Tuomas Nurmi
TURUN YLIOPISTON JULKAISUJA –  ANNALES UNIVERSITATIS TURKUENSIS
Sarja - ser. AI osa  - tom. 508 | Astronomica - Chemica - Physica - Mathematica | Turku 2015
ADAPTIVE DYNAMICS OF 
RESOURCE SPECIALIZATION
Supervised by
Docent Kalle Parvinen
Department of Mathematics
and Statistics
University of Turku
Turku, Finland
Professor Mats Gyllenberg  
and Dr. Stefan Geritz
Department of Mathematics
and Statistics
University of Helsinki
Helsinki, Finland
University of Turku 
Faculty of Mathematics and Natural Sciences
Reviewed by
Professor Andrew White
Department of Mathematics
Heriot–Watt University
Edinburgh, Scotland 
Dr. Martijn Egas
Department of Population Biology
University of Amsterdam
Amsterdam, Netherlands
Opponent
Professor Sebastian Schreiber
Department of Ecology and Evolution
University of California
Davis, California, USA
The originality of this thesis has been checked in accordance with the University of Turku 
quality assurance system using the Turnitin OriginalityCheck service.
ISBN 978-951-29-6036-1 (PRINT)
ISBN 978-951-29-6037-8 (PDF)
ISSN 0082-7002
Painosalama Oy - Turku, Finland 2015
Research director
Professor Marko Mäkelä
Department of Mathematics
and Statistics
University of Turku
Turku, Finland
Abstract
Ecological specialization in resource utilization has various facades rang-
ing from nutritional resources via host use of parasites or phytophagous
insects to local adaptation in different habitats. Therefore, the evolution
of specialization affects the evolution of most other traits, which makes
it one of the core issues in the theory of evolution. Hence, the evolu-
tion of specialization has gained enormous amounts of research interest,
starting already from Darwin’s Origin of species in 1859. Vast major-
ity of the theoretical studies has, however, focused on the mathematically
most simple case with well-mixed populations and equilibrium dynamics.
This thesis explores the possibilities to extend the evolutionary analysis
of resource usage to spatially heterogeneous metapopulation models and
to models with non-equilibrium dynamics. These extensions are enabled
by the recent advances in the field of adaptive dynamics, which allows
for a mechanistic derivation of the invasion-fitness function based on the
ecological dynamics. In the evolutionary analyses, special focus is set
to the case with two substitutable renewable resources. In this case, the
most striking questions are, whether a generalist species is able to coexist
with the two specialist species, and can such trimorphic coexistence be
attained through natural selection starting from a monomorphic popula-
tion. This is shown possible both due to spatial heterogeneity and due to
non-equilibrium dynamics. In addition, it is shown that chaotic dynamics
may sometimes inflict evolutionary suicide or cyclic evolutionary dynam-
ics. Moreover, the relations between various ecological parameters and
evolutionary dynamics are investigated. Especially, the relation between
specialization and dispersal propensity turns out to be counter-intuitively
non-monotonous. This observation served as inspiration to the analysis
of joint evolution of dispersal and specialization, which may provide the
most natural explanation to the observed coexistence of specialist and
generalist species.
Tiivistelma¨
Ta¨ssa¨ tyo¨ssa¨ tutkitaan ekologisten resurssien ka¨yto¨n erikoistumista. Mate-
maattisen mallinnuksen na¨ko¨kulmasta resursseiksi voidaan ravinnon ja
suojapaikkojen lisa¨ksi mielta¨a¨ myo¨s esimerkiksi loisela¨inten isa¨nna¨t tai
sirpaloituneen ympa¨risto¨n erilaiset asuinalueet eli laikut. Ta¨ta¨ monimuo-
toista alaa on tutkittu runsaasti, mutta keskittyen la¨hes yksinomaan mate-
maattisesti yksinkertaisimpiin malleihin, joissa elio¨t ka¨ytta¨va¨t vain yhta¨
homogeenista elinaluetta ja populaatiodynamiikan attraktori on kiintopiste.
Ta¨ssa¨ tyo¨ssa¨ tutkitaan jaksollisen tai kaoottisen populaatiodynamiikan
seka¨ metapopulaatiorakenteen vaikutuksia erikoistumisen evoluutioon.
