to thankmy colleagues at the CERES-ERTI for a stimulating interdisciplinary and generally intellec-tual environment (providing lots general discussions about mathemetics, climate, culture,language, history and geography). In particular, I want to thank Denis Rousseau for sup-porting me in my writing of this memoire, by giving me all the freedom I needed. In advance,I want to thank the members of my jury for accepting to critically read this document, anddiscussing it in January. Not last but not least, I also want to thank my colleagues with whomI have worked together. Without their collaboration, this work would not have been possi-ble. To name a few: Jens, Amaury, Robin, Vincent, Thomas, Francois, Boris, Mick, Chris, Jan,Oystein, Eric, Manuela, Jean-Francois, David, Tim, Tobias, Karen, Par, Ulf, Michael, Andreas,Bernard, Frank.

This memoire 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 impor-tantly of course, the miracles being in love and having a family, miracles that include ourlovely daughters Vera 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) . . . . . . . . . . . . . . . 162.4 Stochastic dynamics of small populations (lizards) . . . . . . . . . . . . . . . . 232.5 Size-structure: conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Adaptive dynamics 293.1 Again: the struggle for existence . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Sympatric speciation in (structured) fish populations . . . . . . . . . . . . . . 323.3 Speciation in dynamic landscapes . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 Adaptive dynamics: conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Current research 514.1 Local adaptation and species range shifts, in a size-structured population . . 524.2 Eco-evolutionary feedbacks between climate and phytoplankton . . . . . . . 55

5 Other work 59

6 Discussion 606.1 The struggle for existence in structured populations . . . . . . . . . . . . . . . 606.2 What next? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7 References 66

A Curriculum Vitae 73A.1 Life history (education and positions) . . . . . . . . . . . . . . . . . . . . . . . 73A.2 Supervision of students and postdocs . . . . . . . . . . . . . . . . . . . . . . . 73A.3 Scientific responsibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75A.4 Teaching responsabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75A.5 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

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 astruggle for existence, either one individual with another of the same species, or with the individ-uals of distinct species, 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 eco-logical vision of evolution. As the driving force of evolutionary change, Darwin identifiesthe ”struggle for existence”, which he considers to be the natural extension of the doctrine ofMalthus to the natural world. He considers this struggle for existence to be the logical resultof two basic observations: the tendency of populations to grow exponentially (”the principleof geometrical increase”); and the tendency of exponentially growing populations to be keptin check by ecological interactions such as food depletion, predation or epidemics. Withoutheritable variation, the struggle for existence simply results in a regulated population, ei-ther at a stable equilibrium or fluctuating. Evolution kicks in as soon as heritable variationbetween individuals of the same population exists. Allowing for heritable variation, somelineages will be more efficient at exploiting the resources or escaping their natural enemies,and are likely to outcompete the less efficient types in the population. I think that Darwin’sterm for the latter process, ”survival of fittest”, is less revealing of the ecological nature ofthe evolutionary process than the ”struggle for existence”.

The idea of the environmental feedback loop, the central theme of this synthesis and de-scribed in the next paragraph, is fully analogous to the Darwin (1859)’s description of thestruggle for existence.

1.2 The environmental feedback loop

The idea of the environmental feedback loop (Fig. 1) is the following. Individuals within apopulation (or a community) interact with each other in direct and indirect ways. Usually,in population biology, this is modelled by direct and indirect density dependence. Density-dependence can be introduced in many different (ad hoc) ways; a catalogue would be im-possible and unuseful. Yet for certain (theoretical) purposes, it may be useful to have ageneric way to represent and model such density dependence; which could pave the way tothe development of more general concepts, theories and numerical methods. For example inthe theories of adaptive dynamics and of physiologically structured populations, a generalconcept of density dependence is very useful (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 in-dividuals (for given environmental conditions) and the dynamics of those environmentalconditions. Connecting these two parts of the system results in a feedback loop since the be-haviour of individuals (and their dynamics) depends on the environmental conditions (e.g.,food abundance), whereas those conditions depend on the individuals that use the environ-ment (e.g., through consumption) (Fig. 1).

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 thedemographic properties of individuals, such as fecundity, survival, individual growth, migrationrate. Arrow 2: These individual properties depend, in part, on the state of the environment (fooddensity, abundance of predators, temperature, etc). The influence of the environment on individualvital rates and life history is called a reaction norm. Arrow 3: Collectively, individuals influence theenvironment, e.g., though exploitation of resources (whereas single individual has a limited impacton its environment).

The idea is to define the ”environment” in such a way, that, knowing the environment,the individuals can be considered independently from each other. That is, if the environ-ment is known, the population dynamically relevant behaviour of an individual (rates ofreproduction, growth, survival, migration) can be known as well. All information aboutdensity dependence is hence captured in the definition of the environment and how the en-vironment and the individuals mutually influence each other. This means that in addition to”individuals” and ”environment”, two processes need to be specified: how do the behaviourand life history of individuals depend on the environment; and how does the environmentchange in response to the (collective) behaviour of individuals. The first process can be seenas a reaction norm, whereas the second process, referred to as the population impact on theenvironment, depends not only on the quality of the individuals (in terms of their age, size,location, trait values, etc) but also on the quantity of individuals. Whereas a single individualoften has negligible impact on its environment, a population or community of individualsgenerally has a strong impact (possibly resulting in a struggle for existence).

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.Assume an unstructured population in which individuals reproduce with rate ρ(E) and diewith rate µ(E), where E refers to the condition of the environment. In continuous time, thepopulation dynamics 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 environment E, the pop-ulation will grow (or decline) exponentially. This corresponds to Darwin’s observation thatall populations tend to grow with a (high) ”geometrical ratio of increase”. To give a concreteexample for sake of illustration, we can assume that the environment affects reproduction

1 INTRODUCTION 8

only, in a linear way: ρ(E) = ρ0E and µ(E) = µ0. This trivial choice of vital rate functionsmay reflect that the reproduction rate depends linearly on food density, here captured by theenvironmental variableE. Then to complete the model, we need an equation for the dynam-ics of the environment. In particular, we need to know how E depends on the state of thepopulation. Assuming that the food dynamics are very fast, and hence that food availabilityis always in steady state with the current consumer population size N , we simply specify adirect relation between population size and food availability. For example:

E = 1− N

κ(2)

The assumption thatE decreases asN increases reflects the struggle for existence, that is, thegeometrical growth of populations leads to a deterioration of environmental conditions (re-source depletion). In this case, the reduction of E withN is a model of intraspecific competi-tion, but similar examples can be given for other ecological interactions such as interspecificcompetition or predation.

Of course in this particular case, the population dynamic model is more easily presentedin the 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 feedbackloop explicit has the advantage of obtaining a general, linear, equation for the populationdynamics. For example, when studying the model above in the context of adaptive dynamicsequation, (1) can be used to find the expression for the invasion fitness, which is an extensionof the function r(E). This is true also for more complex models: writing the ecological modelin the form of the environmental feedback loop (representing all relevant density-dependentinteractions), results in a model formulation that is naturally designed for extending it in thedirection of adaptive dynamics (Diekmann, 2004).

In the context of the more complex physiologically structured populations, this repre-sentation has already paved the way for the development of general numerical techniquesfor the continuation and stability analysis of such models (Kirkilionis et al., 2001; de Rooset al., 2010; Diekmann et al., 2010; Diekmann and Metz, 2010). An example (not the generalmathematical theory) of the latter 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 usu-ally assume that all individuals within a population are identical. This is obviously a veryuseful simplification, which has led to important results on the dynamics of populations andcommunities (see almost any textbook on ecology). Yet we all know that most populationsare structured in different ways: spatially, sexually, by age, size, reproductive status, epi-demiological status, etc. Here I focus on two kinds of population structure: spatial structureand size structure. Spatial structure is, intuitively, likely to have important implications,which are quite different for ecological and evolutionary dynamics, respectively. Ecologi-cally, spatial structure means that individuals interact with resources and other individualsin a small, local part of the environment. In other words, the environmental feedback loop

1 INTRODUCTION 9

depends on the spatial location. This has been shown to affect ecological dynamics in differ-ent ways: pattern formation, dampening of population cycles, invasion waves, etc (Dieck-mann et al., 2000). Evolutionarily, one particular relevance of spatial structure is that it maylead to reproductive isolation of spatially separated sub-populations. For example, the clas-sical distinction between allopatric and sympatric speciation refers to the spatial structure ofpopulations. Some examples of the relevance of spatial structure, in particular for evolution-ary dynamics, are given in section 3.3. Spatial structure also directly influences the fitnessof 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 indi-viduals 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 typicallife history of a Eurasian perch (Perca fluviatilis). At birth an individual has a body mass ofabout 2 mg and a length of 6-7 mm. After finishing its yolk sac, it starts feeding on zoo-plankton. When reaching several cm in length, it starts including macroinvertebrates in itsdiet. It matures at around 10-11 cm (around 10 g). If lucky, the individual can grow further,and soon will no longer feed on zooplankton. Instead, in continues feeding on macroinver-tebrates, and will also include small fish in its diet. In fact, it can include any fish in its dietthat is between about 5 and 45 % of its own body length (which means it can start becomingcannibalistic from about 10 cm in length). Perch can reach lengths up to 80 cm (in Lake Win-dermere for example), and weigh up to several kilograms. Clearly, the position in the foodweb of an individual may change significantly throughout its life time, while body massmay increase by up to a factor of 106. From trophic and metabolic points of view, a newbornand an old individuals are hence quite different ”organisms”. This illustration should makeit intuitive that size structure is likely to have implications for population dynamics.

So it is as with mountain climbing: Why do we want to model population structure?Because it is there. The existence of population structure in natural populations merits thequestion of what are the consequences of this structure. The consequences of size structureand, more generally, physiological structure, on the dynamics of populations and commu-nities has received quite some attention since the 1980s, in particular since the seminal workby Gurney and Nisbet (1985), Metz and Diekmann (1986), de Roos (1997) and Kooijman(2000). Since, it has been shown that size structure may have a wide range of dynamicalconsequences, the most basic of which is the emergence of generation cycles, also knownas cohort cycles (Gurney and Nisbet, 1985; de Roos, 1997; Persson et al., 1998), but also al-ternative stable states (Persson et al., 2007), catastrophic behaviour (De Roos and Persson,2002), ecological suicide (Van de Wolfshaar et al., 2008), population dynamics-induced sizebimodality (Claessen et al., 2000), and others. This work has shown that the range of dynam-ical behaviour of size-structured populations is richer than that of unstructured models suchas the Lotka-Volterra type models. This is no surprise, of course, as size-structured modelsare more complex and, in particular, they feature more complex and higher-dimensional en-vironmental feedback loops than unstructured models. While this complexity explains theinteresting, new types of population dynamics observed in these models, the complexityalso poses some limitations. Studying models of multiple, interacting size-structured popu-lations quickly becomes exceedingly difficult. Yet the qualitatively different types of dynam-ics predicted by size-structured models leads to new research questions as to its occurrenceand importance in natural populations. Physiologically structured population models, de-scribed below, allow us to pose a new kind of questions in population ecology. Turning this

2 SIZE-STRUCTURED POPULATIONS 10

around, for certain questions relevant to ecology and evolution, the use of structured popu-lations is necessary. In section 2 I argue for this point of view with some examples of researchon size-structured populations.

The case of genetic population structure (meaning heritable variation between individu-als), is of course a special case, as suggested above (section 1.1). Without genetic structurethe struggle for existence results in population regulation only. With such structure, it willnaturally induce evolutionary dynamics in conjunction with the ecological dynamics. Ge-netic structure, in the sense of genotypical differences between individuals, is also importantfor a range of dynamical phenomena such as speciation, hybridisation, reproductive isola-tion, and ecological polymorphism, including ecotypes. Genetical structure may depend to alarge extent on the type of inheritance (sexual reproduction, haploid/diploid, single/multi-locus traits, dominance, etc). It may also interact with spatial structure, such as for examplelocal adaptation and maladaptation in spatial gradients. Genetical structure, and the ensu-ing evolutionary dynamics, will be discussed in section 3.

In my work I have used the concept of the environmental feedback loop to study eco-logical and evolutionary dynamics of structured populations. Below is an overview of thiswork, structured as follows. Section 2 presents research on the ecology of size-structuredpopulations. Section 3 presents work on the adaptive dynamics of structured populations,including size-structured populations (section 3.2) and spatially structured populations (sec-tion 3.3). Section 4 presents two lines of ongoing research, both including ecological andevolutionary dynamics, and both including physiological and spatial population structure.Throughout the text, the idea of the struggle for existence and in particular its formalisationas the environmental feedback form a recurrent theme. The idea is to illustrate how this con-cept can be used to help formulating models of eco-evolutionary dynamics. In particular,the text should demonstrate how even in complex situations, this concept can help obtain astraightforward definition of fitness.

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 introduce the modelling framework that I have used to formulate the models, referredto as physiologically structured population models (PSP models). Next, I show how theidea of the environmental feedback loop helps in developing a method for the equilibriumanalysis of such models (section 2.2). Then, two examples of specific questions related tosize 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 individ-uals may differ from each other by their physiological state (e.g. age, size, body condition)(Metz and Diekmann, 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 state of the environment in terms of food availability, abundance ofcompetors and predators, etc. By specifying how, in turn, the environment is affected by theaction 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 tostudy the interaction between population dynamics 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. Variationin life history which is caused by variation in environmental conditions I refer to as lifehistory plasticity. Many environmental factors that influence life history vary in both spaceand time. First, some of these factors (e.g. food density) interact with the population suchthat population fluctuations result in environmental fluctuations which feed back onto lifehistory. In fluctuating populations, life histories of individuals born in different years cantherefore be entirely different (e.g. size-dimorphism in fish: Claessen et al., 2000; Perssonet al., 2003). Second, the environmental factors are often distributed heterogeneously inspace (Hanski and Gilpin, 1997). Life histories of individuals living in different regions maydiffer even in the absence of genetic variability (e.g. lizards: Adolph and Porter, 1993; Sorciet al., 1996).

PSP models make an explicit distinction between state variable at different levels: the in-dividual 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 bothplastic life 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; abundanceand possibly 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 envi-ronment.

• how the state of the environment changes under the collective influence of all the indi-viduals (the population)

The most frequently used formulation consits of a set of ordinary differential equations forthe continuous 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 anequilibrium analysis of a physiologically structured population model. The general theoryfor this method 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 beused to reduce an infinite-dimensional problem to a two-dimensional problem. Althoughthis section may seem a bit tedious with a certain amount of mathmatical detail, I find anexplicit introduction of at least one PSP model a prerequisite for this HDR report.


An example: the Kooijman-Metz model Although nothing new, and certainly not myinvention, I will give a fairly detailed description of the Kooijman-Metz (KM) model sinceit has become something of a standard model in size-structured population modelling, andhas been a frequent starting point in my teaching and modelling work (Claessen et al., 2000;Claessen and Dieckmann, 2002; Claessen and de Roos, 2003, and section 4.1). The KM modelcomes in various degrees of detail, and the one below is quite detailed. The reason is thatthis version is more ”biological”, that is, the functions describing individual-level properties(maintenance, attack rate, digestion rate, lenght-weight relation) can be easily interpretedand even measured in lab experiments. The presentation of the KM model also illustrates therole of the environmental feedback loop in PSP model formulation. Finally, a cannibalisticextension of this model has been used to demonstrate how the environmental feedback loopapproach can be used to arrive at a numerical continuation method for PSP models.

The KM-model (Kooijman and Metz, 1984) describes the dynamics of a size-structuredpopulation and its unstructured resource (food) population. The model description belowis borrow from Claessen and de Roos (2003), but to keep the presentation simple here, themodel below does not include cannibalism. The full (cannibalistic) model can be found in theoriginal publication, including all parameter values, based on piscivorous fish, in particularEurasian perch (Perca fluviatilis), and zooplankton (Daphnia spp.). Assume that the physio-logical state of an individual is completely determined by its body length x. Vital rates suchas food ingestion, metabolism, reproduction and mortality are assumed to depend entirelyon body length and the condition of the environment. The population size distribution isdenoted by n(x) and the density of the alternative resource by R. All individuals are bornwith the same length xb, and are assumed to mature upon reaching the size xf . Reproduc-tion is assumed to be continuous (in time) which implies that the size distribution n(x) iscontinuous.

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, andH(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 isobtained by subtracting the metabolic rate from the energy intake rate. Assuming that thebody weight and the metabolic rate scales both scale with the cube of body length, λx3 andρx3, respectively, 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 reproduc-tion by the energy cost of producing a single newborn:

b(x) =

{cr(1− κ)F (x) 1

λxb3if x ≥ xf ,

0 otherwise,(6)


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

Weight-length w(x) = λx3

Attack rate A(x) =

{αx2 (x− xp)2 if x ≤ xp0 otherwise

Digestion time H(x) = ξx−3

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) 1

λxb3if x > xf

0 otherwise

Total mortality µ(x) = µ0 + µs(x)

Starvation µs(x) =

{s [M(x)− κF (x)] if κF (x) < M(x)

0 otherwise

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 anequilibrium analysis we do not have to consider starvation mortality. However, in popula-tion cycles individuals may go through periods of food shortage and starvation. For suchcases we assume that starvation mortality rate increases linearly with the difference betweenthe metabolic rate and the food assimilation rate (Table 1).

We assume that the alternative resource population is unstructured. In our model it fol-lows semi-chemostat dynamics extended with a term to account for the effect of consump-tion by the structured 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-level model is presented in Table 2. The list of parameters and their values can


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) =

∫ xmax

xf

b(x)n(x) dx

Resource dynamicsdR

dt= r(K −R)−

∫ xmax

xb

A(x)R

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

be found in Claessen and de Roos (2003).PSP models such as this one are often studied by numerical intergration (simulation).

An efficient method for this is the Escalator Boxcar Train (EBT) (de Roos, 1997). In order tosimulate the model, the population size distribution n(x) needs to be discretised, and theEBT method does so in a natural way by keeping track of a (variable) number of cohorts; foreach cohort, the EBT integrates ordinary differential equations for the cohort abundance andthe i-state variables (body length x in this case). There are two typical population dynamicbehaviours of the KM model: a stable equilibrium and generation cycles. Generation cyclesare discussed in more detail below (section 2.3).

The environmental feedback loop How does all this illustrates the principle of the envi-ronmental feedback loop? The interaction environment, denoted by E in the Introduction, ishere defined as the resource density (R). On the one hand, knowing R, the life history of anindividual is entirely specified: its growth trajectory is obtained by integration of the growthrate; its reproductive output is obtained by computing its per-capita, size-dependent birthrate; its survival curve can be obtained by integrating the size-dependent per capita mortal-ity rate. This illustrates the statement that once E is known, individuals can be consideredin isolation (despite the presence of density dependent interactions). On the other hand, theimpact of an individual on its environment can also be computed, by integrating its feed-ing rate along its life history (see below for equations). Since this represents the impact onthe environment by a single individual only, we need a measure of the total population sizein order to complete the description of the environmental feedback loop (Fig. 1). A conve-nient measure is the total population birth rate P (that is, P is the product of the numberof individuals and their total expected reproductive output). Then, multiplying the cumula-tive consumption rate with P gives the consumption rate of the whole population. In otherwords: the impact of the population on its environment (Fig. 1).

Life history as an input-output map In more mathematical terms, the above paragraphcan be made explicit as follows. Elements from the individual-level model outlined abovecan be used to construct a life history if the appropriate input is given. We subdivide the lifehistory into three aspects; 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, denotedx(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,denoted B(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 reproduc-tive output, 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) andR0 define a life history. In other words, the recipe for translating a given environmentalcondition into the corresponding life history (arrow from ”Environment” to ”Individual” inFig. 1) is by solving equations (9-11).

The next thing we need to do, is to find the return map, that is from a given life historyback to the impact on the environment. For that we need two ingredients: the life historyand the number of individuals (since the impact is determined by the whole populationcollectively). For a single individuals with a given life history, the expected, cumulativeconsumption 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 productof P and θ(∞, R). Note that for this computation P is required as an input variable, since itcannot be derived from the life history. In other words, the population impact on the envi-ronment given a certain life history (arrow from ”Individual” passing through ”Population”to ”Environment” in Fig. 1) equals P θ(∞, R).