Evoluution mallintaminen tapahtuu ta¨ssa¨ tyo¨ssa¨ adaptiivisen dynamiikan
keinoin eli johtaen kelpoisuusfunktio mekanistisesti ekologisesta dyna-
miikasta. Tyo¨ssa¨ keskityta¨a¨n ennen kaikkea tapaukseen, jossa organismi
voi ka¨ytta¨a¨ kahta vaihtoehtoista resurssia, ja tutkitaan, milloin monomor-
fisesta populaatiosta alkava evoluutio voi johtaa trimorfiseen yhteiseloon,
jossa generalisti kykenee ela¨ma¨a¨n yhdessa¨ kahden spesialistin kanssa,
vaikka kumpikin spesialisti hyo¨dynta¨a¨ yksitta¨ista¨ resurssia generalistia
tehokkaammin. Trimorfinen yhteiselo ei ole mahdollista yksitta¨isessa¨ ho-
mogeenisessa elinympa¨risto¨ssa¨, jos populaatiodynamiikan attraktori on
kiintopiste. Ta¨ssa¨ tyo¨ssa¨ osoitetaan, etta¨ monomorfisesta populaatiosta
alkava evoluutio voi johtaa trimorfiseen yhteiseloon silloin, jos homogee-
nisen elinympa¨risto¨n populaatiodynamiikka on jaksollista tai kaoottista,
sek silloin, jos homogeenisen elinalueen sijaan tarkastellaankin metapo-
pulaatiota. Lisa¨ksi osoitetaan, etta¨ kaoottinen populaatiodynamiikka voi
joskus johtaa sykliseen evolutiiviseen dynamiikkaan tai jopa koko popu-
laation tuhoon evolutiivisen itsemurhan kautta. Tyo¨ssa¨ tutkitaan myo¨s
ekologisen mallin eri parametrien vaikutusta erikoistumisen evoluutioon.
Muuttoliikkeen vaikutus erikoistumisen evoluutioon havaitaan intuition
vastaisesti epa¨monotoniseksi, minka¨ innoittamana syvennyta¨a¨n myo¨s muut-
totodenna¨ko¨isyyden ja erikoistumisen yhteisevoluutioon, joka todetaan
kenties luontaisimmaksi selitykseksi spesialistien ja generalistien trimor-
fiselle yhteiselolle.
Acknowledgements
This work was initiated at the Department of Mathematical Sciences, Uni-
versity of Turku, around the beginning of the century. The research was
mainly committed at the Department of Mathematics, and at last final-
ized to the form of a dissertation at the Department of Mathematics and
Statistics. It has been a lengthy project, and though the staff of the de-
partment has encountered less variance than the name, many important
persons have been involved.
I am grateful to Professor Mats Gyllenberg for introducing me to the
field of mathematical ecology, and giving me the opportunity to work
within this interesting field. I wish to thank Stefan Geritz for showing
me the way to the world of adaptive dynamics and for guiding me at the
first steps on my path to specialization research. Most of all, I wish to
express my sincere gratitude to my supervisor Kalle Parvinen for his con-
stant support, guidance and collaboration during this work. I also wish
to thank ´Eva Kisdi for valuable discussions on mathematical biology, and
especially for organizing inspiring meetings and summer schools. Spe-
cial thanks are due to the pre-examiners of this thesis, Professor Andrew
White and PhD Martijn Egas.
I acknowledge Professor Juhani Karhuma¨ki, the head of the Depart-
ment of Mathematics and Statistics, for excellent working facilities and
environment. I express my respect and gratitude to Professor Marko
Ma¨kela¨ for his communicative way of leading the Applied Mathemat-
ics group that creates the specially warm and open, but still productive,
atmosphere. Moreover, I wish to acknowledge the Finnish Doctoral Pro-
gramme in Computational Sciences (FICS) and Turun Yliopistosa¨a¨tio¨ for
financial support on traveling.
My sincere thanks belong to the colleagues and personnel at the De-
partment of Mathematics for being part of the unique, warm atmosphere.
I express my warmest gratitude for friendship and insightful discussions
to Anne Seppa¨nen and Hanna Eskola, with whom I have shared the status
of being a PhD-student in the biomathematics group. I wish to thank my
roommates Henri Virtanen, Sanna Ranto and Riku Kle´n for friendship and
endurance. I am grateful to the sporting club Luiskaotsat, especially the
long-term active members Tommi Meskanen, Jyrki Lahtonen and Mikko
Pelto, for enriching the days with physical exercises. I thank Tuire Huus-
konen, Sonja Vanto, Lasse Forss and Laura Kullas for endless patience
and helpfulness whenever I have had problems with the bureaucracy. I
am grateful to Arto Lepisto¨ for technical support; the harder the problem,
the faster its solved by Arto. I also want to thank Napsu Karmitsa, Ville
Junnila, Tomi Ka¨rki, Petteri Harjulehto, Eeva Vehkalahti, Eija Jurvanen,
Roope Vehkalahti, Jarkko Peltoma¨ki and Vesa Halava, among others, for
cheering up the lunch- and coffee-breaks with enjoyable discussions.
I want to express my warm thanks to my parents Mirja and Pauli for
all the love, support and encouragement through my life and education.
Finally, my warmest gratitude belongs to my dearest ones, my wife
Hanna and my beloved children Hilla, Matleena and Linnea. Thank you
for making my life so meaningful.
Turku, January 2015
Tuomas Nurmi
List of original publications
1. Tuomas Nurmi, Stefan Geritz, Kalle Parvinen
and Mats Gyllenberg:
Evolution of Specialization on Resource Utilization
in Structured Metapopulations
Journal of Biological Dynamics 2: 297–322 (2008).
2. Tuomas Nurmi and Kalle Parvinen:
On the evolution of specialization with a mechanistic
underpinning in metapopulations
Theoretical Population Biology 73:222–243 (2008).
3. Tuomas Nurmi and Kalle Parvinen:
Joint evolution of specialization and dispersal
in structured metapopulations
Journal of Theoretical Biology 275: 78–92 (2011).
4. Tuomas Nurmi and Kalle Parvinen:
Evolution of specialization under non-equilibrium
population dynamics
Journal of Theoretical Biology 321:63–77 (2013).
Part I
General theory
Contents
1 Introduction 1
2 Metapopulation models 5
2.1 Spatially heterogeneous models of ecological
dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The Levins’ model and other patch occupancy
models . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Structured metapopulation models . . . . . . . . . . . . 7
3 Adaptive dynamics 13
3.1 Historical background . . . . . . . . . . . . . . . . . . . 13
3.2 The adaptive dynamics approach . . . . . . . . . . . . . 15
3.3 The evolutionary analysis of scalar-valued strategies in
monomorphic populations . . . . . . . . . . . . . . . . 19
3.4 Evolutionary analysis of scalar-valued strategies in poly-
morphic populations . . . . . . . . . . . . . . . . . . . 26
3.5 Joint evolution of several traits (vector-valued
strategies) . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Mechanistic derivation of ecological models for the adaptive
dynamics of resource use 29
4.1 Discrete-time model for local population dynamics . . . 29
4.2 Trade-off between the resource consumption rates . . . . 36
4.3 Metapopulation dynamics . . . . . . . . . . . . . . . . . 40
5 Invasion fitness and environmental interaction variable 43
5.1 Invasion fitness in well-mixed populations . . . . . . . . 43
5.2 Invasion fitness in metapopulations . . . . . . . . . . . . 44
5.3 Environmental interaction variable and the
principle of competitive exclusion . . . . . . . . . . . . 48
6 Evolution of resource specialization 51
6.1 Ways to model specialization . . . . . . . . . . . . . . . 51
6.2 Evolution of specialization in well-mixed populations un-
der equilibrium dynamics . . . . . . . . . . . . . . . . . 52
6.3 Evolution of specialization in the case of well-mixed pop-
ulations with non-equilibrium dynamics . . . . . . . . . 56
6.4 Spatially heterogeneous models for the evolution of spe-
cialization and the joint evolution of dispersal
propensity and specialization . . . . . . . . . . . . . . . 64
1 1 INTRODUCTION
1 Introduction
Ecological specialization is one of the core issues in the study of evolu-
tion. Specialization, often viewed in the form of local adaptation, affects
the evolutionary dynamics of any life-history trait. Therefore, specializa-
tion has been a topic for a wide range of research. Already Darwin (1859)
used the existence of various forms of specialization and local adaptation
as evidence when arguing that species evolve in nature. Since then, the
amount of research work focused on the different aspects of the evolution
of specialization has increased enormously.