A merit of this way to characterise the environmental feedback loop, is that it providesus with a low-dimensional definition of the population dynamical equilibrium, and a toolto compute this equilibrium using continuation techniques. It should be noted that thepopulation-level model (Table 2) is an infinite-dimensional object (i.e., a continuous func-tion). Characterising the population equilibrium with the function n(x) therefore does notlend itself to numerical equilibrium analysis. By contrast, the environmental feedback loopis characterised by only two (unkown) variables: the food density R and the populationbirth rate P . Now observe that the population dynamic equilibrium of the KM model canbe charcterised by two criteria, being the requirements 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 := I2 as the input variables I (where I is the vector of the input variables), and to theequilibrium conditions R0 − 1 := 01 and r(K −R)− P θ(∞, R) := O2 as output variables O(the vector of ourput 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 wehave two input and output variables. See Claessen and de Roos (2003) for another examplewith k ≈ ∞). An equilibrium can be found via an input I∗ ∈ Rk for which the equilibriumand feedback conditions

f(I∗) = 0 (15)

hold. The condition f(I∗) = 0 can now be used in numerical continuation. The continu-ation method of Kirkilionis et al. (2001) can trace the equilibrium as a function of one freeparameter.

More recent development of the mathematical theory has enabled stability analysis ofthe equilibrium along the traced equilibrium curve, and in particular of the detection ofa Hopf bifurcation and the two-parameter continuation of the Hopf bifurcation (de Rooset al., 2010; Diekmann et al., 2010). These are the first steps towards more general numericalcontinuation tools for PSP models. Given the enormous contribution for the study of modelsbased on ODEs with continuation tools such as AUTO and Content/Matcont, this is a verypromising development.

Applications An application of this method is illutrated in Fig. 11 (page 35). The contin-uation method enables us to trace the equilibrium curve of a PSP model while varying amodel parameter. 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 equilibriumcurve, in between two alternative stable states, would be impossible to find by simulationonly. In Claessen and de Roos (2003) we use this method to study the influence of the size-dependent nature of a cannibalistic interaction on the equilibrium. The method allows us todetect two fold bifurcations (similar to the folded curve in Fig. 11) in the equilibrium curve.The found equilibrium curve helps interpreting the population dynamics observed in simu-lations (which are limited to stable equilibria and other attractors). In particular, we are ableto identify the ”biological” process that causes bistability in the cannibalistic model: only ifthe cannibals spare their smallest victims, i.e., if the victims are invulnerable to cannibalismup to a critical size, then cannibals are able to reach giant sizes. If, by contrast, cannibalsare able to include even the smallest indviduals in their diet, then they are bound to reach amaximum body size not much bigger than their maturation size. We called this the ”Hanseland 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),and co-workers at CEES, Oslo and the CEH, Lake Windermere. The modelling work has


led to two publications: Ohlberger et al. (2011a) and Ohlberger et al. (2011b) and is stillongoing. The first objective, described here, is to adapt the PSP model that I developedduring my thesis for cannibalistic freshwater fish (Claessen et al., 2000) in order to answer toa number of questions concerning the dynamics of Lake Windermere fish in particular, andsome questions on the effect of temperature on cohort cycles in general. This work was donein the wider context of a project proposed by Eric Edeline to the Norwegian Science Councilto study the effect of climate change on lake ecosystems, using both population dynamicmodelling and time series analysis. The empirical part of the work is based on the longterm observations of the fish populations in Lake Windermere (UK). The general aim of theproject is to try to disentangle the effects on the fish community dynamics of climate change,ecological interactions, fisheries, short-term evolution, and nutrient loading; processes thatare all known or likely to influence the perch and pike populations of Lake Windermere.In addition to the work on a PSP model, described in some detail below, we also studied amore simple stage-structured population model (Ohlberger et al., 2011b), using the biomassmodelling approach of De Roos et al. (2007, 2008).

A first study of the effect of temperature on population cycles was published by Vasseurand McCann (2005). They used a simple, unstructured bioenergetics model to determine theinfluence of temperature on a consumer-resource interaction in order to predict the conse-quences of temperature changes on the dynamics and persistence of consumer populations.Their results indicate that warming is likely to destabilize consumer-resource interactionsand that the qualitative response of the population dynamics depends on whether individualmetabolic rate increases faster or slower with temperature than ingestion rate. Their modelis a first step toward a bioenergetics theory of the impact of climate change on food webdynamics. The simplicity of their model allows them to analyse the model in quite some de-tail. The drawback is that they cannot address the size-dependent influence of temperatureon organisms, and its consequences. In particular, empirical evidence suggests that smalland large individuals do not respond equally to temperature. For example, in cold environ-ments, large individuals have a higher metabolic efficiency compared to small individuals(Kozlowski et al., 2004). Van de Wolfshaar et al. (2008) present the first size-structured pop-ulation model that accounts explicitly for seasonal temperature effects on vital rates. Theyshow that the combined temperature- and size-dependence of vital rates may have fatal con-sequences for winter survival of both individuals and the population as a whole. However,they did not study the effect of changing temperature on population dynamics, the objectiveof the modelling exercise described here (Ohlberger et al., 2011a).

Life history and population dynamics of a number of freshwater fish species includ-ing roach (Rutilus rutilus), Eurasian perch (Perca fluviatilis), yellow perch (P. flavescens) andNorthern pike (Esox lucius), have been modelled with a model in which the i-state of a fish isdefined by two variables: the amount of irreversible mass (x) and reversible mass (y) (Pers-son et al., 1998; Claessen et al., 2000, 2002; de Roos and Persson, 2001; Persson et al., 2004;Persson and De Roos, 2006). The model assumes that if x and y are known, all ecologicalfunctions can be derived from these two quantities: x and y hence completely define thestate of an individual. For example, total body mass equals w = x + y, gonad mass equalsy − qx (where q is a constant), body length depends on x only, the search rates for differenttypes of food are functions of x only. The state of the population is defined as the distributionof the number of individuals over the individual state (e.g., n(x, y)). Inspired by the biol-ogy of temperate freshwater fish, these models assume that reproduction occurs in a pulsedway during spring. At this moment, the gonad mass of adult individuals is converted into


Figure 2: Population dynamics of themodel of Ohlberger et al. (2011a) at tem-peratures 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)

newborns. Together, the newborns form a new ”cohort” of identical individuals. Thus thepopulation consists of a variable number of cohorts, each described by a set of differentialequations for the dynamics of x, y and Ni, where the latter is the abundance of cohort i. Co-horts disappear from the population when their abundance drops below a trivial threshold(e.g., a single individuals per lake). In these models, the environment is characterised by thepopulation densities of prey (e.g., zooplankton) and, possibly, predators (e.g., cannibalisticconspecifics). Population feedback arises from the assumption that the dynamics of the en-vironment depend on the state of the fish population: consumption by the fish depletes thezooplanton 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 citedabove and in Ohlberger et al. (2011a) .

Generation cycles A very general result from PSP modelling is that intra-specific competi-tion tends 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 kind of population cycles is distinct from the better-known predator-prey cycles, alsoreferred to as ”consumer-resource cycles” or ”delayed-feedback cycles”. The two types ofpopulation cycles can theoretically be distinguished by the cycle periodicity (Murdoch et al.,2002). Predator-prey cycles are expected to have a periodicity of at least 4TC + 2TR, whereTC and TR are the maturation times of the consumer and resources, respectively. Intuitively,this can be understood by considering the classical Lotka-Volterra model and its predator-prey cycles. Each cycle consists a phase of exponential predator growth, depleting the preypopulation; followed by a phase of predator exponential decline; followed by exponentialprey growth. Each bit of exponential growth or decline requires at least a few generationsto complete. By contrast, generation cycles have a periodicity of one generation (exceptionsinclude 0.5 generations or 2 generations per cycle).

The basic mechanism of generation cycles is an inter-generational conflict: each genera-tion replaces its parental generation through intra-specific competition. The mechanism can


be illustrated by considering an example. In Ohlberger et al. (2011a) we present a modelof a size-structured fish population. Under certain conditions (e.g., 17◦C in Fig. 2) the pop-ulations exhibits cycles of which the periodicity corresponds to the maturation time. Thefigure shows that the birth of each new generation (black dots, lower panel) is followed by asudden depletion of the zooplankton population (resource, upper panel). The adult portionof the population goes extinct rapidly, starved to death by the low resource abundance. Thenew generation remains juvenile for several years, during which it declines exponentiallydue to a constant mortality rate, and the individuals grow in size. The resource graduallyincreases (due to the decline of the juvenile cohort). The juveniles mature in their 5th year,and the following spring they reproduce the next generation.

The underlying mechanism of this type of cycles depends on the size scaling of the eco-logical properties of individuals. The basic, size-dependent functional relations in the modelare similar to that of the Kooijman-Metz model (Table 1), i.e., a type II functional response,a hump-shaped attack rate on zooplankton, an increasing maintenance rate with body size,decreasing digestion time with body size. From these basic ingredients we can derive a de-pendent relationship, which is referred to as the ”critical resource density” R∗(x). As its no-tation suggests, this quantity is analogous to Tilman (1982)’sR∗, except that in our case it is afunction of body size rather than a single measure for the whole population. As in Tilman’stheory, an individual’s competitive ability is measured by R∗(x): the lower its value, thebetter the individual can deal with severely competitive situations. In the case of the fishmodel, R∗(x) is an increasing function of body size. This results mainly from the fact thatmaintenance requirements increase faster with body size than the feeding rate. This resultis general for PSP models that have been parametrised for particular fish species (Perssonet al., 1998; de Roos and Persson, 2003; Persson and De Roos, 2006). More theoretically, wecan note that the feeding rate is generally a surface-limited process and hence likely to scalewith the body mass to the power 2/3, whereas maintenance is a mass-limited process andhence likely to scale linearly with body mass (Kooijman, 2000) (even though an alternativetheory postulates a 3/4 scaling rule, Brown et al., 2004). Thus, also based on these simpletheoretical considerations, R∗(x) is expected to increase with body size (even under the 3/4scaling rule!).

Generation cycles result from the asymmetric competition between differently sized in-dividuals, if R∗(x) is sufficiently steeply increasing (or decreasing) with body size (Perssonet al., 1998; de Roos and Persson, 2003). In our example, R∗(x) increases with body size andhence newborns are competitively superior to adults. A sufficiently high fecundity of adultsthen automatically results in generation cycles.

Generation cycles occur even in more complex models that include multiple ecologicalinteractions. Cannibalism has the potential to dampen generation cycles, because it allowsadults to reduce the competition with newborns in two ways: killing newborns reducesthe abundance of competitors, and eating newborns provides extra energy (Claessen et al.,2000). Yet even with strong cannibalism generation cycles may be present, even though theyare modified by the cannibalistic interaction; for example, the period length may be longer,and the population may contain very big, cannibalistic individuals in addition to the cohortof juveniles that drive the periodicity (Claessen et al., 2000; Persson et al., 2004).

The effect of temperature What are the consequences of climate change on populationdynamics? Or more generally, what is the relation between abiotic factors and ecological


Figure 3: Temperature dependence terms (A)for perch consumption (solid), perch metabolism(dashed) and zooplankton growth rate (dotted),and the individual net energy gain (B) as a func-tion of temperature for perch of body weight0.1g (solid), 1g (dashed) and 10g (dotted). Thindotted lines (A) indicate calibration to a valueof 1 at 20◦C (see text). The body size of perchwas set to 8.2g, the optimal size for prey attack.The net energy gain for differently sized perch(B) was calculated at a zooplankton density of 2Ind/L, which resembles rather low resource lev-els where exploitative competition in perch is ex-pected to be high. From: Ohlberger et al. (2011a)

dynamics? Answering these questions may provide clues for testing theoretical predictionsof our ecological theories with empirical data. And it may provide clues for how ecosystemsrespond to environmental change. Modelling dynamical consequences of the physiologicalresponse to temperature change requires the ability to translate individual-level physiolog-ical processes into population level dynamics. The framework of PSP models allow us todo so. The model formulation focusses on the description of individual-level properties ofindividuals such as the maintenance requirements, feeding rate, etc. Parameters are oftennumerous in such models (see the long tables in many of the above cited articles). Yet mostof these parameters are fairly ”easy” to measure in lab settings. The model then provides in-dependent population-level predictions of the emergent dynamics. This contrast with moresimplistic modelling frameworks, such as the Lotka-Volterra and derived models, whichform the basis of theoretical ecology. Finding the temperature dependence of parameters ofsuch simpler models may be more difficult than finding all parameters of a PSP model. Forinstance, the parameters r and K of the logistic growth equation (a frequent ingredient ofLotka-Volterra type models) are essentially population-level quantities, and are hence inher-ently 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-physiological functions in the above described ”fish” model (Claessen et al., 2000, 2002)temperature dependent; the consumption rate (parametrised by the attack rate and the di-gestion time), the metabolic rate, and the zooplankton renewal rate. To do so, we multipliedeach function by a temperature-dependent scaling factor (Fig. 3). We us empirically-basedscaling relations rather than the more commonly used theoretical model for temperaturedependence of chemical reactions, referred to as the Boltzmann factor or the Van ’t Hoff-


Figure 4: The regime diagram of thePSP model in Ohlberger et al. (2011a).GC=generation cycles. FP=fixed point dy-namics. CD=cannibal-driven dynamics. CD-GC=alternation of CD and GC dynamics(both types of dynamics are unstable). FPor GC=coexistence of two attractors (corre-sponding to FP and GC).

Arrhenius equation (Brown et al., 2004). The latter equation is one of the elements of themetabolic theory of ecology. However, looking into the literature on fish ecology, we foundthat the actual measurements of the temperature dependence of these relations deviates sig-nificantly from the theoretical (chemical) model. In particular, all three empirical relations(Fig. 3) display a maximum (at different optimum temperatures) rather than a monotonicallyincreasing 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 re-sults: increasing temperature tends to increase the level of intra-specific competition andtends to result in the onset of generation cycles. That is, on a temperature gradient we ex-pect generation cycles at high temperatures and stable populations (”fixed point dynamics”)at low temperatures. This result is illustrated with the example of the predicted populationdynamics at two different temperatures (Fig. 2). At 15◦C, the population displays so-calledfixed point dynamics (FP), which means that the within-year year dynamics are more or lessthe same from year to year. (Note that reproduction is pulsed at the beginning of the growingseason). Such FP dynamics are characterised by the coexistence of a large number of cohorts(age classes). Intra-specific competition is too weak to result in the exclusion of some cohortsby other ones. By contrast, at 17◦C, the model displays typical generation cycles, in this casecharacterised by the existence of a single cohort during most of the cycle (except during ashort period following reproduction).

While this provides only two examples, a more general result can be found in Fig. 4. Infact the model we studied includes cannibalism as well as competition (in Fig. 2 cannibalismis assumed to be absent). Cannibalism was included in the analysis to assess the generalityof the effect of temperature on generation cycles, since we know generation cycles occurwithout and with cannibalism, although in modified form (see above). The figure clearlyshows that increasing temperature tends to result in generation cycles irrespective of thelevel of cannibalism. The amplitude of population fluctuations is predicted to increase withtemperature, in both cannibalistic and non-cannibalistic populations Ohlberger et al. (2011a).

Throughout the studied temperature range (12-22◦C), the model assumes that the re-source population growth rate increases with temperature. Yet the model predicts an overalldecrease of the mean resource density (i.e., the long-term average given a fixed temperature),caused by increased food intake by consumers, and increased consumer total reproduction


rate (Ohlberger et al., 2011a). The lower resource level at high temperatures reflects the in-creased level of intra-specific competition. A direct consequence of the different temperaturedependencies of the vital rates at the individual level is that energy gain increases faster withtemperature for small individuals (Fig. 3). Therefore, cool conditions favour big individu-als, while warm conditions favour small ones. Increasing temperature hence reinforces thecompetitive advantage of small over large individuals, which enhances the mechanism thatcauses generation cycles (Persson et al., 1998).

Above we have seen that generation cycles are often the result of the fact that the criticalresource density R(x) increases with body size (which is the case for all fish for which thedata exist, Persson and De Roos, 2006). The size dependence of intraspecific competition,however, changes with temperature. The net energy gain of an individual increases fasterwith temperature at smaller sizes (Fig. 3), thereby magnifying the competitive advantage ofsmall over large individuals. This effect is reflected in the general observation that optimumgrowth temperatures (which for immature fish can be assumed to be equal to those of the netenergy gain) decrease with increasing body size (Kozlowski et al., 2004). This has been re-ported for several fish species (Karas and Thoresson, 1992; Bjornsson and Steinarsson, 2002;Imsland et al., 2006) and other ectotherms such as amphipods (Panov and McQueen, 1998).Thus, this functional form of the temperature-size relationships may be valid for other fishspecies and possibly ectotherms in general.

Our conclusion on the effect of temperature on intra-specific competition and the re-sulting generation cycles are in line with those of Vasseur and McCann (2005): higher tem-peratures are predicted to destabilise population dynamics. Recall that that the model ofVasseur and McCann (2005) concerned consumer-resource (predator-prey) cycles. Interest-ingly, these two types of population cycles (generation cycles vs predator-prey cycles) arecaused by ecologically and dynamically very different processes (size-asymmetric competi-tion vs delayed feedback through the predator-prey interaction), and characterised by verydifferent cycle periodicities (Murdoch et al., 2002). Whether these predictions hold for morecomplex food webs than the ones modelled in these studies remains to be seen. Yet thesepredictions can be put to the test relatively easily, by comparing population dynamics acrosstemperature gradients, albeit natural ones or in the laboratory (Ohlberger et al., 2011a).

The environmental feedback loop In the model of Ohlberger et al. (2011a), the environ-mental feedback loop includes two different ecological interactions: competition via deple-tion of the zooplankton resource; and cannibalism. Competition for zooplankton is an indi-rect density-dependent interaction, that operates through the E-variable R. The impact of Ron individual life history is via the size-dependent functional response (similar to equation(4)). The impact of the population on this E-variable is through the population-level, totalfeeding rate, a very similar expression to equation (8). Cannibalism is a direct-dependentinteraction, in the sense that cannibalistic mortality and feeding rates depend directly onthe current population abundance and size distribution. For this interaction, the E-variablehence contains the entire population size distribution. The impact of this E-variable on theindividual life history is mediated by a complex function that takes into account the bodysizes of any pair of interacting cannibals and victims (Claessen et al., 2000, 2002, 2004), seeequation Ac(c, v, x, y, T ) in Table 2 of Ohlberger et al. (2011a)), as well as the abundances ofvictims and cannibals.


Figure 5: Thecommon lizard(Lacerta vivipara)in its experimen-tal environment:the semi-naturalenclosures at theCEREEP fieldstation.

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 twopublications: Gonzalez-Suarez et al. (2011a) and Gonzalez-Suarez et al. (2011b). The goal ofthis work is to study the consequences of demographic stochasticity for populations of whichlife history is subject to plasticity. Demographic stochasticity is randomness that results fromthe discreteness of individuals and the probabilistic nature of demographic processes suchas birth, death, clutch size, etc. Its influence is expected to be inversely proportional topopulation abundance: it should hence 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 ManuelaGonzalez-Suarez as well as other members of the laboratory, and complemented with litera-ture data. The objective is to obtain a model of population dynamics for this species that canbe used to predict or interpret the dynamics of semi-natural enclosures at the CEREEP ex-perimental station2. The model is used to infer general aspects of the population dynamicsof small, structured populations, subject to demographic stochasticity.

General context Spatio-temporal stochastic variability is ubiquitous in ecological systems.It is particularly important in extinction and invasion which are inherently stochastic pro-cesses due to low population numbers. To understand extinction and invasion dynamics it ishence crucial to understand the consequences of stochasticity. Heterogeneity in the environ-ment is expected to result in life history variability between individuals (Van Kooten et al.,2004, 2007). Owing to the mutual dependence of life history and population dynamics, suchvariability is bound to have consequences for extinction dynamics of small populations andfor mutant invasion in evolutionary dynamics.

To date, stochasticity and plastic (density-dependent) life history have rarely been stud-ied in conjunction. On the one hand there is a large body of theory on the role of stochas-ticity on structured-population dynamics which has yielded powerful tools to predict pop-ulation growth rate, extinction risk, evolutionarily stable strategies, etc (e.g. Tuljapurkar,1997; Orzack, 1997; Haccou et al., 2005). However, most of this theory assumes density-

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


independent population dynamics (e.g. Tuljapurkar, 1997) or non-plastic life history (e.g.age-structured populations, Bjornstad et al., 2004; Le Galliard et al., 2005) and therefore can-not account for plastic life histories. On the other hand, a dynamic theory about plastic lifehistory has been developed based on physiologically structured population models (de Rooset al., 2003). The message from this theory is twofold: population feedback on life history(i.e., density-dependent effects on life history) is critical to the realization of life historieswhile in turn population dynamics depend on realized life histories. However, current the-ory of PSP models assumes deterministic population dynamics (but see some exceptions:Claessen and Dieckmann, 2002; Van Kooten et al., 2004, 2007; De Roos et al., 2009).