When evolutionary biology is popularized, the term ”specialized” of-
ten refers to species with extraordinary or bizarre features, e.g., tremen-
dous horns or other extravagant armament. As noted by Amadon (1943),
these extraordinary traits may sometimes be evolutionarily extremely im-
portant by allowing the species, or clades, to obtain abilities to utilize
completely new niche types. Most obvious example of this process was
presented by Huxley (1868, 1870) who viewed birds as glorified dino-
saurian reptiles with the extraordinary ability to fly.
However, usually in the evolutionary biology literature, the term ”spe-
cialization” is used in the situations where a species is, in principle, ca-
pable of using two or more alternative resources, but there is a trade-off
between the abilities to use these resources (Futuyma and Moreno, 1988;
Jaenike, 1990; Scheiner, 1993; Abrams, 2000b; Ravigne´ et al., 2009;
Poisot et al., 2011; Forister et al., 2012). Generalists use all, or several,
of these resources whereas specialists exclude some of the resources in
order to be more efficient in using the others. The term ”resources” may
here be interpreted rather generally. It may refer to, for example, nutri-
tional resources such as different plants eaten by a herbivore, different
prey species captured by a predator, different hosts of a parasite, different
possible habitats in a spatially heterogeneous landscape, etc. For field-
biologically inclined discussions concerning the concepts of specialism,
generalism and the nature and existence trade-offs, see, e.g., Fry (1996,
2003), Kneitel and Chase (2004), Loxdale et al. (2011) and Dennis et al.
(2011).
Specialization, in this wide sense, is a part of the evolutionary dy-
namics of any other life history trait. Most of all, the evolution of spe-
cialization, in the form of local adaptation, interacts with the evolution of
1 INTRODUCTION 2
dispersal: the better an individual is adapted to its prevailing local con-
ditions, the higher is the risk that this individual, if dispersing, ends in a
habitat with less favorable conditions (Clobert et al., 2001). The relation
between specialization and dispersal, however, is more complicated than
this simplification, especially in the presence of local disasters or other
temporal variations that may harm, or even wipe out, local populations
(Nurmi and Parvinen, 2008, 2011).
From the point of view of conservation biology, it is important to un-
derstand this relation, since, on one hand, increasing habitat fragmen-
tation makes the species and their local populations more vulnerable to
temporal disorders (Schoener and Spiller, 1987; Root, 1998; Casagrandi
and Gatto, 1999), and on the other hand, habitat loss and fragmentation
have an outstanding effect on the loss of biodiversity worldwide (Bar-
bault and Sastrapradja, 1995; Debinski and Holt, 2000; Sih et al., 2000;
Fahrig, 2003), and the degree of specialization affects crucially both the
consequences of habitat fragmentation and the global extinction risk of
species (Turner, 1996; McKinney, 1997; Henle et al., 2004; Colles et al.,
2009; Bru¨ckmann et al., 2010). Altogether, it is of great importance to
understand the complex interplay between the evolutionary dynamics of
specialization and dispersal in the presence of temporal variations in order
be able to study their evolutionary dynamics in heterogeneous and frag-
mented environments. This thesis aims to explore this interplay, and thus,
to provide tools for understanding the possible evolutionary responses for
habitat degradation and fragmentation.
However, this thesis has also another, equally important, objective:
understanding the origins of biodiversity. This objective is targeted, in the
case of two alternative resources, via one specific theoretical question: un-
der which conditions can an initially monomorphic species (i.e., a species
that comprises one type of individuals only) evolve to the trimorphic co-
existence of a generalist type with two specialists types. This question
has recently been vividly discussed (Wilson and Yoshimura, 1994; Egas
et al., 2004; Abrams, 2006a,b). In this thesis, two mechanisms are being
demonstrated and analyzed that allow an initially monomorphic popula-
tion to evolve to the trimorphic coexistence of generalists and specialists.