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

1. Life history provides the population with a “memory” of past stochastic fluctuations.What is the role of such delayed effects on the dynamics of physiologically structuredpopulations?

2. What is the role of plastic life history in the population dynamics of small popula-tions? First, density-dependent compensatory effects are expected to increase growthrate and fecundity at low densities. Second, demographic stochasticity is expected toresult in life history variability between individuals. Do these effects alter the predictedextinction risk and 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-Suarezet al. (2011a). There are three i-state variables: age, structural mass (i.e., bone, organs), andreserves mass (i.e., adipose and reproductive tissues). We assume that energy acquisition,growth, survival, and reproduction are functions of body mass defined by an energy bud-get model. Food intake and metabolism also depend on environmental conditions, that is,sunshine duration, to reflect the importance of weather on lizard life history (Adolph andPorter, 1993). Food intake is also a function of a density-dependent scaling function D(B)that provides feedback from population dynamics to the individual process of food con-sumption. Note that D(B) represents hence the environmental feedback loop in this model(Fig. 1). Whereas PSP models usually model the dynamics of the resource population explic-itly, we cannot accurately model prey dynamics because the common lizard feeds on a largevariety of prey and its functional response is not well understood (Avery, 1971; Gonzalez-Suarez et al., 2011b). In the absence of quantitative empirical data to adequately define theconsumer-resource interaction in this species (in terms of the functional response, prey re-newal rates, etc), we model density dependence in a phenomenological way, using a simplefunction D(B) that reflects our general knowledge of the species feeding biology. An indi-viduals feeding rate is obtained by multiplying its empirical, size-dependent feeding rateunder standard conditions (Gonzalez-Suarez et al., 2011b) by the function D(B), which is adecreasing function of the populations weighted abundance B.

Understanding how assimilated energy is actually channelled in an organism is compli-cated, and numerous energy allocation rules have been proposed (Kooijman, 2000; Claessenet al., 2009). We assume that individuals follow a ”net-production allocation rule” beforefirst reproduction and a ”gross-production allocation rule” (Kooijman 2000) after the first re-production event. These two allocation models reflect observed differences in prioritization


between reproductive lizards, which prioritize reproduction, and nonreproductive individ-uals, which prioritize structural growth (Andrews, 1982).

The environmental feedback loop The above model is an example of a PSP model withdirect density dependence, that is, the interaction is not mediated through a resource. (Notethat the cannibalistic interaction in Claessen et al. (2000, 2002); Claessen and de Roos (2003);Ohlberger et al. (2011a) is also a direct density dependent interaction). In more detail, theenvironmental feedback loop is modelled as follows. First, the environment E impacts indi-vidual life history through 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 func-tion γ1xγ2 is the empirical relation between feeding rate and body size under standardizedconditions. Thus, under ”typical” population densities, the feeding rate of an individualdepends linearly on sunshine duration, and depends allometrically on body size. Wheneverpopulation density exceeds the ”typical” density, denoted by B0, the feeding rate shouldbe below the standard rate, and vice versa. We modelling this in a phenomenological wayusing 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 populationdensity B. To complete the description of the feedback loop we need to specify the ”popu-lation density” B. Even though this is a phenomenological model (i.e., non-process based,non-mechanistic), the idea behind it is that the lizards interact mainly through the depletionof their resources. Therefore, we assume that the contribution of an individual to the collec-tive impact on the environment depends on its individual feeding rate. We define B as thetotal, 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 ofindividual i.

Thus, in terms of Fig. 1, the arrow from environment to individual is captured by equa-tions (16) and (17); the arrow from population to environment corresponds to equation (18).The arrow from individual to population represents the contribution of individuals to theircollective dynamics: 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 increas-ingly stochastic. The first version is a fully deterministic model, in which the abundance ofeach year class j is described by a differential equation dNj/dt. All other model versions arestochastic. In model 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 onthe same day in the year), grow continuously in time, reproduce discretely in time, and then


Figure 6: Comparison of population-level model predictions (A) and individual-level predictions of survivalrate (B). Model versions are version 1=Det, version 2=Dis, version 3=Food, version 4=Birth, version 5=F&B.Emp=empirically observed. 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 estimatesare not available. Shorter error bars illustrate SD among 10 stochastic replicates. From: Gonzalez-Suarez et al.(2011a)

die. Model version 3 modifies the second model with a stochastic component to the foodintake rate. This reflects that the function D(B) describes the average environment, whereaseach individual experiences its local environment in a more heterogeneous way. One day,some individuals get lucky and find a large prey, whereas others find almost nothing. Eachday, an individual’s local environment is randomly drawn from a normal distribution with amean of D(B) and a standard deviation reflecting observed variability in consumption ratesof a lizard population. An alternative extension of the second model (version 4) relaxes theassumption that all individuals of a year class are born on the same day. Rather, a femalemay reproduce on any day in a given birthing period of about a month (drawn from a trun-cated normal distribution). Finally, version 5 combines versions 3 and 4: discrete individualswith stochasticity in food intake and in birthdays. We chose these two stochastic processes(food intake rate and birthday) because they are thought to be important heterogeneouscomponents of the life history of the studied species.

The model has been thoroughly parametrized to laboratory, field and literature data. Afull list of model parameters, their values and how the data was obtained is given in Table 1of Gonzalez-Suarez et al. (2011a). All versions of the model thus parametrized, predict rela-tively stable population dynamics, in the sense that generation cycles or predator-prey cyclesare not observed. One explanation of the absence of generation cycles is the relatively lowpopulation-level reproduction rate, which is insufficient to cause the level of size-dependentcompetition required for generation cycles. The dynamics of the stochastic model versionsare quite similar to the deterministic version, although of course population dynamics andgrowth trajectories display more variability in the latter cases. Yet the stochasticity does notqualitatively affect the population-level population dynamics. It should be noted that up todate we have focused on non-extinction population dynamics, with fairly high populationnumbers (around 100 to 200 individuals). It remains an open question whether closer toextinction the stochastic model versions may display qualitatively different behaviour.

Note that even though the model is heavily parametrized with empirical data, in particu-


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

lar on individual-level aspects, many aspects of life history and population dynamics remaindynamic (output) variables of the model. In particular, model predictions of life history in-clude growth trajectories, maximum body size, age at maturation, realized fecundity, thesurvival function. Population-level predictions include the emerging population structure,for example the proportion of age-0, age-1 and older individuals, proportion of juvenilesand adults, level of fluctuations. The stochastic models also provide predictions of the levelof variability in any of these life history 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 observedin natural populations (Massot et al., 1992). The predicted age structure (Fig. 6A) agrees wellwith empirical 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 ageclass, for which the survival rate is strongly size-dependent, and hence the model predictionsdepend on the realised population dynamics through the environmental feedback loop. Thesurvival rate of older individuals merely reflects our assumption of the size-independentcomponent of the mortality rate.

More interesting are the life history comparisons of fecundity and the growth trajecto-ries (Fig. 7). Although our model underestimates the level of variability in fecundity, themean fecundity is well captured (Fig. 7A). The growth trajectories predicted by the modelare very similar to empirically observed ones, although the stochastic model versions tendto overestimate the average growth rate and the maximum body size (Fig. 7B), especially themodel versions 3 and 5. The increase in mean snout-vent length (SVL) in older individualsin model versions 3 and 5 is accompanied by a decrease in body condition, so that theseadults are characterized by being longer and skinnier. Variation in birth date alone (model


Figure 8: Illustration of how Jensen’s inequality influences the energy budget for adults (body mass 4.5 g; panelA) and for juveniles (body mass 0.4 g; panel B). Thick lines: assuming a constant food intake rate, the relationshipbetween the growth rates of reserves mass (grey line), or structural mass (black line), with food intake rate ispiece-wise linear. The arrows indicate the transition in growth rates that occurs when the assimilated energy isinsufficient to cover metabolic costs (to the left of the arrow). Dotted lines: the average growth rates assumingdaily 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” changesfrom zero to one. From: Gonzalez-Suarez et al. (2011a)

version 4) results in differences in SVL among individuals born early in the birthing period(first 10 days) vs. those born late (last 10 days), with the former being significantly largerat all ages and reaching larger maximum sizes (Student t tests P < 0.031). Although signif-icant, the actual differences in size are generally small (< 1 mm). Interestingly, variabilityin food consumption eliminates these differences: there are no differences in SVL betweenindividuals born early or later in the F&B version (Student t tests P > 0.10).

The most striking result of the above modelling exercise is that stochastic variation in thefood intake rate results in long and skinny individuals even though the mean food intakeremains constant. Skinny individuals are also less fecund, and thus the number of offspringper female decreases, which reduces competition among the newborn class and leads tohigher survival of young individuals. This predicted change in individual morphology canbe explained by Jensen’s inequality and the non-linear relationship between body growthand 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 av-erage f(E(xi)) (Ruel and Ayres, 1999). Here, the non-linearity in body growth is caused bythe transition that occurs in the energy allocation when the assimilated energy is not suffi-cient to cover metabolic costs. This transition leads to a concave up relationship betweenstructural-mass growth and resource availability because growth is halted when energy in-take is below maintenance costs (Fig. 8). As a result the mean growth rate of structural masswhen food intake varies daily is higher than the growth rate predicted for the mean foodintake. Conversely, reserves mass growth has a concave down relationship as reserves areconverted back to energy used to cover maintenance costs when food intake is insufficient.As a result the mean growth rate of reserves mass is lower in a stochastic environment.

Transitions in growth are expected whenever individuals are able to survive for some

3 ADAPTIVE DYNAMICS 29

time using energy reserves and body growth is reduced or stopped at the time when foodintake is not sufficient to cover maintenance costs. These simple requirements are met bya wide variety of taxa (Kooijman, 2000); thus, the non-linear relationship between bodygrowth and food availability should be very widespread. However, morphological changesmay not be apparent if food availability always remains above or below the transition point.Changes in morphology will become apparent only when food intake falls below mainte-nance costs for some individuals at some point in time. This is likely to occur in food-regulated populations when population size is near carrying capacity, or in habitats withhigh intrinsic stochasticity in food availability. Confirming our predictions, laboratory stud-ies have shown that changing the temporal variance in food availability, while keeping themean constant, results in morphological changes in sticklebacks and sea urchin larvae (Aliand Wootton, 1999; Miner and Vonesh, 2004). Whether the observed morphological changeshave demographic consequences in natural populations remains to be clarified. However,our results suggest population sizes may change, even though slightly in our case, therebyaffecting overall resource levels.

2.5 Size-structure: conclusions

Above I have described and illustrated a theoretical framework (PSP models) that allowsus to model the dynamics of size-structured populations. Size-dependent interactions maygive rise to a number of dynamical behaviours that go beyond the scope of dynamics ofunstructured models such as Lotka-Volterra type models. I have given the example of gen-eration cycles: asymmetric competition between small and large individuals may give riseto population cycles with a periodicity of one generation.

A characteristic of populations with plastic life history is that the environment (and hencethe environmental feedback loop) determines, to a certain extent, the realised phenotypeof individuals, in terms of their growth trajectory and possibly other physiological traitssuch as corpulence. If the environment is constant, all individuals will have the same lifehistory. If it is periodic (as in generation cycles), life histories will vary periodically, too.If the environment is stochastic and variable between individuals (as in the lizard model),this will result in between-individual variability in life histories. In the case of the lizardmodel, the stochastic food environment influences life history in an unexpected way, shiftinglife history systematically towards taller and skinnier phenotypes. Whereas such within-cohort variability may have little effect on the population dynamics (as in the lizard model),its effect on realised phenotypes may have implications for fitness and hence evolutionarydynamics.

In other words, the struggle for existence determines not only population dynamics, butalso realised phenotypes. The relation between phenotype and environment is called a reac-tion norm. It is, of course, not a new idea, but the above illustrates how PSP models accountfor such reaction norms, and rather, how they emerge from the intricate relation between thestate of the environment, and the state of the population (i.e., the environmental feedbackloop).

3 Adaptive dynamics

Let is also be borne in mind how infinitely complex and close-fitting are the mutual relations of


all organic beings to each other and to their physical conditions of life; and consequently whatinfinitely varied diversities of structure might be of use to each being under changing conditionsof life. Can it, then, be thought improbable, seeing that variations useful to man have undoubtedlyoccurred, that other variations useful in some way to each being in the great and complex battleof life, should occur in the course of many successive generations. If such do occur, can we doubt(remembering that many more individuals are born that can possibly survive) that individualshaving any advantage, however slight, over others, would have the best chance of surviving andof procreating their kind? Darwin (1859), page 108

3.1 Again: the struggle for existence

How can we use the idea of the environmental feedback loop to study evolutionary dynam-ics? As I have argued in the Introduction, the struggle for existence can be formalized usingthe concept of the environmental feedback loop. A struggle for existence as depicted in Fig.1 ”merely” leads to population regulation and non-linear dynamics, as long as all variabil-ity between individuals is non-heritable, as was the case in all physiologically structuredpopulation models discussed above. What happens if we allow for heritable variation? Animmediate answer would be that we’d have to extend our models to include not only intra-specific competition (and other ecological interactions), but also inter-specific competition,which would reflect competition between different heritable types within the same popu-lation (of course they are not different species, but inter-type competition is analogous tointer-specific competition if the types are reproductively isolated). Although this is doable,there is an easier alternative, which is to start by looking at the invasion process of a newtype into an established population of a resident type. Any new type (usually thought of asa new mutation) that arrives in a population first needs to ”invade” the population, before itcan actually start ”competing” with the previous types in the population. That is, initially itsdensity is sufficiently low that its impact on the environment and hence the resident type isnegligible. Now it becomes clear how the environmental feedback loop can be used: for in-vading types (mutants), the feedback loop is not closed. That is, while they are subject to thesame influence of the environment on their individual level behaviour as the resident types,they do not themselves impact the environment. Hence for the mutant type, one arrow ismissing in the feedback loop (Fig. 9). Instead, the mutant type is a ”slave” of the resident’sfeedback loop: the mutant’s life history depends on an environment that is determined bythe resident type (Fig. 9). The fact that its loop is not closed means that the populationdynamics of the mutant type are linear (i.e., exponential growth). The question whether amutant can invade or not therefore boils down to computing its exponential growth rate.This exponential growth rate is hence a robust fitness definition, which is referred to as inva-sion fitness (Metz et al., 1992, 1996; Geritz et al., 1998). Only if the invasion fitness is positive,is a mutant expected to be able to invade and possibly establish itself. Note that from the def-inition it follows that the resident’s fitness is necessarily zero, reflecting that its populationis being regulated (dN/dt = 0).

Assume that a population is characterised by an evolving trait u (that takes real values),and that the population is currently monomorphic in this trait, i.e., all individuals have traitures, where ”res” refers to ”resident”. Assume as well that the resident population is atits ecological attractor defined by the environmental feedback loop. In other words, thedensity-dependent interactions in the resident population have regulated the environmentalvariable at its equilibrium value E(ures). The invasion dynamics of a mutant type can now


Figure 9: The environmental feedback loop representing the struggle for existence during the inva-sion of a mutant type (red) into the population of a resident type (blue).

be described by an ODE analogous to equation (1):

dNmut

dt= r(umut, E(ures))Nmut (19)

whereNmut is the abundance of mutants (assumed to be small), ui is the evolutionary trait ofthe mutants and residents. This notation emphasizes that the mutant’s population growthrate depends on its own trait value but also on the environmental conditions as set by theresident.

Since we have identified fitness by r(umut, E(ures)), this notation reveals an importantcontribution of game theory and adaptive dynamics to evolutionary theory: the fitness of atype is not some given, static quantity but it depends on the mutant’s environment (and inparticular on the identity of the resident). One immediate consequence of this is that fitnesslandscapes are inherently dynamic: as soon as the current resident will be replaced by afuture successful mutant type, the fitness landscape will be different.

Conceptually, the advantage of using the environmental feedback loop is that it providesa straightforward method to incorporate the role of density dependence in the definitionof fitness, even in complex models. As soon as one has succeeded in writing a populationdynamical model in terms of the environmental feedback loop, the model is ready to beextended to study its evolutionary dynamics, as we have seen by going from equation (1) toequation (19). For example, for the KM model discussed above, such an extension is now astraightforward exercise: given the environment R set by the resident, the life history of amutant can be computed by integration of equations (9-12).

The above is nothing new, if not a new perspective. Of course the basic and advancedtheory of adaptive dynamics can be found in the literature. A lot of the above is inspired bya great introduction to adaptive dynamics written by Odo Diekmann (2004). I have focussedon the definition of invasion fitness, for which the idea of the environmental feedback loop is,in my opinion, very helpful. Invasion fitness is at the basis of one of the main contributionsof adaptive dynamics which is the classification of singular points (”equilibrium points”of the evolutionary dynamics) in terms of evolutionary attainability and of evolutionaryinvasibility (Metz et al., 1996; Dieckmann, 1997; Geritz et al., 1998). Yet it should not beoverlooked that invasion fitness tells only part of the story: following successful invasion,


competition between the types becomes decisive for the question whether the successfullyinvaded mutant will outcompete the resident and replace it; or whether another outcome islikely: either coexistence or even extinction of the mutant type (Mylius and Diekmann, 1995).Once the mutant population is sufficiently large in order to have a significant impact on theenvironment, Fig. 9 is no longer valid since the missing arrow needs to be restored. Then,the struggle for existence takes it full meaning in the sense that several competing types areimpacting the environment with only the best adapted types being able to survive. Yet forquite general conditions (basically: being away from bifurcation points), Geritz et al. (2002)have shown that a successful invasion is indeed followed by replacement of the resident bythe mutant population, except for the case of mutual invasibilty. Geritz et al. (2002) call this”attractor inheritance”. The case of mutual invasibility (i.e., the mutant type u1 can invadethe resident type u2, and a mutant type u2 can also invade the resident type u1) generallyleads to coexistence of types u1 and u2 (Geritz et al. (1998); see below).

3.2 Sympatric speciation in (structured) fish populations

This section describes the work I’ve done during a postdoc project, in Amsterdam and Paris,which follows up on some of the work of my PhD project, and which was done in collab-oration with Jens Andersson and Lennart Persson of Umea University, Sweden, and Andrede Roos, University of Amsterdam. The work in this section is published in Claessen andDieckmann (2002); Andersson et al. (2007); Claessen et al. (2007, 2008).

The biological context A considerable number of freshwater fish species display resourcepolymorphism, i.e., the coexistence (in the same lake) of a number of distinct morphologicaltypes, that are specialized on different resources (Andersson et al., 2007). Resource poly-morphism is not restricted to this group of species, in fact it occurs in a wide range of taxa(Skulason and Smith, 1995). Resource polymorphism may be an early stage in a processeventually leading to ecological speciation (Schluter, 2000; Skulason and Snorrason, 2004).In lake fish, such polymorphisms often include a pelagic morphotype feeding mainly onzooplankton and a benthic morphotype feeding mainly on macroinvertebrates (Smith andSkulason, 1996; Skulason and Snorrason, 2004). Examples are sticklebacks, whitefish pump-kinseed, Arctic char and bluegill sunfishes (Andersson et al., 2007).

Different morphotypes may have allopatric origins, but once in sympatry through re-peated invasion of the same lake, they remain polymorphic due to disruptive selection thatresults from ecological interactions, such as competition for food (Svardson, 1979; Nymanet al., 1981; Schluter and McPhail, 1993; Rundle and Schluter, 2004). Alternatively, the mor-photypes may originate sympatrically as a consequence of adaptive sympatric speciation.Evidence for sympatric (within-lake) origin of Arctic char morphotypes stems from bothgenetic (Gıslason et al., 1999) and morphological studies (Alekseyev et al., 2002). In bothscenarios, disruptive selection is an essential force driving and maintaining the sympatricdivergence and coexistence of morphotypes.

Fish are particularly interesting in this respect since the resource polymorphism mayhave a size-dependent, ontogenetic origin, since fish often use different resources at dif-ferent sizes; a phenomenon sometimes referred to as ”ontogenetic niche shifts” (Fig. 10).Ontogenetic shifts in food use are often associated with habitat shifts, but also with morpho-logical changes. A species which has such a repertoire of feeding habits and morphologiesin its genome may be able to evolve into several resource specialists. For example, through a


Figure 10: A schematic representation of ontoge-netic niche shifts. As an individual grows in bodysize, it will change its food choice. For exam-ple, a Eurasian perch (Perca fluviatilis) will startout feeding on zooplankton, shift to macroinver-tebrates at intermediate body sizes, and eventu-ally to feeding mainly on fish, including cannibal-ism. Such shifts in prey types are often associatedwith shifts in habitat use. The increasing heightsof the curves’ maximums reflects the increasingoverall feeding rates of bigger individuals.

process similar to neoteny, one can imagine that morphological changes are delayed (whilegrowing in body size) such that the green curve in Fig. 10 would be inflated to the right, atthe expense of the blue curve. The contrary is imaginable as well, if the blue curve extendsits range towards smaller body sizes, which corresponds to precocious ontogenetic onset ofthe corresponding morphological changes.

A illustrative example of these ideas is the Arctic char (Salvelinus alpinus), which has in-spired much of my research during this postdoc period. This fish species has a circumpolardistribution and occurs in lakes and streams, and also includes an anadromous form. Arcticchar are highly polymorphic; the degree of within-lake polymorphism varies between oneand four morphotypes (Alekseyev et al., 2002; Adams et al., 2003; Skulason and Snorrason,2004). The four morphotypes are two benthic types; a pelagic zooplanktivore specialist; anda pelagic piscivorous specialist (Skulason and Snorrason, 2004). The benthic types retainthe morphology typical for juveniles throughout their lifetime. The pelagic types obtainan ”adult” morphology early in life and maintain it; the main difference between the twopelagic types and between the two benthic types being their body size (Skulason and Snor-rason, 2004).

An adaptive dynamics approach During my PhD project, I have formalized the hypoth-esis that an ontogenetic niche shift may result in the evolution of resource polymorphismthrough evolutionary branching (Claessen and Dieckmann, 2002). This was done by extend-ing a standard Kooijman-Metz (KM) model in two directions: (1) to incorporate two foodresources, instead of one, that are exploited in a size-specific manner such as in Fig. 10; and(2) to allow for evolutionary dynamics by including heritable variation. The latter was doneby introducing a heritable i-state variable denoted by u which is the ”resource utilisationstrategy”. In Claessen and Dieckmann (2002), u is the body size at which the ontogeneticniche shift takes place: individuals smaller than u feed mainly on resource type 1, whereasbigger individuals feed chiefly on the second resource type. We have showed that mostoften an intermediary strategy u∗, corresponding to a generalist strategy that exploits bothresources in a balanced way, is an evolutionary singular point. In particular, this generaliststrategy is under most conditions an attractor of the monomorphic adaptive dynamics. Theevolutionary stability of this singular point depends on the trade-off between the foragingrates on the two food types. Depending on the model, the trade-off can be determined bythe size-scaling of the foraging rates (Claessen and Dieckmann, 2002) or by the phenotypicplasticity in the feeding rates (Claessen et al., 2007, 2008). Depending on the trade-off, u∗ is


either a Continuously Stable Strategy (CSS) in which case the population will evolve to u∗

and remain their indefinitely. Or else u∗ is an Evolutionary Branching Point (EBP) in whichcase the population first evolves to u∗ upon which the population may branch into two sis-ter populations that will diverge (in terms of u), although what exactly happens at the EBPdepends on the assumptions with respect to mating and genetics (clonal vs sexual reproduc-tion; haploid vs diploid; number of loci; dominance; assortative mating; etc) (Claessen et al.,2008).

Conceptually, the central question of this part of my research, or at least its starting point,is: do population fluctuations facilitate or inhibit evolutionary branching? This questionfirst surfaced in discussions with Jens Andersson about the speciation in Arctic char. Forthis species, a proposed hypothesis for explaining the different levels of resource polymor-phisms in different lakes, is that the ”stability” of the different populations influences thefacility of the evolution of the polymorphism: more stable systems are more likely to con-tain high levels of polymorphism (Skulason and Snorrason, 2004). We wanted to put thishypothesis to the test, first by simulating the evolutionary branching under different lev-els of intrinsic population stability; both deterministic and stochastic. First, the idea is tocompare cycling vs stable populations (do generation cycles facilitate or inhibit evolution-ary branching?). Second, to compare small and large non-cycling populations, to assess theeffect of absolute population size and the associated level of demographic stochasticity onevolutionary branching. We also wanted to study the effect of externally imposed stochas-ticity (referred to as environmental stochasticity), but we did not have time to investigatethis aspect. However, this has recently been done by Johansson et al. (2010), who showedthat external stochastic forcing has a similar effect on evolutionary branching as internal,demographic stochasticity.

The effect of generation cycles on evolutionary branching According to Hans Metz (Lei-den University, The Netherlands and IIASA, Laxenburg, Austria), Lotka-Volterra predator-prey cycles do not qualitatively influence the predictions of adaptive dynamics based onthe equilibrium conditions (personal communications). The idea is that what matters forevolution on the time scale of adaptive dynamics and evolutionary branching (i.e., manygenerations) is the average environment experienced by the individuals. Since in Lotka-Volterra type models the average environment, computed over the duration of a populationcycle, is the same as the environment at the (unstable) equilibrium, it follows that predictionsbased on the equilibrium conditions hold for the predator-prey cycles. I have not checkedthat these conditions are true for non-Lotka-Volterra models of predator-prey interactions.

For generation cycles the situation is different. In a generation cycles there is a strongcorrelation between the age and size of an individual, and the environmental conditions.For example in the Kooijman-Metz (KM) model (see section 2.2), most small individuals ex-perience very severe competition, whereas most intermediate sized individuals experiencelittle competition. Most individuals reproduce under favourable conditions, but are thenconfronted with severe competition. Given this strong correlation between life history stageand resource levels, we cannot use the equilibrium conditions to predict the ”average” lifehistory. We cannot use the equilibrium conditions to predict the fitness of possible mutantsarriving in the population.

Here I present results based on analysis of a series of related models. The first is an ex-tension of the KM-type model studied in Claessen and Dieckmann (2002), by introducing a


Figure 11: A schematic representation of ”evo-lutionary cycling” caused by ecological hystere-sis in a Kooijman-Metz model with a juveniledelay and an ontogenetic niche shifts (i.e., be-tween two resources). The ecological attractorsare generation cycles driven by the interaction inthe first niche (blue attractor) or generation cy-cles driven in the second niche (green attractor).The evolutionary singular point is located on theunstable equilibrium curve connecting the greenand blue attractors (black dot). The generationcycles disappear through a bifurcation close tothe saddle-node bifurcations (”tipping points”).The black arrows indicate direction of evolution-ary change of the trait u. The dotted arrows in-dicate ecological attractor shift.

juvenile period necessary to obtain generation cycles (model 1). The second model is similar,except that the two resources are exploited throughout the life history (model 2). There ishence no ontogenetic niche shift, but rather the utilisation of the two resources is governedby a simple time splitting argument: an individual spends a fraction u of its time (per day)in one habitat, the remainder (1− u) in the other habitat. The third model is an unstructuredvariant, which drops all size-dependent elements, and keeps only the time splitting argu-ment (model 3). In addition, we study the effect of phenotypic plasticity, which is done byassuming that the more time an individual spends in a habitat (per day), the more efficientit becomes on the resource in that habitat. This ”learning” assumption has been studied inboth the structured model (referred to as model 2+) and in the unstructured model (model3+).

It should be noted that for these models, in equilibrium conditions, the prediction of theevolutionary stability of u∗ depends solely on the parameters of the trade-off, and not onany of the values characterising the ecological equilibrium (Claessen and Dieckmann, 2002;Claessen et al., 2007, 2008).This feature allows us to make predictions about the evolutionarystability of singular points without having to simulate the system to compute the actualfitness landscape (referred to as a pairwise invasibility plot, PIP). Of course, the question iswhether this still holds under non-equilibrium conditions.

Do generation cycles prevent evolutionary branching? Our simulations show: not at all.In fact, in simulations with model 2+ we found that evolutionary branching occurred in allcases for which the equilibrium conditions predicted u∗ to be an EBP, and even for certaincases for which the prediction is CSS (i.e., no branching predicted).

However, in our investigation of the adaptive dynamics of the size-structured models,we ran into a number of surprising dynamical effects, which complicated the analysis. First,in order to get generation cycles in a KM model, we need a juvenile life stage. In model 1,the combination of a juvenile period and an ontogenetic niche shift results frequently (for awide range of parameter values) in bistability: the equilibrium curve is folded at two saddle-node bifurcations (Fig. 11) (unpublished results obtained with the continuation techniquedescribed in section 2.2). The problem is that, on top of this complication, the evolutionarysingular point lies then invariably on the unstable equilibrium. The resulting evolution-ary dynamics are hence unending cycles of directional evolution towards a ”tipping point”


(saddle-node bifurcation), followed by a shift towards the other ecological attractor and areversal of the direction of evolution (Fig. 11). This model is hence not very well suited toanswer to our question.

To circumnavigate this problem, we designed model 2, without ontogenetic niche shift,but with a rather more gradual change in the foraging rates on the two resources, result-ing from different exponents of the allometric equation for the attack rates. As the previousmodel, this model exhibits three types of dynamics: stable equilibrium, classical genera-tion cycles with a period of one generation, and generation cycles with a periodicity of 1/2generations (Fig. 12A). The latter cycles are governed by a similar mechanism as normalgeneration cycles, except that two cycles are squeezed into a single generation; yet for eachgeneration, the same mechanism is at work. Each ecological attractor has a convergent stablesingular point (i.e., CSS or EBP), which lies at an intermediate value of u on the equilibriumcurve (u∗ = 0.36, 0.4 and 0.45 for the three attractors, resp.; Fig. 12B, C, D). The three fitnesscurves show that the evolutionary stability is not the same on the three attractors: whereasthe stable equilibrium and the period-1/2 cycles are CSS (the generalist strategy u∗ is evo-lutionarily unbeatable, similar to an ESS), the singular point on the generation cycles is anEBP. In the latter case, evolution is expected to converge the population to u∗, at which pointselection becomes disruptive (Fig. 12). This is hence radically different from the situation inequilibrium, and is solely due to the type of population cycles. We formulated an individual-based equivalent model to simulate the stochastic adaptive dynamics of this system. In thestochastic model, we assume clonal reproduction (no sex, no genetics). The simulations allshowed the same pattern (Fig. 12E and F): convergence to the EBP, evolutionary branching,followed by an ecological attractor shift and the extinction of the upper branch. The twosister species coexist for a relatively long period (ecologically speaking; 100s of generations)on the new attractor before going extinct.

Two things are interesting in this example: (i) generation cycles are indeed capable ofchanging the shape of the fitness function (at the singular point) from stabilizing selectionto disruptive selection; and (ii) the eco-evolutionary dynamics of the system with two com-peting populations (after branching) lead to an attractor shift, followed by the loss of onespecies. Although I have not verified the deterministic prediction of the trajectory of thepair of evolutionary traits u1 and u2 in the dimorphic case (which would be possible bymodelling two structured populations in competition), the pattern observed in Fig. 12 is con-sistent in a large number of stochastic simulations. It thus appears that the attractor shift isan inherent aspect of the evolution in this system. The take home message is hence twofold:population structure matters in evolutionary dynamics through its effect on the shape of thefitness function and through the richness of its dynamical behaviour.

The effect of demographic stochasticity on evolutionary branching In trying to answerthe question ”do generation cycles impede evolutionary branching?”, we ran into anothertechnical problem (in addition to the folded equilibrium curve and the attractor shifts). Innumerous simulations with the individual-based equivalents of models 1 and 2, we foundthat the population did not branch while the computed fitness curve showed disruptiveselection at the branching point (as indicated the associated PIP). This appeared to be con-sequence of small, finite population size, as a systematic study of the effect of the absolutepopulation size on the success of evolutionary branching showed. We decided to pursuethis line of ”variability”, in addition to the deterministic variability of generation cycles. The


Figure 12: An example of the effect of generation cycles on the fitness landscape, in the size-structured model with two resources and the time splitting trade-off (model 2, see text). A: the re-lation between ecological dynamics and the evolutionary trait u. The system has three attractors: astable equilibrium (blue curve); period-1/2 generation cycles (green) and period-1 generation cycles(red). The dots indicate the amplitude of the cycles. B: The fitness curve at the singular point u∗ onthe blue attractor. C: The fitness curve at the singular point u∗ on the green attractor. D: The fitnesscurve at the singular point u∗ on the red attractor. E and F: A stochastic realisation of the adaptivedynamics, starting in period-1 generation cycles. E shows the convergence to the EBP on the period-1generation cycles. The evolutionary branching is followed by an attractor shift to the period-1/2 gen-eration cycles. This ecological attractor does not permit the coexistence of the two incipient speciesand one of them goes extinct. The remaining branch converges to the CSS of the green attractor. Fshows the amplitude of oscillations in the total population abundance, along the evolutionary trajec-tory in E. Note the sudden reduction in amplitude at the moment of the attractor shift. E and F havethe same time scale: the extinction of the upper branch occurs long after the attractor shift.


plan was to get back to the issues of population cycles once the effect of finite populationsize well understood.

A much simpler way to assess the effect of absolute population size is to use an unstruc-tured population (models 3 and 3+). This avoids confounding factors, e.g., of the interplaybetween small population size and population structure. The simplest equivalent model isa simple one-consumer-two-resources model. Such a model can be studied in quite somedetail using standard techniques from ecological and adaptive dynamics theory. In model3, the ”time-splitting” assumption alone amounts to a linear trade-off in the sense of Levinsfitness sets, as formalised by Rueffler et al. (2004, 2006). Introducing the ”learning effect” (de-scribed above, model 3+) introduces non-linearity in the fitness curve at the singular point.Assuming that individuals improve their performance if they spend more time foraging theresource in question, the fitness curve is disruptive at the singular point (the PIP correspondsto an EBP).

Absolute population size in finite populations may influence the adaptive dynamicsin different ways. One line of studies on evolutionary dynamics in finite populations isgame-theoretic and investigates the consequences of the fact that a single mutant cannotplay against itself (Riley, 1979; Schaffer, 1988). The evolutionarily stable strategy (ESS) thenappears to depend on absolute population size: the smaller the population, the more spite-ful the ESS (Schaffer, 1988). A second line of research addresses the effect of demographicstochasticity on evolutionary dynamics. Proulx and Day (2001) argue that the expectedgrowth rate of a small mutant population (the standard definition of fitness in adaptive dy-namics theory (Metz et al., 1992)) may not accurately predict the direction and endpointof evolution in finite populations subject to environmental stochasticity. In the absence ofdemographic stochasticity, alleles with a negative expected growth rate have zero probabil-ity to reach fixation. Proulx and Day (2001) show that in a finite population they may yethave a fixation probability that is greater than that of a neutral allele. They argue that it ishence more correct to use the fixation probability of rare alleles to describe the evolutionarydynamics of small populations. Cadet et al. (2003) and Parvinen et al. (2003) study the evo-lution of the dispersal rate in a metapopulation model and demonstrate that accounting forfinite population size in local patches alters the evolutionary prediction. They propose twoexplanations for the difference. First, when local populations are small, the relatedness ofindividuals is high, leading to kin competition. Second, demographic stochasticity resultsin variation in local population size such that a disperser from a non-empty patch alwayshas a chance to find a patch with fewer competitors. Both explanations favour the evolutionof a higher dispersal rate under the influence of demographic stochasticity. In conclusion,these studies show that the direction of evolution in finite populations may differ from theexpectation based on a deterministic model.

In the context of the resource polymorphism in lake fish, absolute population size has astraightforward interpretation: lake size. Identifying lake size with absolute population sizeis a strong simplification, since lake size influences fish populations in different ways (num-ber of habitats available, complexity of ecological communities and hence interactions, etc),lake size and population abundance are likely to be strongly correlated (all else being equal).One reason why pursuing the effect of demographic stochasticity on evolutionary branch-ing was tempting, is that we might be able to obtain results that are empirically testable withdata on lakes of different sizes.

Our model results are straightforward (Fig. 13) (Claessen et al., 2007): in large popula-tions, evolutionary branching occurs quickly after the population has evolved close enough


Figure 13: The duration of two phases ofadaptive dynamics computed for ten simu-lations per lake volume, assuming the sin-gular point is an EBP. Time is expressed inunits of the average life span. All runs withV = 1 are extinct before approaching theattractor. (a) Time needed to converge tothe singular point u∗ = 0.5 from the initialcondition u = 0. (b) Branching delay; thetime elapsed between approaching the EBPand the onset of (successful) evolutionarybranching. From: Claessen et al. (2007)

to the singular point. In small populations, there can be a very long delay between converg-ing to the singular point, and the onset of evolutionary branching. Below a certain mini-mum population size, evolutionary branching is delayed indefinitely (> 106 generations).The same results have been obtained with the size-structured model (models 1 and 2), andwith other unstructured models (Stephane Legendre, personal communications) suggestingthis is a rather general phenomenon. We argue that two processes cause the delay (Claessenet al., 2007). First, random drift allows the population to drift away from the singular pointu∗, over periods of thousands of generations. Second, we have observed frequent extinc-tions of incipient branches. Such extinction can be the result of random drift towards theextinction boundary of one of the two species. Alternatively, the extinction can be due tothe strong, ecological similarity of the incipient branches, which results in ”near-neutral”stability of the equilibrium with two sister populations: whereas the total abundance of thetwo sister populations is well regulated by the interaction with the resources, the relativeabundance of the two sister populations is nearly neutral and poorly regulated. This allowswild fluctuations in the proportion of either sister population, making them prone to go ex-tinct. The contrast in time scales between (slow) relative dynamics of similar phenotypesand (fast) aggregate dynamics of the total consumer density was analysed by Meszena et al.(2005) who studied the dynamics of a number of similar clones in a (unimodal) distribution.Our result suggests that their result is relevant even for the dimorphic dynamics soon afterbranching.

Speciation vs evolutionary branching Evolutionary branching is not yet speciation. Atleast, that depends on your species definition. The most commonly used definition is thatof reproductive isolation (in sexually reproducing populations). In order to argue that the


Figure 14: Time to genetic poly-morphism and to speciation, de-pending on lake volume V, withthree loci. For each simulation run,the time at which a complete poly-morphism at one locus arises (tri-angles) is indicated. The approachtime (plus symbols) and the timeat speciation (filled circles) are alsoplotted. Note the equilibrium pop-ulation size of a monomorphic pop-ulation with u∗ = 0.5 equals n =18V . From: Claessen et al. (2008)

above results apply to sexually reproducing population such as the Arctic char, it was re-quired to test the idea in a model of diploid, sexually reproducing populations. We testedthe effect of the number of loci, the level of recombination (crossing over), the mutation rate,the mutation step size and the level of assortative mating, as well as the curvature of the fit-ness function (Claessen et al., 2007, 2008). The influence of all of these factors on speciationand on the delay to speciation are very unsurprising: speciation gets more difficult (and thedelay longer) with increasing levels of the number of loci and/or recombination, and withdecreasing levels of assortative mating and curvature of the fitness function. Smaller muta-tion rates and step sizes slow down the adaptive dynamics but do not prevent speciation.So, overall, the results described above for the clonal model (and illustrated in Fig. 13) carryover to sexual populations (Fig. 14).

However, the investigation of the sexual model did show an unexpected result, intrinsicto the more complex underlying genetics. If the phenotypic trait u is determined by severalloci, it is possible that the evolutionary branching reaches a partial end result; that is, evolu-tionary branching occurs at only one or two loci, producing two alleles for these loci, but theremaining loci remain monomorphic. The population remains panmictic. Such polymor-phism can be maintained for very long periods (> 105 generations), and may be followedby speciation (branching of the remaining loci and reproductive isolation) or they can col-lapse (return to a lower level of polymorphism). The explanation of this dynamic behaviouris that, in general, the different loci do not branch simultaneously. Each successive branch-ing at the level of a single locus reduces the curvature of the fitness landscape. It becomestherefore more difficult to branch successfully for the subsequent loci, for the same reasonsas the ones explaining delayed branching in clonal populations. The formation and subse-quent collapse of polymorphisms in multilocus trait evolution has been analysed in detailby van Doorn and Dieckmann (2006). Their description of different phases of formation andcollapse of polymorphisms closely matches the dynamics observed in our model. With adeterministic approximation of the evolutionary dynamics at the allele level based on thecanonical equation of adaptive dynamics they show that the loss of polymorphism in all butone locus can be expected even in very large populations. A similar result has been foundby Kopp and Hermisson (2006), who showed that frequency-dependent, disruptive selec-


tion favours such concentration of genetic variation at one or a few loci. We have shownthat the level of stable polymorphism depends on the strength of disruptive selection andon the system volume. Only with sufficient curvature and in sufficiently large lakes do wefind multiple polymorphism that allows for speciation. With weaker curvature or in smallerlakes, the population genetic structure tends to get trapped in a collapsed polymorphism.Because the strength of disruptive selection is smaller in such partially polymorphic popula-tions than in monomorphic populations at the EBP (Kopp and Hermisson, 2006), this processis likely to increase the delay to speciation.