One of the mechanisms is based on the joint evolution of dispersal propen-
sity and resource specialization (Nurmi and Parvinen, 2011), whereas the
other builds on non-equilibrium population dynamics (Nurmi and Parvi-
3 1 INTRODUCTION
nen, 2013).
The main focus of this thesis is in the understanding of the evolution
of resource usage in the case of two alternative resources and spatially
heterogeneous environments. When the population is not well-mixed,
the evolutionary analysis of any trait becomes rather cumbersome. Here,
population dynamics in heterogeneous space are modeled by structured
metapopulation models. The evolutionary analysis utilizes the adaptive
dynamics approach. Below, these main tools, metapopulation models and
adaptive dynamics are introduced.
In order to build metapopulation models suitable for evolutionary anal-
ysis, one first has to derive a model for the dynamics of the local popula-
tions based on individual-level processes, and then lift this model to the
metapopulation level by book-keeping. After the general introductions of
the metapopulation models and adaptive dynamics, this modeling process
is introduced together with a metapopulation-level proxy for the invasion
fitness.
Finally, the results of the evolutionary analyses are reviewed in the
light of current conceptions of evolutionary dynamics of specialization.
These results concentrate on the evolution of resource utilization in meta-
populations, on the joint evolution of specialization and dispersal, and on
evolution under non-equilibrium ecological dynamics.
4
5 2 METAPOPULATION MODELS
2 Metapopulation models
2.1 Spatially heterogeneous models of ecological
dynamics
In traditional models of ecology, it is assumed that all the individuals un-
der consideration interact equally with each other, independent of their
exact location. Based on this assumption, it is possible to assume that
contact rates between individuals follow the law of mass action. For ex-
ample in the case of predator–prey relationships, the rate at which prey
is captured by the predators is often assumed to be linearly proportional
to the rate of encounters between the prey individuals and the predator
individuals. This rate, in turn, is proportional both to the prey density and
to the density of prey-searching predators. Thus, all the prey individuals
encounter identical predation pressure independent of the area they in-
habit. This kind of population is often called well-mixed. Note that in the
predator–prey example, the number of prey-searching predators is gener-
ally not directly proportional to the number of predators since the preda-
tors need time to capture, handle and digest the prey (nonlinear functional
response).
However, natural populations are usually not well-mixed, and the en-
vironment, in which they live, is neither homogeneous nor of uniform
quality. Sometimes, for example in the case of marine organisms, changes
in the environmental quality occur continuously. In this case, the spatial
heterogeneity encountered by the organisms can be described by a par-
tial differential equation, and the modeler ends up using, e.g., reaction-
diffusion models (Skellam, 1951; Levin, 1976; Gurtin and MacCamy,
1977; Holmes et al., 1994; Okubo and Levin, 2001). However, when
terrestrial organisms are considered, changes in the environment rarely
occur continuously. Instead, the suitable grazing and breeding areas of
any species are usually distributed to patches surrounded by unsuitable
areas. These suitable patches are called local habitats. Individuals within
a local habitat interact almost exclusively with each other, and thus, form
a local population. Only rough estimates of local population dynamics
can be presented on the basis of models that deal solely with well-mixed
populations. This is because the local populations interact by dispersal,
which usually affects the local population dynamics in the patch. Once
2 METAPOPULATION MODELS 6
this dispersal is taken into account, one ends up with metapopulation dy-
namics (Hanski, 1998, 1999).
The term ”metapopulation” was first used by Levins (1969, 1970). In
his terminology, a metapopulation is a collection of partially isolated lo-
cal populations living in discrete habitat patches connected by dispersal.
Levins assumed that the local habitat patches are prone to local disasters
that may occasionally wipe out the local population. This results in empty
habitat patches that may again become recolonized by immigrants arriv-
ing from the other patches. In the Levins’ metapopulation model, the local
population dynamics within patches are completely omitted. Therefore,
a patch may only have two alternative states: either the patch is occu-
pied or it is empty and colonizable by immigrants. Moreover, the spatial
configuration of the habitats, as well as differences between them, are ne-
glected in the dispersal process. Furthermore, because of mathematical
tractability, it is assumed that there are infinitely many local habitats.