Lake size and Arctic char resource polymorphism We can make two qualitative predic-tions. First, in a given species, resource polymorphism (whether the result of speciation ora genetic polymorphism) is more likely to occur in large lakes than in small lakes, since ab-solute population size is expected to be proportional to lake size. Second, given that themorphological trait underlying the resource polymorphism is most likely a multilocus trait(Skulason and Snorrason, 2004), our model further predicts that the extent of reproductiveisolation in observed cases of resource polymorphisms depends on lake size, with completereproductive isolation and hence speciation underlying the resource polymorphism in largelakes, while in intermediate lakes the resource polymorphism is more likely to be a geneticpolymorphism in a panmictic population (Fig. 14).

The first prediction is confirmed by our analysis of empirical data on Arctic char poly-morphism in 22 lakes in Transbaikalia that showed that, indeed, the number of morphotypesis positively related to lake size (Alekseyev et al., 1998, 1999, 2002; Claessen et al., 2008). Inaddition, the data analysis revealed a negative relation with the number of other fish speciespresent. We hypothesize that the presence of other species may reduce the likelihood of evo-lutionary branching in two ways. First, additional species reduce the availability of nichesand hence limit the maximum level of diversification. The presence of another fish speciesin one of the available niches reduces the resource associated with that niche and hencechanges the curvature of the fitness function, resulting in directional selection toward spe-cialization in the other niche. Second, the presence of a predator species would not affect therelative abundance of the two resources but rather reduce the population size of the focalspecies. Our results suggest that a predator thus decreases the likelihood of evolutionarybranching in its prey by reducing its absolute population size. This suggests that higher-dimensional environments (e.g., including competitors and predators) may modulate therelationship that the model has revealed; in both cases, evolutionary branching is inhibitedby the presence of other species. A negative relation between the number of other speciesand the number of morphotypes is, however, consistent with our hypothesis of a sympatricorigin of the morphotypes, depending on lake size.

It should be stressed, however, that alternative hypotheses may lead to the same predic-tion. First, if the probability of invasion of a lake increases with lake size, the hypothesis ofmultiple invasions of lakes by allopatrically diverged morphotypes would lead to the ob-served correlation between lake size and diversity. Second, it can be argued that larger lakesare likely to harbour more available niches or habitats and may therefore support higherlevels of diversity. The data (table 1 in Claessen et al. (2008)) therefore cannot differenti-ate between these hypotheses. However, analysis of meristic data from a subset of theselakes suggests that different sympatric morphotypes are more closely related than allopatricpopulations of the same morphotype, thus supporting a sympatric origin of morphotypes


(Alekseyev et al., 2002). This observation seems to rule out the invasion hypothesis butleaves the second alternative hypothesis intact.

The environmental feedback loop In the work described above, the idea of the environ-mental feedback loop is not discussed explicitly. However, as explained in section 3.1, thisidea is central to the whole approach of adaptive dynamics, and to the definition of fitnessin particular. Furthermore, although it has been stated above that the shape of the fitnesscurve (in equilibrium) at the singular point does not depend on the state of the environment,this is only partially true. In fact, the environmental feedback loop is the driving force of thedirectional selection as long as the population is not yet at the singular point. It is only af-ter the resources have been ”balanced” by this directional selection, that the environmentalfeedback loop moves into the background, and that physiological trade-offs move into theforeground. Thus, the CSS or EBP is characterised by a very precise state of the environment,which has been obtained by the operation of the feedback loop.

Another interesting aspect of the feedback loop is the role of the ecological struggle forexistence in what could be called ”macroevolution” (Fig. 12F). The way in which the pop-ulation is being regulated by ecological processes appears to have potential consequenceson the shape of the evolutionary tree. The struggle for existence forces two sister speciesto diverge from each other, yet their ecological interaction remains sufficiently strong thatonce the bifurcation point has been reached, and the populations settle on a different ecolog-ical attractor, one population is doomed through the competitive interaction with the otherspecies.

By contrast, the processes resulting in delayed evolutionary branching seem to stem froma weakness of the environmental feedback loop. It seems that a population can drift againstthe fitness gradient for thousands of generations only if the feedback is not very strong. Also,the strong fluctuations of two sister populations in a ”near-neutral” equilibrium, leading toa high extinction probability of one of them, is possible due to a very weak feedback loopgoverning the relative dynamics (despite a strong feedback for the aggregate dynamics).

3.3 Speciation in dynamic landscapes

The inhabitants of the Cape de Verde Islands are related to those of Africa, like those of the Gala-pagos to America. I believe this grand fact can receive no sort of explanation on the ordinary viewof independent creation; whereas on the view here maintained, it is obvious that the GalapagosIslands would be likely to receive colonists, whether by occasional means of transport or by for-merly continuous land, from America; and the Cape de Verde Islands from Africa; and that suchcolonists would be liable to modification; –the principle of inheritance still betraying their originalbirthplace. Darwin (1859), pages 538-539

This section describes the work done by Robin Aguilee during his PhD project, co-supervised by Amaury Lambert (UPMC) and myself. The work has led to three publications(Aguilee et al., 2009, 2011a,b) and a submitted manuscript. The populations discussed in thissection are structured spatially and genetically, but not physiologically as was the case of themodels discussed in sections 2 and 3.2.

The central topic of Robin’s PhD thesis is the effect of landscape changes on evolution-ary dynamics. Here, landscape dynamics refer to changes in the connectivity of differentparts of the geographic range of a population or group of populations. The simplest case of


landscape dynamics studied in his thesis is a repetitive cycle of a population’s spatial rangebreaking up into two parts and later re-connecting into a single range (Aguilee et al., 2009,2011b). This can be thought of as a geographical barrier that repeatedly appears and disap-pears inside the geographical expanse of a species. Such alternation between ”sympatric”and ”allopatric” stages in the evolution of species is a phenomenon that is probably quitecommon in evolutionary history, but has rarely been studied theoretically. Robin has stud-ied the evolutionary dynamics on different temporal scales, ranging from microevolutionaryprocesses of the invasion dynamics of a single allele (Aguilee et al., 2009), to the intermedi-ate time scale of character displacement (Aguilee et al., 2011a), to the mesoevolutionary timescale of ecological speciation (Aguilee et al., 2011b) and adaptive radiation (Aguilee et al.,unpublished manuscipt).

Micro-evolution On the shortest time scale, landscape dynamics influence the populationgenetics through two distinct processes: repetitive bottlenecks leading to founder events,and a refuge effect (Aguilee et al., 2009). Whereas the first is well-known in population ge-netics, the second one is specific to this kind of landscape dynamics. Each process influencesthe fixation probability and the time to fixation in a specific way. A succession of founderevents decreases the fixation probability of an advantageous mutation, but accelerates its fix-ation (conditional on fixation). However, the coexistence of two temporarily disconnecteddemes generates a refuge effect which can strongly delay fixation. If population fusions arerare, refuge effects are the principal factor determining fixation times which are longer thanin a static landscape. In contrast, if fusions are frequent, founder effects are the principal fac-tor and fixation times are then shorter than in a static landscape. Note that founder effectsare only observed in the case of asymmetrical fragmentation (i.e., when one of the fragmentsis small). These results are similar to earlier results on the effect of bottlenecks on fixationprobability (Otto and Whitlock, 1997; Wahl and Gerrish, 2001; Heffernan and Wahl, 2002),despite the fact that these models do not consider spatial dynamics.

Our question was to understand how dynamic landscapes alter mutant fixation. Onemain modification is that founder effects and refuge effects are repetitive and cumulative.These effects can exist in static landscapes, but can occur only once. Landscape dynamicsthus strengthen alterations of fixation probabilities and times to fixation. This is particularlytrue for times to fixation which can be increased by a factor of more than 10 (Aguilee et al.,2009). This stresses the importance of considering the isolation of some population and theirfusion after a possibly long time. Depending on the characteristics of the dynamics, fixationcan be disfavoured or unaffected, delayed or accelerated. Therefore, one needs to describethe whole dynamics of the landscape, and to specify characteristic time scales. Fast dynamicsare appropriate to model ecological processes such as dispersal and recolonization events(establishment of new colonies and their later fusion because of their expansion or becauseone habitat becomes unsuitable), or to model geographical processes such as changes inthe fragmentation of habitat due to human action. In that case, beneficial mutations have”one small chance of doing very well”: fixation is unlikely (frequent founder effects) butvery fast if it occurs (limited refuge effects). In contrast, slow dynamics are appropriate togeographical events such as the separation of populations due to glacial events followedby postglacial contact or such as repetitive fragmentation and fusion of islands (or lakes)due to water level variations caused by climatic events. In that case, fixation of a beneficialallele would be more likely (rare founder effects) but very slow when it occurs (strong refuge


Figure 15: Evolution in dynamic landscapes,with stabilizing selection (singular points areCSS): typical evolutionary trajectories of theecological trait over time. Population densityis indicated by the level of gray. Black horizon-tal lines indicate the values of singular pointsin sympatry (solid lines), in the first patch in al-lopatry (dashed lines) and in the second patchin allopatry (dotted lines). Vertical gray linesindicate shifts of the landscape from sympa-try to allopatry (solid lines) and conversely(dashed lines). The two subpopulations are re-productively isolated by assumption. A: simi-lar population sizes in allopatry, long allopatricphases but short sympatric phases. Long termcoexistence of incipient species. B: similar pop-ulation sizes in allopatry and long allopatricand sympatric phases. One of the incipientspecies goes extinct in sympatry due to limit-ing similarity. C: As B but different populationsizes in allopatry. The branch with the lowestabundance (evolving towards the dotted line inallopatry) rapidly goes extinct upon secondarycontact. From: Aguilee et al. (2011b)

effects).The above results have been obtained by using a simple model for the population and

allele dynamics: a population genetics haploid model with two types, mutants and residents,representing individuals carrying two different alleles, respectively. This model, referred toas the Moran model or Moran process (Moran, 1962), is embedded into a model of landscapedynamics, specified below. The Moran process is similar to the Wright-Fisher model (Wright,1931), but in continuous time (overlapping generations). It is a stochastic process whichdescribes a finite population of constant size and based on the following mechanism: duringan infinitesimal time dt, a birth or death event can occur or not; if it does, the population attime t+ dt is updated from that of time t by randomly selecting an individual to reproduceand then, independently, randomly selecting an individual to be removed. Each individualwith birth rate b has a probability bdt to reproduce during dt. Each resident reproduces at rateb = 1 and each mutant at rate b = 1 + s where s is its selective advantage. For an undividedisolated population whose allele frequency fluctuates via a Moran process, classical resultsand approximations are known for the fixation probability and time to fixation (Wright, 1931;Kimura, 1962; Kimura and Ohta, 1969; Ewens, 2004). Clearly, this approach does not accountfor any environmental feedback. Fitness is not a consequence of ecological interactions.Rather, the simplifying assumptions made for this model permit a formal analysis of itspopulation genetics in more detail than is common in adaptive dynamics (Aguilee et al.,2009).

Meso-evolution A next step is to study the effect of the above described landscape dy-namics in a more ecological context, i.e., accounting for the environmental feedback loop.We chose to use a simple, well-studied base model, and to place it in the context of land-


scape dynamics. Specifically, we use the unstructured model (one consumer, two resources)described in section 3.2 (model 3+, p. 35), and investigate the effect of landscape dynamicsof the type defined above on speciation (Aguilee et al., 2011b). The idea is to use a model ofwhich we know under what conditions (in a static landscape) its adaptive dynamics are ex-pected to lead to evolutionary branching (singular point is an EBP) or not (singular point isCSS), and to subject this model to the mentioned landscape dynamics in either case. Interest-ingly, our model hence allows for both sympatric speciation (as in section 3.2) and allopatricspeciation. Obviously, in order to be able to speak of ”speciation”, the model has to accountfor diploid genetics and mating behaviour.

The model thus obtained is relatively simple: ecologically, it amounts to one or two con-sumer populations competing for two resources. The exploitation strategy of the consumersis the evolutionary trait u, as defined above, with the non-linear trade-off that can be eitherweak or strong (Rueffler et al., 2004, 2006), leading either to stabilizing selection at a singularpoint (which is a CSS) or to disruptive selection at a singular point (which is an EBP), respec-tively. Behaviourally, the individuals mate according to a second, mating trait, which maybe chosen to evolve or to be a fixed parameter. Dynamically, a number of interesting aspectsappear due to the landscape dynamics. When the habitat breaks up, the two subhabitats arecharacterised by different resource distributions. Thus, each (allopatric) subpopulation willevolve in such a way to specialise on the local resource availability. If such specialisation isassociated with a divergence in the mating trait, it could be imagined to lead to allopatricspeciation. Upon fusion of the habitats and its resource distributions, the two subpopula-tions are in secondary contact. The eco-evolutionary dynamics that follow depend criticallyon the level of ecological divergence and of reproductive isolation that has evolved in sepa-ration. It becomes clear that the evolutionary dynamics of the mating trait, and the level ofhybridisation at secondary contact, are of importance in this model.

For the case of stabilizing selection, we make the a priori assumption of allopatric spe-ciation: subpopulations that have evolved in allopatry are assumed to be reproductivelyisolated when they meet in sympatry (upon fusion of the landscape) (Aguilee et al., 2011b).(Reproductive isolation does not evolve intrinsically, in this case). The model results showthat ecological divergence occurs in allopatry, but the coexistence of the two new species,although ecologically stable, is evolutionarily unsustainable in a sympatric landscape (i.e.,after secondary contact) (Fig. 15B and C). We showed that landscape dynamics (allopatry-sympatry oscillations) may facilitate their long-term coexistence (Fig. 15A). In particular,landscape dynamics preserve allopatric speciation given certain characteristic time scalesof the landscape dynamics (i.e., long allopatry, short sympatry). Also, the maintenance ofspeciation is facilitated by similarly sized subpopulations upon secondary contact.

When selection is disruptive at singular points (which are then EBPs), we know that spe-ciation can occur in sympatry (standard adaptive dynamics theory, see Dieckmann (1997);Geritz et al. (1998)). Yet even under these conditions speciation is all but guaranteed, since anumber of factors may impede the evolutionary branching to succeed. One such factor is de-mographic stochasticity in small populations (Claessen et al. (2007), see section 3.2). Anotherone is sometimes referred to as the ”Mendelian mixer” (Hans Metz, Leiden University andIIASA): recombination in sexual populations with diploid multilocus genetics may producesuch a high rate of hybridisation, that the disruptive selection is too weak to produce evo-lutionary branching (Waxman and Gavrilets, 2005; Claessen et al., 2008). Here we assumea number of independently segregating loci (L = 6) sufficiently large to delay speciationalmost indefinitely; in a static landscape the model hence does not predict speciation, even


Figure 16: Disruptive selection (singularpoints are EBP). A: Specific evolutionary trajec-tories of the ecological trait. B: the mean mat-ing trait of the population over time. Popula-tion density is indicated by the level of gray.Vertical solid lines indicate shifts from allopa-try to sympatry with a window of partial sec-ondary contact (vertical dotted lines indicatethe end of the windows of partial secondarycontact). Vertical dashed lines indicate shifts ofthe landscape from sympatry to allopatry. Spe-ciation is successful at the second secondarycontact and is maintained in both allopatry andsympatry afterwards. From: Aguilee et al.(2011b)

if the singular point is an EBP.Our results show that, under these conditions (i.e., EBP and Mendelian mixer) landscape

dynamics can generate diversity where this is not possible in a fixed, sympatric landscape.A shift from sympatry to allopatry stops gene flow, allowing ecological divergence that wasimpossible in sympatry due to the genetic constraints. Speciation in a dynamic landscape ismore likely than in a static, allopatric landscape, if the transition from allopatry to sympatryis ”soft”, that is, if there is an intermediate phase in which the mating probability betweenindividuals that belonged to the same allopatric habitat is higher than between those comingfrom different habitats. In other words, if upon removal of the geographical barrier betweenthe allopatric populations, the sympatric population is not immediately well-mixed. Theexistence of a ”hybrid zone” between the two formerly allopatric populations favours a pro-cesses referred to as ”extrinsic ecological reinforcement” (Noor, 1999; Servedio and Noor,2003). Hybrids between specialists have reduced fitness, and are hence selected against,while at the same time assortative mating is selected for. Yet if the rate of hybrid produc-tion is too high (for example if the new sympatric population is panmictic at once), then thereinforcement is too weak to have an effect on the evolution towards stronger assortativemating. We have found that a single event of secondary contact is rarely sufficient to resultin sufficiently high rates of assortative mating for the two ex-allopatric populations to re-main reproductively isolated. Yet the model results show that the system keeps a memoryof previous landscape events. Over a sequence of allopatry-sympatry transitions, the levelof assortative mating will gradually increase, and may reach a level sufficient to result in ef-fective speciation (Fig. 17). Thus landscape dynamics facilitate the evolution of reproductiveisolation between ecologically differentiated subpopulations by offering many opportunities(at each secondary contact) for reinforcement to be successful.

The struggle for (co-) existence An interesting aspect of the above work is the notion thatcoexistence of two species can be considered on both ecological and evolutionary time scales,with possibly different outcomes. This system has two renewable resources so we knowfrom ecological theory that a maximum of two consumer species can coexist in stable equi-librium conditions (Tilman, 1982). Ecologically, coexistence can be defined as the mutual


Figure 17: The effect of landscapedynamics on speciation in the EBPcase. Top: Mean assortative mat-ing trait in simulations leading tospeciation (circles) and in simulationswithout speciation (squares). Bottom:probability of speciation. Whiskersrepresent the 95% confidence intervalsof the estimated means and proba-bilities over the simulation replicatesFrom: Aguilee et al. (2011b)

Figure 18: A: Example of a pairwise invasibility plot (the CSS case in Aguilee et al. (2011b)). Whiteareas indicate positive invasion fitness of the mutant into the equilibrium of the resident; blue areasto negative fitness. B: The corresponding trait-evolution plot (TEP). The white areas correspond tothe set of traits that can coexist; red areas to competitive exclusion. From: Aguilee et al. (2011b)

invasibility of the two consumer populations. That is, if consumer A is at its steady state,does a tiny group of consumer B individuals have a positive population growth rate (andvice versa)? In other words, do both consumers have a positive invasion fitness in the en-vironmental conditions corresponding to the ecological equilibrium of the other species? Ifyes, then the two populations are likely able to coexist (mutual invasibility is neither neces-sary nor sufficient for coexistence, in general, but in many particular models such as Lotka-Volterra type models, it is).

We can hence use pairwise invisibility plots (PIP) to predict which combinations of con-sumer traits are able to coexist. The PIP tells us for a given (resident) population with traitx, which is the set of traits y that have positive invasion fitness (Fig. 18A). Inverting the roleof resident and mutants, we can determine the set of traits y that can invade trait x. Super-imposing these two sets results in a new plot, referred to as a trait evolution plot (TEP, Fig.18B), which displays the set of mutually invasible trait combinations, which corresponds tothe set of traits that can coexist ecologically (Metz et al., 1996; Geritz et al., 1998).

The examples shown in Fig. 15 correspond to the case illustrated in Fig. 18. Upon sec-ondary contact, the pair of traits of the allopatrically diverged subpopulations is in the co-existence area. Ecologically, these species are hence able to coexist. Yet on an evolutionary


time scale, this coexistence is not sustainable. The reason is that, given the environment setby the combined impact of the two species on the two resources, the mutants that have pos-itive invasion fitness have traits which are intermediate between the two resident species.This is a consequence of our assumption of a weak trade-off in this case. On an evolutionarytime scale, the simultaneous struggle for existence between the two residents residents andtheir mutants will thus lead to a convergence of the trait values of the two coexisting species(indicated by the trajectories of the stochastic process, Fig. 18B). This convergent evolution islikely to result in the extinction of one of the species, due to limiting similarity and competi-tive exclusion: one species will lose the struggle for existence (in a (too) static landscape!).

For a scenario in which the singular point is an EBP (disruptive selection), however, thesituation is different. In that case the coevolution of the two coexisting species results indivergent evolution (away from the diagonal in the TEP). That is, for each resident, the mu-tants that have positive invasion fitness have traits that are more specialised than the residentitself. Successive invasions and replacements will hence lead to a resident that is less similarto the competing resident. In this case, the struggle for existence favours the coexistence ofthe two species: in fact, the evolution tends to decrease the level of interspecific competition,and hence increase the scope for coexistence.

Darwin’s finches The above ideas were applied to a classical problem in evolutionary bi-ology: the radiation of the Galapagos finches. The proposed scenario for the radiation ofDarwin’s finches on the Galapagos islands (Grant and Grant, 1996b) hypothesizes the di-vergence of an ecological trait in two populations that came into secondary contact after amigration event (Schluter et al., 1985; Grant and Grant, 2006). This character displacementis thought to be a mechanism that prevents competitive exclusion. As a critical investigationof this scenario, we have studied a model, similar to the one described on page 45, to assessunder which conditions the divergence of an ecological trait in two populations that are insympatry is a likely evolutionary response to interspecific competition during the secondarycontact, thereby avoiding competitive exclusion. The currently accepted speciation scenariohypothesizes that the speciation process is initiated in allopatry and completed in sympatryaccording to three steps. First, migrants colonize one island of the Galapagos archipelagofrom the mainland. Second, some individuals disperse onto another island and find a newcolony. This step may be repeated several times. Third, migrants from a secondarily colo-nized island come into sympatry with the original colony. This secondary contact results insuccessful speciation if two conditions are fulfilled: (a) immigrant and ancestral populationsdo not interbreed and (b) they stably coexist that is, no competitive exclusion (Gause, 1934).

The founding of new colonies may ensure that condition (a) is satisfied: song divergencein allopatry induces pre-mating isolation in sympatry (Grant and Grant, 1996a, 2002). Somemorphological differences evolved in allopatry, on which mate choice is based, may alsocontribute to pre-mating isolation in sympatry. Condition (b) may be difficult to satisfy.One solution would be that migrants occupy an as yet unoccupied ecological niche. Grantand Grant (1996a, 1997, 2006), however, showed that secondary contact causes a divergencein beak size, allowing immigrant and ancestral populations to gradually feed on differentresources. This implicitly suggests that no significant ecological divergence had occurred inallopatry. At the time of secondary contact, immigrant and ancestral populations share thesame niche and thus one of the populations risks competitive exclusion. According to theaccepted scenario, this is avoided because character displacement takes place, in sympatry,


such that immigrant and ancestral populations form two different species that can coexist,even on evolutionary time scales.

Analysing this model, we have shown that sympatric character displacement in two eco-logically similar populations is a mechanism that prevents competitive exclusion under tworestrictive conditions (Aguilee et al., 2011a). First, populations should be stuck at a fitnessminimum, i.e. at an EBP. Second, pre-mating isolation between the two populations shouldbe very strong. We have also shown that character displacement is easier when disruptiveselection is strong due to intense competitive interactions; when the variance in the off-spring trait distribution is high; when ecological divergence has been initiated in allopatry;and when immigrant and/or resident populations are large.

While coexistence of resident and immigrant population on an ecological time scale isa prerequisite for character displacement, their coexistence on an evolutionary time scale isa consequence of character displacement. In particular, when the resident is at a CSS, thecoexistence of ecologically differentiated immigrants and residents may be possible (on anecological time scale), but convergent evolution of residents and immigrants will eventuallyresult in the extinction of one of the populations through competitive exclusion (cf., Fig.15B, C and Fig. 18B). Only when the resident is at an EBP is coexistence probable on bothecological and evolutionary time scales, the latter mediated by character displacement.

This exercise demonstrates how the ideas from adaptive dynamics can be applied to sce-narios that have previously been developed in verbal terms only. Thinking in terms of theenvironmental feedback loop as the basis of evolutionary change (the struggle for existence),invites us to consider ecological processes as well as genetic, morphological, phylogenetic,mating behavioural and spatial processes. Whereas the scenario proposed by (Grant andGrant, 1996a, 2002) can indeed be the correct processes behind the radiation of the Darwin’sfinches, we should at least complete the scenario with the restrictions imposed by the eco-logical interactions.

Radiation in Lake Victoria The final chapter of Robin’s PhD thesis considers landscape dy-namics in a much more complex setting (Aguilee et al., unpublished manuscipt). Whereasso far we have considered two allopatric subpopulations at the most, we can extend the de-scription of the landscape to include a large number of potentially separated habitats. Thelandscape model is inspired by Lake Victoria (and its cichlids) and by the complex dynamicsof its spatial structure over recent geological history (Sturmbauer, 1998). We consider a cen-tral lake basin, surrounded by a number of satellite lakes. We assume that any pair of lakes(central or satellite) can be connected or not. The (dis-) appearance of barriers between sitesis thought to be caused by water level fluctuations. There are three habitat types (Pelagicfor the central lake, Estuary and Rocky for the satellite lakes). Again we assume an ecolog-ical model similar to above (defined on page 45), except that the resources are now charac-terised by a continuous distribution (instead of 2 discrete resources). We consider uniquelythe case of disruptive selection (i.e., all singular points are EBPs). Finally, we assume againthat genetic constraints make normal sympatric speciation impossible (multilocus trait andrecombination).

The analysis of this model shows that landscape dynamics can facilitate a radiation.During a landscape-dynamics-induced radiation, diversity is generated by the joint actionof allopatric, ecological divergence under directional selection towards different ecologicaloptima, and of disruptive selection in sympatry leading to reinforcement or character dis-


placement. Landscape dynamics generate also other mechanisms producing or restrictingdiversity: hybridization, sympatric evolutionary branching (induced by hybridization), andextinction. In a dynamic landscape, all these mechanisms are combined repetitively. Becauseof their interactions, they constantly alter the (co-) existing species. This results in a differentconfiguration (in terms of species, reproductive isolation, assortative mating) at each sec-ondary contact: although secondary contact always occurs between the same three habitattypes, its outcome (i.e., coexistence or not) changes each time. Consequently, after severalsecondary contacts, the conditions necessary to reach the various ecological ”niches” arelikely to have been fulfilled, so that the number of species in each habitat can be significantlyhigher than the number of habitat types (and even than the total number of habitats) in thewhole landscape. In addition, our results show that landscape dynamics may enable radia-tion in cases where sympatric speciation is unlikely. The resulting diversity is higher whenradiation is caused by landscape dynamics than when it results from sympatric speciationin a static landscape: some ecological niches that are attainable by character displacementoccurring in a dynamic landscape, are not attainable by sympatric evolutionary branchingonly.

Interestingly, our modelling study revealed that the characteristic time scales of the land-scape dynamics influence the properties of the emerging community in terms of total biodi-versity, robustness of the diversity, and the rate at which the number of species increases. Inparticular, the highest asymptotic species diversity is generated by fast landscape dynamicsin a landscape almost always fragmented. However, diversity is then unstable: it is likely tocollapse into a hybrid swarm. The quickest increase in diversity is obtained under fast land-scape dynamics in a fragmented landscape subject to fusion of many sites at the same time.In this case, the level of diversity remains relatively low, but is very stable. A landscape thatis rarely fragmented generates only little diversity.

These results are complex and intriguing. They demonstrate that the processes identifiedin the simpler models described above (on pages 45 and 48, Aguilee et al. (2011a,b)) operatein a more complex context as well. Furthermore, the same processes may interact to pro-duce quite surprising effects. This modelling exercise shows that observed radiations suchas the cichlids in Lake Victoria may be the consequence of landscape dynamics, in concertwith more established processes such as allopatric and sympatric speciation. The work alsoshows that certain restrictive conditions (e.g., small number of loci), necessary to obtain spe-ciation in a static landscape, are not required once the spatial context of the system is takeninto account. The different modelling studies described here also show that disruptive se-lection is an important element of generating diversity, even outside the context of standardsympatric speciation.

3.4 Adaptive dynamics: conclusions

First of all, I have argued that, having formulated an ecological model using the idea of theenvironmental feedback loop, its extension to including evolutionary dynamics is straight-forward. It suffices to include genetic structure, and the feedback loop can be used to defineinvasion fitness. I have exploited this feature of the feedback loop to study adaptive dy-namics in size-structured populations. The take home message is of that work (section 3.2)is twofold: size structure matters in evolutionary dynamics through the richness of the dy-namical behaviour it induces, but also through its effect on the shape of the fitness function(in different types of population dynamics).

4 CURRENT RESEARCH 51

In the landscape dynamics models, the idea of the environmental feedback loop has alsobeen used to derive invasion fitness, which allowing us to predict the evolutionary dynamicsin each temporary spatial configuration (sympatric, allopatric patches) by drawing pairwise-invasibility plots (PIPs). The complexity introduced by landscape dynamics is such that wehave not been able to derive PIPs for the overall dynamics of the model discussed in section3.2 (page 45). In fact, deriving the invasion fitness in non-equilibrium, spatially structuredpopulations is a non-trivial issue (Dieckmann et al., 2000; Ferriere and Le Galliard, 2001),and will be elaborated on in the Discussion (section 6). The landscape dynamics models wehave used are individual-based and simulate the struggle for existence explicitly.

The research discussed in this section highlights the strong link between ecological andevolutionary dynamics. On the one hand, the ecological dynamics determine the fitnesslandscape (for example, when fitness curve depends on the type of population cycle). Onthe other hand, an evolutionary analysis of the system (is the singular point a CSS or anEBP?) is a good predictor of the possibilities of long-term coexistence between competingpopulations.

4 Current research

When we travel from south to north, or from a damp region to a dry, we invariably see somespecies gradually getting rarer and rarer, and finally disappearing; and the chance of climatebeing conspicuous, we are tempted to attribute the whole effect to its direct action. But this isa false view; we forget that each species, even where it most abounds, is constantly sufferingenormous destruction at some period of its life, from enemies or from competitors for the sameplace and food; and if these enemies or competitors be in the least degree favoured by any slightchange of climate, they will increase in numbers; and as each area is already fully stocked withinhabitants, the other species must decrease. Darwin (1859), page 96.

From abiotic factors to eco-evolutionary dynamics (and back) My current research fo-cusses on the influence of abiotic factors on ecological and evolutionary dynamics. Develop-ing theory of the link between abiotic factors and eco-evolutionary dynamics may be used topredict the consequences of environmental change on the dynamics and evolution of ecosys-tems. Furthermore, given the impact of ecosystems on their abiotic environment, we mayexpect the existence of feedback mechanisms operating in this context. For example, weknow that increasing temperature or the concentration of CO2 in the oceans is likely to in-fluence phytoplankton dynamics and evolution. In turn, we also know that the CO2 con-centration in the atmosphere depends in part on the growth of a number of phytoplanktonfunctional groups; some groups are more efficient in exporting CO2 to the deep ocean thanothers, depending, among other things, on their cell size. The response of the phytoplanktoncommunity, in terms of ecology and evolution, to environmental change, is hence likely tofeedback to the climate system. Other groups of organisms are also known to have stronginteractions with climate, in particular terrestrial plants (Bonan, 2008). Theory on the linkbetween abiotic factors and eco-evolutionary dynamics is also likely to lead to testable pre-dictions, which would allow us to falsify our basic hypotheses. For example, in section 2.3we have seen that temperature has an impact on the dynamics of size-structured popula-tions, in particular on the occurrence of generation cycles. Such results can be compared toempirical data on the behaviour of populations along climatic gradients.


Figure 19: Left: An adult springtailFolsomia candida (Collembola). Right: aphoto of an experimental population in apetri dish. The red ovals highlight threeindividuals of different sizes.

My current research activity is mainly funded by two ANR projects (Evorange (Dec 2009- Dec 2013), and Phytback (Oct 2010- Oct 2014)). A large part consists of the co-supervisionof two PhD students, Vincent Le Bourlot (with Thomas Tully, UMR 7625) and Boris Sauterey(with Chris Bowler, IBENS). Boris and Vincent both started their PhD project in September2010, and have three years to finish their thesis. Below I give a description of these two linesof research, that are both situated in the context given above.

4.1 Local adaptation and species range shifts, in a size-structured population

The geographical distribution of a given species is set by physiological limitations but alsoby interactions with other species (Sexton et al., 2009). Changing abiotic conditions, suchas those associated with climate change, will likely result in species range shifts, as has al-ready been documented (Parmesan, 2006). The majority of current research on the effectof climate change on species range shifts tends to focus on correlations of ranges with cur-rent abiotic, climatic factors. Such statistical approaches, referred to as ”climate envelopemodels” are in fact quite successful (Parmesan, 2006). However, there are two importantprocesses that may result in unexpected dynamics as compared to predictions based on cli-mate envelope models: (i) local adaptation to changing conditions; and (ii) species rangesbased on ecological interactions (Lavergne et al., 2010). First, local adaptation may result inwhat is called ”niche lability”: the niche of a species is changes over time due to evolution.From an evolutionary point of view, one can say that nothing is more normal than nichelability (Pearman, 2008). Yet it is often assumed that on the time scale of climate change, eco-logical niches of species can be assumed to be constant. Second, in the cases that a species’range is determined by species interactions such as interspecific competition or predation,the response will depend on how both species’ physiology responds to the changing abioticfactors, as well the response of the resource(s) for which the species are competing. The eco-physiological processes underlying ecological interactions may be complex, and hence theoutcome of a change in, for example, temperature, may be surprising.

The research described here is part of a wider research project, called Evorange (”Howdoes evolution affect extinction and species range dynamics in the context of global change?Implications for ecological forecasting”), coordinated by Ophelie Ronce (Universite Mont-pellier 2), and funded by the ANR. The central idea of the Evorange project is to study whathappens if (i) and (ii) are the case.

The specific question studied by Vincent Le Bourlot during his PhD project is how thesize-structured dynamics of the chosen model species (Fig. 19) is affected by temperature


and genotype (i.e., clone). The idea is, first, to characterise time series of experimental pop-ulations, under different temperatures and for different clones. Second, to try to model thesize-structured population dynamics and its temperature dependence in a fairly mechanis-tic way using PSP models. We hope to be able to reproduce realistically looking time series,and better even, to be able to predict population dynamics as a function of temperature fordifferent clones. Third, to predict adaptive dynamics using these results, in which differentclones represent different mutant vs resident pairs. Fourth, the idea is to simulate ”speciesranges” by letting different clones interact along a spatial temperature gradient. If possible,these predictions will be tested empirically using the clones and an artificial temperaturegradient.

Vincent has started his PhD project in September 2010. The first results are intriguingbut we cannot answer to the main questions and issues posed above. Here I give a shortdescription of some preliminary results. The model species used is the springtail Folsomiacandida (Collembola), which is a parthenogenetically reproducing (clonal) species (but sexualstrains exist). Our lab (Ecology & Evolution, UMR 7625) possesses a facility for raising andmonitoring many populations of this species at the same time, under controlled conditions(temperature). This facility has been set up by Thomas Tully, co-supervisor of Vincent’s PhDproject. A full description of the species, the clones and the set up can be found in Tully(2004).

The experimental determination of time series under different temperatures is still on-going and it is too early to present results on the effect of temperature or genotype. Anexample of such a time series is presented in Fig. 20 (top panel). Several aspects of thesedynamics are intriguing. First, the population size structure is bimodal for most of the timeperiod, featuring an ”adult” mode (around 1.2 mm) and a ”juvenile” mode (around 0.3 mm).Second, the dynamics depicted in the figure do not display typical generation cycles, quitegenerally predicted for size-structured populations (see section 2.3). Yet the dynamics arenot strictly ”equilibrium” either, since we can observe a cohort recruiting (the growth curvein the middle of the time series, highlighted by a fitted straight line). Third, the juvenilemode fluctuates. These features are common to a large number of time series obtained fromthe experimental set-up.

The environmental feedback loop We have tried to model these population dynamics.The logical starting point is the Kooijman-Metz (KM) model (section 2.1), since it is the sim-plest and best-studied PSP model for species with continuous reproduction. Vincent hasformulated a version of the PSP model in which the environmental feedback loop is mod-elled in a phenomenological way, similar to the model of the common lizard (section 2.4).The reason for using a phenomenological model is that the feeding behaviour of the collem-bolans is not sufficiently well-known quantitatively. Rather, using a more abstract model ofthe environmental feedback loop allows us to play with several hypothesis as to how thepopulation is being regulated. We study two alternative hypothesis, that, based on obser-vation (by Thomas Tully, Francois Mallard and Vincent Le Bourlot) are both candidates forthe population regulation of the experimental populations: (a) exploitation competition fora single resource; and (b) size-dependent interference competition. Assuming (a) results inthe classical KM model in which the environment is characterised by a single variable forthe limiting resource, sayR. Individual energy intake is a positive function ofR and of bodysize. A depletion of the resource is felt by all individuals, although small individuals are


Figure 20: Population structure-time plots. Top: An experimental time series (Folsomia candida). Theblack line is a fit to the growth curve for the recruiting cohort. Middle: The individual-based ver-sion of the Kooijman-Metz model, modified for Folsomia candida, assuming implicit competition for asingle unstructured resource. Horizontal line: size at maturation. Bottom: the model but assumingsize-dependent interference competition.


better able to deal with resource rarity due to lower energetic requirements (see section 2.1).Assuming (b), the energy intake depends directly on the presence of bigger individuals. Thatis, the access to the resource is supposed to be reduced by the presence of larger individu-als (this reflects ”social” behaviour that has been observed in Folsomia populations). Theseassumptions have been studied in a deterministic PSP model as well as in a discrete-event,stochastic, individual-based version of it (as has been done in section 2.4).

A first question to address can now be formulated as follows: which model of the envi-ronmental feedback loop can explain the observed type of population dynamics? Some pre-liminary results are presented in Fig. 20, (middle and bottom). The middle panel shows thatwith hypothesis (a), combined with a high mortality rate for small individuals, a bimodalpopulation structure can be obtained. (Generation cycles are predicted for lower mortal-ity rates.) The lower panel shows an example of dynamics with interference competition.This model displays a different type of dynamics, in which the competition severely limitsthe growth of small individuals. Interestingly, the dynamics include incidental recruitmentevents, not unlike the recruitment event observed in the experimental data (Fig. 20, top).Also, the dynamics include a large variation in the growth curves of adult individuals. Thetwo simulations demonstrate clearly the importance of the dimensionality of the environ-mental feedback loop. If the feedback is one-dimensional (such as R, hypothesis (a)), lifehistories tend to converge and resemble each other (Fig. 20, middle). If the feedback is high-dimensional (such as the size-distribution of competitors, hypothesis (b)), then life historiesmay diverge from each other (Fig. 20, bottom). The life history divergence due to a com-plex feedback loop in the model with interference competition, is similar to the occurrenceof ”dwarfs and giants” in cannibalistic populations (Claessen et al., 2000), equally due to acomplex (high-dimensional) environmental feedback loop.

Although these preliminary results do not give conclusive answers to the question howto model the Folsomia populations, let alone the questions posed in the context of the project,they do give interesting clues to which elements may need to be considered.

4.2 Eco-evolutionary feedbacks between climate and phytoplankton

The Phytback project (”Ecology-climate feedbacks due to evolution of phytoplankton cellsize and shape”; coordinated by myself, funded by the ANR; see annexe) focusses on phyto-plankton, a group of organisms that participates considerably in the regulation of the atmo-spheric CO2 concentration. For these species, the environmental feedback loop hence getsan additional meaning. Whereas on the short term the feedback loops such as depicted inFig. 1 and Fig. 9 work in a similar way as for many species (e.g., an environment definedby nutrients and predator abundance), on a longer time scale the evolution and dynamicsof phytoplankton populations may influence abiotic conditions which are normally not con-sidered to be under influence of populations: temperature, atmospheric CO2 concentration,ocean acidity. This second layer of feedback has been of vital importance in the geologicalhistory, for instance by producing a oxic atmosphere, but also for sequestering a significantportion of the CO2 emitted by human activity in the deep ocean (Falkowski et al., 1998).

The time scales considered here, though, are fairly short (10s-100s years). The goal ofthe Phytback project is to study the role of adaptation in the response of phytoplankton toclimate change. The focus is on the evolution of cell size and cell shape, which are known toinfluence both the eco-physiology of phytoplankton as well as its capacity to sequester CO2

though its influence on the sinking rate of algal cells (Aksnes and Egge, 1991; Litchman et al.,


2007; Litchman and Klausmeier, 2008). The initial idea for this project came from readingan article on how global circulation models (GCM) can be used to predict the ecological andbiogeographic response of oceanic phytoplankton to climate change, and how this ecologicalresponse feeds back to the climate system through its influence on the carbon cycle (Cermenoet al., 2008). Whereas a model defined with a fixed set of phytoplankton types (as all GCMsare) can be used to predict a purely ecological response to climate change, this predictiondoes not take into account any evolutionary change in the phytoplankton. Given the shortgeneration time of phytoplankton (of the order of hours or days), the time scale of climatechange corresponds to an evolutionary time scale for these organisms. The question thatemerges is hence, how are predictions of the phytoplankton’s response to climate changeinfluenced by evolutionary change?

Current global circulation models (GCM), used to predict climate change and its conse-quences, account for ecology in a rather simple way, and do not account for evolution. Suchmodels usually include a single compartment for each functional group of phytoplankton(diatoms, coccolithophores, dinoflagellates, cyanobacteria, etc) (Le Quere et al., 2005). Suchsimplicity is of course a necessity in order to obtain a tractable model. Yet it poses both con-ceptual and methodological problems. Conceptually, the within-functional group diversitymay respond to environmental conditions in such a way that the dynamical behaviour of thefunctional group is not the same in different parts of the world, or under a changing climate.Other model formalisms, such a size-spectrum approach (Armstrong, 1999), may be able toenlighten this kind of questions.

Cermeno et al. (2008) used the GCM developed at MIT (referred to as MITgcm), for whicha rather sophisticated method to include phytoplankton types has been developed recently(Follows et al., 2007). In this model, the phytoplankton community, at each geographicallocation, is seeded at time t = 0 with a large number of randomly defined phytoplanktontypes. For each type, the model parameters characterising its eco-physiology in terms of cellsize, affinity for nutrients, maximum growth rate, etc, are randomly drawn from realisticranges for each parameter. Their population dynamics is simulated in conjunction with theglobal circulation of the ocean. The struggle for existence between these types then results ina very limited number of types on each given location, and globally consistently in around10 dominant types (in a number of independent simulations). The emerging biogeography(when interpreted in terms of functional groups) resembles empirical distributions of func-tional types (Follows et al., 2007).

Although this model simulates the struggle for existence between the initially presentphytoplankton types, once the initial variation has eroded away, new genetic variation can-not arise in the system. Hence, simulating climate change scenarios with this model suffersfrom the same rigidity of the definitions of phytoplankton types as the other GCMs. Forexample, species may be predicted to go extinct in such scenarios, whereas allowing themto adapt to the changing climate may allow them to persist. The different predictions of(changing) phytoplankton diversity, made by models that do or do not allow for adaptation,may have consequences for the carbon cycle and hence the feedback to the climate system.

In collaboration with Mick Follows and his group at MIT, we are currently working onextending the MITgcm to include phytoplankton evolution. This is done by Boris Sautereyas part of his PhD project, co-supervised by Chris Bowler (Molecular Plant Biology, Institutde Biologie, ENS) and myself. Compared to the original Follows et al. (2007) publication,Boris is working on a more recent version of the MITgcm in which the phytoplankton typesare modelled not only by their (spatially distributed) biomass, but also in terms of internal


nutrient quota (based on the ”Droop” or ”variable-internal-stores” model; Grover (1991);Kooijman (2000)). Initially, we are working on a simplified version that includes only a1D water column at a specific location in the ocean (i.e., the Ocean Weather Station Mike(OWSM) in the Norwegian Sea), instead of the world ocean in 3D. Introducing evolutionin this model is done using ideas from adaptive dynamics, by applying the idea of inva-sion fitness in the following way. The system is run over a number of years with a giveninitial set of (possibly randomly defined) ”resident” phytoplankton types, to reach the (non-equilibrium) attractor of the ecological dynamics. The within-year dynamics are inherentlynon-equilibrium since several properties of the water column are driven by abiotic factors(wind-driven turbulence, temperature fluctuations), that are periodic with a periodicity ofexactly one year. The year-to-year dynamics, however, are (very close to) steady state (oncethe transient phase is over, each year the same dynamics are repeated). These seasonal forc-ings have been parametrised to the available weather data from the OWSM station (by S.Dutkiewicz). For each resident type, a number of ”mutant” types is then introduced into thesystem. Their dynamics are driven by the environmental feedback loop of the resident types,as in Fig. 9. Their population structure, in terms of the spatial distribution over the watercolumn and of nutrient quota, is allowed to converge to a steady state (which still fluctuateswith an annual period). Once in their stable population structure, the long-term populationgrowth rate is estimated based on the annual multiplication factor (i.e., the geometric rate ofincrease of the total biomass at any given point within the annual cycle). This can be donesince we know that the mutant types are capable of only two types of population dynam-ics: exponential growth or exponential decline, due to the absence of population regulation(Fig. 9). We obtain the exponential growth rate, equivalent to the Malthusian parameter r inequation (1), as follows:

ru =1

Tlog

(Bu(t+ T )

Bu(t)

)(20)

which is a measure of invasion fitness of a mutant with trait value u, in the environmentset by the set of currently present residents. Here T is the duration of one cycle (one year),Bu(t) is the biomass at time t of the mutant with trait value u, summed over the water col-umn. Assuming a 1-dimensional trait u, then this method can be used to simulate evolutionrelatively efficiently. If two mutants are introduced that have trait values either just belowor just above the resident trait, then this method will identify which of the mutants can suc-cessfully invade. In the algorithm, the resident is then replaced by the successful mutant,and the procedure can be repeated for a new round of mutant invasions. Of course herewe apply the assumption that ”invasion implies replacement”, which is commonly made inadaptive dynamics, and which is backed up by the theory of attractor inheritance (Geritzet al., 2002). While this assumption is not universally valid (Mylius and Diekmann, 1995), itavoids simulating the much more time-consuming explicit inter-type competition betweenresident and a successful mutant, which is expected to lead to competitive exclusion.

The environmental feedback loop This fitness definition and the algorithm outlined aboveare a direct application of the concept of the environmental feedback loop. The idea of thefeedback loop leads to a straightforward fitness definition, despite the complexity of themodel in terms of spatial structure, temporal forcing, and physiological structure (in termsof nutrient quota).


Figure 21: An example of a simula-tion of the MITgcm (1D version). T=0-600 months: without evolution. At timeT = 0, the model is seeded with 78randomly defined species. After an ini-tial, transient phase, two phytoplanktontypes dominate the system (blue and vi-olet curves). A few species persist atvery low densities. T=600-1200 months:with evolution. In response to evolu-tion in al resident species, the blue dom-inant species nearly goes extinct. Theevolutionary trait (cell volume) of theblue and red species converge duringthe evolutionary phase.

Figure 22: Schematic representation of whyevolution reduces biodiversity in the simu-lations. A. The fitness landscape after thefirst phase of the simulation, without evolu-tion. The competition between the initially nu-merous phytoplankton types has resulted ina coexistence of three species. Note that allhave zero invasion fitness. B. After evolutionhas taken place, convergent evolution of onespecies pair has resulted in the loss of one res-ident species. The two remaining species areContinuously Stable Strategies (CSS).

5 OTHER WORK 59

Preliminary result An interesting, preliminary result is based on simulations which consistof two parts: a first phase using the algorithm as in Follows et al. (2007), without evolution.Subsequently a second phase with evolution of the remaining resident populations. Withoutgoing into the details, what we found is that the number of species that remain following thesecond period is consistently lower than the number of species that are selected by the firstperiod. It appears hence, that the degree of biodiversity estimated by the method of Followset al. (2007) is likely to be an overestimation (if we want to allow for evolution). The processleading to the loss of species can be interpreted as follows. At least some of the residents thathave emerged as ”winners” after the first phase, are in fact ”suboptimal”. That is, they donot occupy summits of the fitness landscape (Fig. 22A). The phase with evolution results inconvergence of a number of species that ”climb” the same mountain in the fitness landscape(Fig. 22B). This process is the same as the evolutionarily unsustainable coexistence discussedin section 3.3.

Of course this phenomenon is a consequence of a relatively sparse coverage of the fulltrait space by the initially present phytoplankton types. In the illustration (Fig. 22) it seemsunlikely that none of the initial phytoplankton types would be closer to the second summitin the fitness landscape, if indeed the initial number of types was high. Yet if the evolu-tionary trait space is multi-dimensional (as in the case of the Follows et al. (2007) model), thecoverage of trait space is likely to be rather sparse, even if the initial number of types is large.Note that in multi-dimensional trait space, more than two populations can be climbing thesame mountain. In a 1-dimensional trait space such as Fig. 22, the maximum is two speciesper mountain.

5 Other work

This memoire does not cover all my publications since my PhD thesis. It covers the part of mywork which is clearly related to the central theme of the ”struggle for existence in structuredpopulations”. Trying to include the remaining articles would be artificial. For the sake ofcompleteness, here is a very brief synthesis of the omitted work, which falls into two parts.

First, my research at Rothamsted Research (UK) was about the role of environmentalstochasticity for the invasion and extinction dynamics of genetically modified feral oilseedrape. The goal was to assess the invasion risk of genetically modified crops (through theestablishment of wild populations from seeds lost during harvest or transportation) usingstochastic matrix models (another way to represent dynamics of structured populations). Iwrote two articles that reported the applied results of this project (Claessen et al., 2005a,b),and one theoretical article in which I demonstrated that so-called ”loop analysis” is validfor stochastic matrix models (as was previously established for deterministic matrices only)(Claessen, 2005).

Second, I have participated in a collaboration to develop a simplification of PSP models(De Roos et al., 2007, 2008). This idea behind this work is that a more simple formulationwould allow to study more complex food webs than is practically possible in the frameworkof PSP model (coupling more than two size-structured populations gets really quite difficultto analyse). This idea was to formulate a stage-structured model that is, in equilibrium, fullyequivalent to a simple PSP model. That is, the equilibrium densities of stage-specific biomass(say, juveniles and adults), are identical to the equivalent quantities in the full PSP model.The advantage is that the thus obtained stage-structured model can be parametrised in the

6 DISCUSSION 60

same, process-based manner as PSP models (which is not generally true for unstructuredor stage-structured models). The obtained stage-structured biomass model is formulatedas a set of ODEs, and is hence easy to analyse with existing tools from dynamics systemstheory. In the context of the Windermere project (section 2.3), I have recently contributedto an application of this method, including cannibalism (Ohlberger et al., 2011b). Anotherapplication of this method concerns the importance of certain model assumptions for thequalitative predictions of the efficacy of marine protected areas, a subject important in fish-eries management (Claessen et al., 2009).

6 Discussion

It has been seen in the last chapter that amongst organic beings in a state of nature there is someindividual variability: indeed I am not aware that this has even been disputed. It is immaterialfor us whether a multitude of doubtful forms be called species or sub-species or varieties, (...), ifthe existence of any well-marked varieties be admitted. Darwin (1859), p 87.

6.1 The struggle for existence in structured populations

I have argued that the environmental feedback loop, as defined in section 1.2, is a formal-isation of Darwin (1859)’s concept of the struggle for existence. As pointed out in the in-troduction a,d in section 2, the idea of the environmental feedback loop is useful for modelformulation as well as theory development in the context of physiologically structured pop-ulation (PSP) models. It invites us to first think about how life history depends on an indi-vidual’s interactions with its environment, in terms of resources, conditions and individualsof the same and other species. From a technical point of view, a model formulated as suchcan be ”decoupled” in the sense that one can cut the feedback loop in two parts: fixing theenvironment, the population dynamics become linear (exponential growth); and fixing thepopulation, we can try to find an environmental condition that is consistent with this stateof the population. On the one hand, this opens the way to defining equilibria of PSP models,and to develop continuation and bifurcation methods (see section 2.1). On the other hand,this provides a straightforward recipe for the definition of invasion fitness and hence for theanalysis of (possibly complex) models in the context of adaptive dynamics (section 3.1).

Noteworthy results An important message that, I think, emerges from my work as sum-marized in sections 2, 3 and 4, is that a number of feedback loops, that are hypothesised toexist on different scales, all have the potential to cause unexpected results. This is a rathergeneral conclusion, which is by no means unique to my work: it is well known that feedbackloops lead to non-linear dynamics, and non-linear dynamics are well-known to be full of sur-prises. But what I try to stress here, is that such feedback loops may be inherent to ecologicaland evolutionary dynamics. Examples of feedback loops that have been treated here are,in order of increasing time scale, the mutual influence between life history and populationdynamics; between resident genotype and invasion fitness; between climate, phytoplanktonecology and the carbon cycle. One of the most striking striking results in this respect is illus-trated in Fig. 12, which reflects the interplay between ecological and evolutionary dynamics.The feedback between life history and population dynamics results in a range of ecological

6 DISCUSSION 61

attractors. The feedback between resident genotype and invasion fitness determines the di-rection of evolution. Evolutionary dynamics eventually result in a qualitative change of theecological dynamics (attractor shift), which feeds back to the evolutionary dynamics (extinc-tion of one species).

Another interesting aspect that emerges from the sections 2 and 3, is that the complex-ity of the feedback loop influences the range of possible dynamics exhibited by the model.Again, this may not be very surprising, but it is nevertheless noteworthy, in particular hav-ing a range of examples available that illustrate this idea. The simplest feedback loop dis-cussed here is the one in the logistic growth equation. One-dimensional as it is, it allowsfor a stable equilibrium dynamics only. Slightly more complex, the feedback in the Lotka-Volterra model is two-dimensional (prey self-limitation and the predator-prey interaction),and may give rise to population cycles. Almost equally simple, the feedback in the size-structured models with a single resource (e.g., the standard Kooijman-Metz model) leadsto either stable fixed points, or population cycles. Yet the time delay, that is brought aboutby the population structure and in particular by the juvenile period, results in a feedbackloop that is more complex than in the Lotka-Volterra model. Adding another resource, andanother delay effect via a size-dependent resource use, increases the complexity further andresult in the possibility of multiple equilibria Fig. 11. Adding the evolutionary feedbackloop obviously results in a more complex feedback structure. The results in section 3 showthat this can give rise to evolutionary branching, one of the most striking results of adaptivedynamic.

This list of varying complexity concerns intrinsic complexity of the feedback loop (i.e., thecomplexity of the feedback resulting from the ecological interactions and the system’s dy-namics). In section 3.3 and section 4.2 we have seen that the feedback loop can also be forcedby extrinsic factors such as landscape dynamics, fluctuating abiotic conditions, and complexspatial processes such as ocean currents. It has become clear that the externally imposedcomplexity should also be counted when considering the outcome of eco-evolutionary dy-namics: landscape dynamics can ”liberate” a panmictic population from a fitness miminumresulting in combined allo-sympatric speciation (Aguilee et al., 2011b). In more complexlandscapes, spatial dynamics can result in higher biodiversity than would be obtained with-out this complexity (Aguilee et al., unpublished manuscipt).

Although the observation that ”complex systems do complicated things” is perhaps lessexciting than ”simple systems do complicated things”; understanding the implications ofcomplexity is relevant, in particualar since natural systems are often complex.

Invasion fitness in complex models Here I will elaborate on the problem of defining inva-sion fitness in complex models, and how the concept of the environmental feedback loop canhelp. As said, for simple models, invasion fitness is relatively easy to compute and to usein simulation or analytical models. Yet in more complex models, for example in structuredmetapopulation models (Metz and Gyllenberg, 2001), its derivation from the underlyingecological model can be a non-trivial challenge. In particular, in spatially structured popu-lations with externally forced, varying abiotic conditions, such as the MITgcm discussed insection 4.2, it may be a real challenge to derive a coherent definition of invasion fitness.

In the most complex cases, the only straightforward solution may be to simulate explic-itly the ”fate” of resident and invader, i.e., to simulate the entire invasion and competitionprocess between invader and resident, but this is extremely inefficient from a computational

6 DISCUSSION 62

point of view. In particular, this could be the only solution for the most complex of mod-els considered in the Phytback project: the 3D ocean circulation and phytoplankton ecologymodel of Follows et al (2007). For this model it is challenging to define invasion fitness otherthan by explicit simulation. To illustrate the problem, consider following reasoning. At agiven time and place in the ocean, the model computes the local rate of increase of a phy-toplankton population. Locally, integrated over an annual period, this can be considered tobe its invasion fitness. However, the long term rate of increase of the type under consider-ation depends on immigration from other locations, as well as on non-periodic fluctuationsin the abiotic and biotic environment. For example, while its local rate of increase due toreproduction and mortality may be negative, the net rate of change may be positive if thesinking rate from above exceeds the local rate of decrease. Hence the invasion fitness shouldbe evaluated over the entire spatial range of the invader, and over the entire periodicity offluctuation of its environment. Of course, for aperiodic (or chaotic) fluctuations, this meansthat the invader must be tracked over a very long period. This example illustrates that bluntsimulation of the mutant-resident interaction to assess invasion fitness is computationallyextremely costly. For this purpose, defining invasion fitness for complex ecological scenar-ios is a central objective to the Phytback project.

The above example of fitness in the MITgcm highlights an important limitation of the in-vasion fitness concept which relates to issues of scale. Invasion fitness is defined as the expo-nential growth rate of a small invader population. Mathematically, this definition supposesthat the invader population has already reached its stable population structure. During thetransient period prior to reaching the stable structure, the population does not yet grow ex-ponentially (Caswell, 2001). Durinx et al. (2008) derive a general formulation of the invasionfitness for physiologically structured populations (using the same concept of the environ-mental feedback loop). The population structure of the mutant population is computed asa perturbation from the resident stable population structure. Above, in section 3.2, we havedone a similar thing: we introduce mutant populations next to a resident populations, let themutants reach their stable population structure, and then compute their exponential popu-lation growth rate (for example to compute the data in Fig. 12B-D). This assumption makessense if the mutant population reaches its stable population structure relatively quickly. Thisimplicitly supposes that the spatial scale of the system is relatively small or at least well-mixed. If the spatial scale is small, then the mutant population will spread quickly to theentire range, even if living conditions are heterogeneous throughout space. For example,in the 1D version of the MITgcm, a mutant population quickly reaches a stable distributionacross the vertical water column, in which parts of the column act as sources, others (deeperdown) as sinks. (The initial distribution of a mutant is a copy of the resident’s distribution,not unlike the approach of Durinx et al. (2008)). Across the whole range, the ”depth profile”of phytoplankton abundance reaches a stable, year-to-year distribution. This distribution asa whole is associated with a single value of invasion fitness. This can be compared to thedominant eigenvectors and corresponding eigenvalue of a matrix model (Caswell, 2001).

The full (3D) MITgcm offers a different kind of spatial environment. The spatial scale ofthe model (the entire global ocean) is such that a mutant, arising in a certain spot on Earth,needs very many generations to spread and reach its stable spatial structure. This suggeststhat the ecological time scale (of dispersal and converging to the stable population structure)is in fact longer than the evolutionary time scale (of local adaptation), which is contrary tothe basic assumption of the separation of ecological and evolutionary time scales made inadaptive dynamics (Metz et al., 1996; Geritz et al., 1998). Interestingly, we may expect that

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a mutant may be able to establish itself (and oust the resident) in certain oceanographic re-gions, but not in others. Does this mean that it has a different fitness in different parts of theglobe? In this context, it seems rather artificial to define a mutant’s initial distribution as aperturbation of the resident’s distribution. For example, for a mutant of a circumpolarly dis-tributed resident such as certain diatom types in the Souther Ocean, its initial distributionwould span about 21,000 kilometres at the moment it starts growing exponentially. Obvi-ously, here the spatial scale is of such magnitude that the assumption of a population-wideinvasion fitness becomes quite artificial. It may, nevertheless, provide the only feasible wayto predict the evolutionary dynamics of phytoplankton in the context of this type of models.

A less mathematical approach to modelling adaptive dynamics, is by using individual-based simulations. In this case, we do not make any assumptions on the initial populationstructure of the mutant; we simply simulate the whole process. Of course, in the individual-based simulations, the environmental feedback loop exists just as well as in a determinis-tic differential equations model. Yet it operates in a more ”natural”, self-organising way,whereas in order to analyse the feedback loop mathematically, we need to make assump-tions such as time scale separation, in order to arrive at any conclusions. Such individual-based simulations are very useful to check the importance of such simplifying assumptionsmade in the mathematical theory. For instance, the individual-based simulations as the onepresented in Fig. 12 have confirmed that predictions based on the deterministic model, andthe associated assumptions of time scale separation, are correct. The adaptive dynamicsconverge to the predicted singular point, and the singular point has the evolutionary char-acteristics (CSS or EBP) as predicted by the deterministic model (in particular, by the PIP).Of course seeding the MITgcm with discrete individuals that spread stochastically throughspace is quite impossible. For this model, therefore, using the resident’s stable populationstructure as a starting point for mutants seems quite a tempting procedure. Alternatively, wemay be able to use other approaches to the problem of invasion fitness in spatially structuredpopulations, such as modelling invasion waves (Dieckmann et al., 2000).

Heritable vs non-heritable individual variability Lamarck and Darwin recognized thatwhat counts in nature are individuals, not species. Before them, the species concept wassomething of a Platonic ”Idea”, of which the individuals were mere imperfect incarnations.Lamarck and Darwin turned this around, recognising that individual variation cannot be ig-nored, and rather, that it is at the basis of evolutionary change. (The above citation illustratesthat Darwin was not, as many modern scientists, obsessed with the definition of the speciesconcept). In the context of population dynamics, individual variation is synonymous withpopulation structure. We have seen that population models that recognize individual vari-ation predict different types of population dynamics than population models of identicalindividuals. That is, the range of dynamics of the former is richer, since the scope of dy-namics of unstructured populations is often included in the range of dynamics of structuredpopulations. A number of these types of dynamics have been observed in a range of naturalpopulations, implying that the importance of individual variation is not limited to mathe-matical models alone (Claessen et al., 2000; Murdoch et al., 2002; Persson et al., 2003, 2007).There is, however, a fundamental difference between heritable and non-heritable individ-ual variation. Non-heritable individual variation, as modelled by physiologically structuredpopulation models and illustrated in section 2, can only directly influence the ecologicaldynamics of a population. Obviously, only heritable variation can result in evolutionary

6 DISCUSSION 64

change. Yet, as I try to demonstrate, evolutionary change is driven by ecological dynam-ics, and hence even non-heritable individual variation is likely to influence the evolutionarydynamics (as illustrated in section 3.2).

In my opinion, the main merit of adaptive dynamics has been to put ecology back at theheart of evolutionary theory.

6.2 What next?

Short term In the next few years, a great deal of my research will focus on the two projectsdescribed in section 4: Evorange and Phytback. In collaboration with the two PhD students(Vincent Le Bourlot and Boris Sauterey), and with other colleagues, I will focus on the fol-lowing aspects. First, the Collembola project needs to be developed in terms of (i) adaptivedynamics of the basic model; (ii) extension of the basic model to a spatially-structured modelon a temperature gradient. The latter will be studied both on ecological time scales (to de-termine the emergence of species ranges through competition) and evolutionary time scales.For the latter, similar questions as to the definition of invasion fitness need to be addressedas above. In addition, experiments need to be performed and analysed, to test the predic-tions based on the above model exercises. One goal is to perform time-series analysis on thedata obtained under different temperatures, which may allow to test some of the predictionsmade in section 2.3, as well as additional predictions based on Vincent’s springtail model.

In the context of the Phytback project, and Boris’ PhD project, I intend to focus on theevolution of cell size and shape in a range of models of increasing complexity. The pro-posed sequence of increasingly complex models is as follows. Model 1: The well-knownDroop model (Grover, 1991) will serve as the reference model. This model has already beenstudied in both ecological and evolutionary terms, including applications of adaptive dy-namics theory (Klausmeier et al., 2007; Yoshiyama and Klausmeier, 2008; Litchman et al.,2009; Verdy et al., 2009). Model 2: A physiologically structured version of model (1), ac-counting for two dimensions of population structure; cell size and cell nutrient quota. Forthe within-cell energy and nutrient dynamics, we might consider using the approach of so-called ”synthesising units” (Kooijman, 2000; Poggiale et al., 2010). Model 3: As model (2)but in an explicit water column (1D vertical space). Model 4: As (3) but with forced abioticconditions, in particular with an annual temperature and nutrient influx cycle, assuming pe-riodic forcing which enables us to use Floquet theory to define invasion fitness (Klausmeier,2008). Boris’ PhD project also includes an experimental part that focusses on parametrisingthe effect of cell shape on the eco-physiology of phytoplankton, using Phaeodactylum tricor-nutum as a model species (Bowler et al., 2010). Whereas many authors have modelled theconsequences of cell size on phytoplankton ecology and evolution (e.g., Litchman et al., 2009;Verdy et al., 2009), the ecological and evolutionary aspects of cell shape have received verylittle attention.

One line of research that I wish to develop (partly associated with the running projects)is that of time-series analysis. In particular, I benefit from a close, interdisciplinary collabora-tion at CERES-ERTI (where I have my office), with mathematicians, economists, physicists,climatologists, even ecologists, and others, many of whom share an interest in the analysisof noisy time series. We have set up an informal working group at CERES-ERTI (AndreasGroth, Michael Ghil, Denis Rousseau, Eric Edeline and Bernard Cazelles, and myself), us-ing time-series analysis (wavelets and singular spectrum analysis (SSA)) to disentangle thenoisy signals in ecological, economic and climatic data sets. One analysis focusses on time

6 DISCUSSION 65

series from the Lake Windermere project, but we equally aim to analyse the data from theCollembola experiments (with Vincent Le Bourlot, Francois Mallard and Thomas Tully). Thisinterdisciplinary collaboration started during the teaching of a workshop on ”data analysisin environmental issues”, an undergraduate course at ENS, taught by Michael Ghil, AndreasGroth and myself. As an intriguing example, the analysis of the Lake Windermere data sug-gest that the main predatory fish species in this lake (perch and pike) do not interact stronglywith each other.

The longer term To finish, here are some thoughts on where future research on the strugglefor existence in structured population may go.

• Time-series analysis of structured populations and parameter estimation of PSP mod-els with time series. The collaboration at CERES-ERTI, mentioned above is a startingpoint for this objective.

• Development and application of numerical tools of bifurcation analysis of PSP models.Developments requires advanced mathematicians and skilled programmers (not me).I think such development will have great impact on future research of PSP modelling,since the numerical analysis of these systems would be significantly more efficient.Next, application of the above mentioned tools to specific research questions and casestudies. This is where I could make my contribution.

• Food webs of size-structured populations. Currently, PSP models include at maximumtwo size-structured populations, which is of course rather limited in the context ofmany ecosystems and research questions. The above mentioned tools could facilitatethis line of research, but it may proof too hard even with efficient bifurcation analysis.The alternative is to use the stage-structured simplification of PSP models (De Rooset al., 2008; Ohlberger et al., 2011b) as a building block for complex food webs.

• Exploring the evolutionary dynamics in PSP models. We have now made a start in thisdirection, but a lot remains to be done. In particular: Co-evolution of multiple, inter-acting size-structured populations. The interplay between ecological and evolutionarydynamics in multiple size-structured populations may have even more surprises instock.

• A classical and ongoing challenge is to develop testable predictions that would allowus to verify the importance of the eco-evolutionary feedback loop, as defined in themodels, for field systems. One example already mentioned is the comparison of pre-dicted dynamics over a temperature gradient. The predictions could be extended toinclude evolutionary dynamics, thus trying to predict latitudinal patterns of ecologicalproperties.

• Related to the work on evolutionary branching in size-structured populations, is thequestion of how to define the conditions for coexistence of two competing, size-structuredpopulations. Can we develop a size-structured extension of Tilman (1982) R∗ theory?As demonstrated in section 3.3, invasion fitness can help determining the conditionsfor coexistence.

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Postface

A different HDR story Vera, my elder daughter of four years, plays that she is living inLondon with her sister Naomi. They are both working. They are now coming to Paris tovisit us. Corinne, her mother, asks:

- Quelles sont les nouvelles de Londres?- Elles ne sont pas encore arrivees. Elles vont arriver demain.- Et tu fais quoi comme travail ?- Je fais mon abecedaire et mon HDR

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But we have better evidence on this subject than mere theoretical calculations. Darwin (1859),page 92

A CURRICULUM VITAE 73

A Curriculum Vitae

David Claessen60, rue de Gergovie75014 Paris

Birth date and place: 27 June 1972, Leiderdorp, The NetherlandsPacse to Corinne Robert. Two children: Vera (4 years) and Naomi (71

2 months)

Current position: Maitre de conferences (assistant professor)Ecole Normale SuperieureCERES-ERTI24 rue Lhomond75005 Paris

CNRS affiliation: Laboratory ”Ecology & Evolution”UMR 7625 CNRS-UPMC-ENSEcole Normale Superieure45 rue d’Ulm75005 Paris

A.1 Life history (education and positions)

1990-1996 BSc and Msc Biology, University of Amsterdam1996-2001 PhD University of Amsterdam (and Umea University), date of defense: 3

June 20022001-2004 Postdoc at Rothamsted Research, Harpenden, UK2004-2006 Postdoc at University of Amsterdam on a VENI fellowship of the Nether-

lands Organisation of Scientific Research (NWO)2006-present Assistant professor at the Ecole Normale Superieure (maitre de

conferences).Researcher in the Laboratory Ecology & Evolution (UMR 7625 CNRS-UPMC-ENS)

A.2 Supervision of students and postdocs

Robin Aguilee, PhD student (Sept 2007 - 2010), co-supervised with Amaury Lambert(UPMC). Funded by a bourse de these (ministere), Ecole Doctorale Diversite du Vivant.Soutenance de these: 30 septembre 2010. Publications: Aguilee et al (2009, 2010, 2011). Onemanuscript in preparation.

Boris Sauterey, PhD student (Sept 2010 - Sept 2013), co-supervised with Chris Bowler(IBENS Molecular Plant Biology, CNRS-ENS). Funded by a bourse de these from the ecoledoctorale Forntires du Vivant (FdV) graduate school.


Vincent Le Bourlot, PhD student (Sept 2010 - Sept 2013), co-supervised with Thomas Tully(Lab Ecology & Evolution UMR 7625). Funded by Vincents ENS Allocation Specifique. Reg-istered in the ecole doctorale Diversit du Vivant graduate school.

Manuela Gonzalez Suarez, postdoc (Jan 2009 - Feb 2010), co-supervised with JF le Galliard(Lab Eco-Evo, UMR 7625). Funded by Region Ile de France (R2DS). Manuelas one-yearpostdoc consisted of the formulation of a stochastic, physiologically-structure populationmodel (PSP model) of the common lizard (Lacerta vivipara). JF le Galliard is an expert ofthe species and supervised the empirical and parameterization (stats) part of the project,whereas I supervised the model design and development. Publications: Gonzalez-Suarez etal (2011a, 2011b)

Jan Ohlberger, postdoc (since Jan 2009 for three years). Funded by Norwegian ResearchCouncil. Jans postdoc project is part of a wider collaboration between myself, Eric Ede-line (BIOEMCO, UPMC-ENS), and Oystein Langangen, N-C Stenseth and L.A. Vollestadat the Centre for Ecological and Evolutionary Synthesis (CEES, Oslo, Norway) in particu-lar, and Ian Winfiled at the Centre for Ecology and Hydrology (CEH, Windermere UK). JanOhlberger is based at CEES, Oslo, but spends several weeks per year in Paris to collaboratewith Eric Edeline and me. His project is to model the effect of temperature on size-structurefish populations, in particular the perch-pike fish community in Lake Windermere. I am themain modeling advisor in this project. Publications: Ohlberger et al (2011a and 2011b).

Loic Chalmandrier, Master 2 ENS (semestre blanche) (Sept 2009 - Jan 2010). Loics masterproject ”Adaptive dynamics of marine phytoplankton communities” was a pilot study for aninnovative approach to phytoplankton dynamics and evolution in global circulation mod-els. Loics project consisted of a first attempt to extend the ecological compartment (nutrients,phytoplankton, zooplankton) of the MIT global circulation model in order to include evo-lution of phytoplankton functional groups, using the adaptive dynamics conceptual frame-work. His work was picked up and continued by Boris Sauterey.

Boris Sauterey, Master 2 EBE (feb - june 2010) (now PhD student since sept 2010), co-supervised with Chris Bowler. Boriss master project ”Eco-evolutionary feedbacks in marinephytoplankton communities under climate change” built on Loics stage. Boris objective wasto finalise the implementation of adaptive dynamics into the MIT global circulation modeland to perform the first numerical experiments with it. This work is being continued duringhis PhD project (co-supervised by Chris Bowler and myself) in the context of the ANR projectPHYTBACK.

Vincent Le Bourlot, Master 2 EBE (feb - june 2010) (now PhD student since sept 2010),co-supervised with Thomas Tully. Vincents project aimed at developing a physiologicallystructured population (PSP) model for a model species (collembola Folsomia candida) thatwill be used to predict the species dynamical response (population dynamics and adaptivedynamics) to climate, in particular as a function of temperature. His work is being continuedduring his ongoing PhD project (co-supervised by Thomas Tully and myself) in the contextof the ANR project EVORANGE.


Benoit de Becdelivre , Master 1 ENS (feb 2009-june 2009). Benoits master project, ”DarwinFinches : Explaining coexistence with adaptive dynamics” used adaptive dynamics theory tocritically re-evaluate a standard hypothesis of the radiation of the Darwin finches. The mainresults were further generalised by Robin Aguilee during the latter’s PhD project. Publica-tion: Aguilee et al (2010).

A.3 Scientific responsibilities

• Coordinator (PI) of the ANR-blanc project (2010-2014) ”Ecology-climate feedbacks dueto evolution of phytoplankton cell size and shape - PHYTBACK”. For this project, Ihave invited researchers from three ENS laboratories (Ecology & Evolution, MolecularPlant Biology, Bioemco) and the Massachusetts Institute of Technology (MIT, Boston)to work on an innovative project that aims at understanding the role of evolutionarydynamics of phytoplankton for the feedbacks between climate change, phytoplanktondynamics and evolution, and the carbon pump. PHYTBACK unites a unique groupof researchers with diverse backgrounds that each brings unique skills and knowledgeto the table, which together offers a truly interdisciplinary and synergistic approach tothe subject matter. The subject of Boris Sautereys PhD project is part of this project.

• Participant and Task leader of the ANR (6e extinction programme) project EVOR-ANGE, a collaborative project with ISEM, LECA, MNHN and CEFE, led by OphelieRonce (ISEM Montpellier). The subject of Vincent Le Bourlots PhD project is part ofthis project.

• Member of the comite de pilotage of the Ecotron-Ile-de-France (ENS-CNRS).

A.4 Teaching responsabilities

• The main courses I teach at the Biology Department, at MSc level (M1 and M2), are:Evolutionary ecology (M1, 6 ECTS), Ecology of structured populations (M2, 6 ECTS),and Adaptive dynamics (M2, 3 ECTS). At the Environmental Research and TeachingInstitute (CERES-ERTI) I teach two courses per year, at BSc level (L3, 4 ECTS), on vary-ing, interdisciplinary, environmental issues, including sustainable agriculture, marinereserves, ecological modeling and time-series analysis.

• Directeur des etudes of the CERES-ERTI, meaning that I am responsible for the coordi-nation of the teaching at CERES-ERTI (around 50 students yearly in total; four coursesper year offered by the CERES-ERTI).

• Member of the bureau pedagogique of the Master programme Ecologie-Evolution-Biodiversite(EBE), in which I represent the ENS together with Regis Ferriere

• Co-organiser of the summer school (ecole thematique CNRS) ”feedbacks in environ-mental systems”, organised by CERES-ERTI (ENS). The first edition took place in 6-11 June 2011, with an emphasis on climate-economy interactions. The next edition isplanned for 3-10 June 2012, with an emphasis on climate-ecology interactions.

• Co-organiser of the seminar series ”Biodiversite : des questions scientifiques aux enjeuxsocietaux”, orgnised by CERES-ERTI (ENS)


Invited teaching

Insituto Veneto di Scienze, Lettere ed Arti Invited lecture and working group leader atthe summer school on ”Biological and Earth System Sciences”, (June 11-18, 2010)

A.5 Publications

Peer-reviewed international journals

1. Aguilee R., A. Lambert and D. Claessen (2011). Ecological speciation in dynamic landscapes.Journal of Evolutionary Biology 24: 2663-2677

2. Ohlberger J, Langangen, E Edeline, D Claessen, I Winfield, NC Stenseth, and A Vollestad (inpress) Stage-specific biomass overcompensation by juveniles in response to increased adultmortality in a wild fish population. Ecology.

3. Gonzalez-Suarez, J.F. Le Galliard and D. Claessen (2011) Population and life-history conse-quences of within-cohort individual variation. The American Naturalist 178(4): 525-537

4. Ohlberger, J, E Edeline, LA Vllestad, NC Stenseth and D Claessen (2011) Temperature drivenregime shifts in the dynamics of size-structured populations. The American Naturalist 177(2):211-223

5. Gonzalez-Suarez, M., M. Mugabo, B. Decenciere, S. Perret, D. Claessen, J.F. Le Galliard (2011)Disentangling the effects of predator body size and prey density on prey consumption in alizard. Functional Ecology 25:158-165.

6. Aguilee, R., B. de Becdelivre, A. Lambert and D. Claessen. (2010). Under which conditions ischaracter displacement a likely outcome of secondary contact ? Journal of Biological DynamicsVol. 5, No. 2, March 2011, 135-146.

7. Edeline E, TO Haugen, FA Weltzien, D Claessen, IJ Winfield, NC Stenseth and LA Vollestad(2009) Body downsizing caused by non-consumptive social stress severely depresses popula-tion growth rate. Proceedings of the Royal Society B

8. Aguilee, R., D. Claessen and A. Lambert. (2009). Allele fixation in a dynamic metapopulation :founder effects vs refuge effects. Theoretical Population Biology 76(2) : 105-117.

9. Claessen, D., A. de Vos and A.M. de Roos. (2009). Bioenergetics, overcompensation and thesource/sink status of marine reserves. Canadian Journal of Fisheries and Aquatic and Sciences66(7) : 1059-1071.

10. Claessen, D., J. Andersson, L. Persson and A.M. de Roos. (2008). The effect of population sizeand recombination on delayed evolution of polymorphism and speciation in sexual popula-tions. The American Naturalist 172(1):E18-E34.

11. de Roos, A.M., T. Schellekens, T. van Kooten, K. van de Wolfshaar, D. Claessen and LennartPersson (2008). Simplifying a physiologically structured population model to a stage-structuredbiomass model. Theoretical Population Biology 73 : 47-62

12. de Roos, A.M., T. Schellekens, T. van Kooten, K. van de Wolfshaar, D. Claessen and LennartPersson (2007). Food-dependent growth leads to overcompensation in stage-specific biomasswhen mortality increases : the influence of maturation versus reproduction regulation. Amer-ican Naturalist 170(3) : E59-E76

13. Andersson, J., P. Bystrom, D. Claessen, L. Persson and A.M. de Roos. (2007). Stabilization ofpopulation fluctuations due to cannibalism promotes resource polymorphism in fish. Ameri-can Naturalist 169(6):820-829


14. Claessen, D., J. Andersson, L. Persson and A.M. de Roos. (2007). Delayed evolutionary branch-ing in small populations. Evolutionary Ecology Research 9(1) : 51-69

15. Claessen, D., C.A. Gilligan, P.J.W. Lutman and F. van den Bosch. (2005). Which traits promotepersistence of feral GM crops ? Part 1 : implications of environmental stochasticity. Oikos 110(1) : 20-29.

16. Claessen, D., C.A. Gilligan and F. van den Bosch. (2005). Which traits promote persistence offeral GM crops ? Part 2 : implications of metapopulation structure. Oikos 110 (1) : 30-42.

17. Claessen, D. (2005). Alternative life-history pathways and the elasticity of stochastic matrixmodels. American Naturalist 165 : E27-E35

18. Claessen, D ; de Roos, AM ; Persson, L. (2004). Population dynamic theory of size-dependentcannibalism. Proc. Roy. Soc. B : Biol. Sci. 271 (1537) : 333-340.

19. Persson, L ; Claessen, D ; De Roos, AM ; Bystrom, P ; Sjogren, S ; Svanback, R ; Wahlstrom,E ; Westman, E. (2004). Cannibalism in a size-structured population : Energy extraction andcontrol. Ecological Monographs 74 (1) : 135-157.

20. Claessen, D ; de Roos, AM. (2003). Bistability in a size-structured population model of canni-balistic fish - a continuation study. Theoretical Population Biology 64 (1) : 49-65.

21. Persson, L ; De Roos, AM ; Claessen, D ; Bystrom, P ; Lovgren, J ; Sjogren, S ; Svanback, R ;Wahlstrom, E ; Westman, E. (2003). Gigantic cannibals driving a whole-lake trophic cascade.P.N.A.S. 100 (7) : 4035-4039.

22. Claessen, D ; Van Oss, C ; de Roos, AM ; Persson, L. (2002). The impact of size-dependentpredation on population dynamics and individual life history. Ecology 83 (6) : 1660-1675.

23. Claessen, D ; Dieckmann, U. (2002). Ontogenetic niche shifts and evolutionary branching insize-structured populations. Evolutionary Ecology Research 4 (2) : 189-217.

24. Vala, F ; Weeks, A ; Claessen, D ; Breeuwer, JAJ ; Sabelis, MW. (2002). Within- and between-population variation for Wolbachia-induced reproductive incompatibility in a haplodiploidmite. Evolution 56 (7) : 1331-1339.

25. Claessen, D ; de Roos, AM ; Persson, L. (2000). Dwarfs and giants : Cannibalism and competi-tion in size-structured populations. American Naturalist 155 (2) : 219-237.

26. Claessen, D ; de Roos, AM. (1995). Evolution of virulence in a host-pathogen system with localpathogen transmission. Oikos 74 (3) : 401-413.

Book chapter

• Gilligan, C.A., D. Claessen and F. van den Bosch. (2005). Spatial and temporal dynamics ofgene movements arising from deployment of transgenic crops. Book chapter in J.H.H. Wesseler(Ed.) . Environmental Costs and Benefits of Transgenic Crops pp. 143-161.

modelling the struggle for existence in structured populations1 introduction 6 1 introduction a...

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