2.2 The Levins’ model and other patch occupancy
models
If one denotes the fraction of occupied patches by p and assumes that
dispersers colonize empty patches with the rate c (”colonization rate”),
and that occupied patches become empty with the rate d (”catastrophe
rate”), then one can write the Levins’ metapopulation model as
dp
dt
= cp(1− p)− dp,
where 1 − p is the fraction of empty patches (available for coloniza-
tion). It is thus assumed that the amount of dispersers colonizing empty
patches is directly proportional to the fraction p of occupied patches. The
Levins’ model is obviously an oversimplification, and its main signifi-
cance is that it provides an easily accessible viewpoint to the most im-
portant metapopulation-scale phenomenon: a species may persist even
though all its subpopulations in local habitats are occasionally, but not
simultaneously, destroyed by randomly occurring disasters (Hanski and
Gilpin, 1997).
The Levins’ model, however, often maintains its mathematical tractabil-
ity even with more realistic functional forms of colonization and catas-
7 2.3 STRUCTURED METAPOPULATION MODELS
trophe rates (Gotelli, 1991; Gotelli and Kelley, 1993; Hanski and Gyl-
lenberg, 1993) or several different patch types (Horn and Mac Arthur,
1971; Levin, 1974, 1976). The assumption of infinitely many uniformly
connected patches is more essential for the mathematical tractability. If
it is dropped, the models usually can be analyzed only via simulations.
Nevertheless, it is relatively easy, for a field biologist, to observe habi-
tat patch connectivities, and to distinguish occupied and empty habitats.
Thus, models based on the patch occupancies and non-uniform dispersal,
such as the incidence function model by Hanski (1992, 1994a,b), provide
widely used tools for field biology.
In the evolutionary analysis, however, the main drawback of the patch
occupancy models is that they are usually built completely phenomeno-
logically directly to the metapopulation level without considering the in-
dividual level processes at all. Therefore, these models enable only evo-
lutionary analysis that is completely based on group selection (see, e.g.,
Van Valen (1971)). Natural selection, however, takes place at the level
of individuals such that the membership of a group may only affect, but
not completely determine, the fitness of an individual (Williams, 1966;
Rueffler et al., 2006). Therefore, the analysis of evolutionary dynamics in
heterogeneous landscapes is reasonable only in structured metapopulation
models that are mechanistically underpinned on individual-level ecolog-
ical dynamics (Geritz and Kisdi, 2012). Moreover, structured metapop-
ulation models offer a unified and clarified approach to the situations in
which multi-level selection takes place and the phenomenological defini-
tion of fitness functions is less straightforward (Wilson and Sober, 1994).
In addition, the structured models allow also biologically more realistic
theoretical analysis of the ecological dynamics.
2.3 Structured metapopulation models
Structured models involve at least some level of spatial heterogeneity, but
still model explicitly the local population dynamics, which, in turn, are
affected by dispersal. In a metapopulation model, each local population
is assumed to be well-mixed. Simplest structured models comprise only
two habitat patches. Let xi and fi(xi) denote, respectively, the local popu-
lation density and the density-dependent per capita population growth rate
in patch i, and furthermore, assume that individuals migrate from patch i
2 METAPOPULATION MODELS 8
to patch j with rate eij and survive migration with probability pi. Then,
one can write down a continuous-time two-patch metapopulation model
as (Freedman and Waltman, 1977; Hastings, 1983; Holt, 1985)
dx1
dt
=f1(x1)x1 − e12x1 + pie21x2
dx2
dt
=f2(x2)x2 − e21x2 + pie12x1.
(1)
Note that the generalization of these models to include any finite number
of different patches is mathematically rather straightforward, but the anal-
ysis of the model and the field-biological determination of the ecological
parameters become cumbersome.
Alternatively, a two-patch model may have discrete-time dynamics
described by difference equations (Hastings, 1993; Gyllenberg et al., 1993).
Despite its simplicity, a two-patch model may exhibit extremely complex
ecological dynamics, which enables one to study the effects of dispersal
on the stability of the population dynamics (Hastings, 1993; Gyllenberg
et al., 1993; Ruxton et al., 1997; Kisdi, 2010). Moreover, the two-patch
models offer the simplest possible framework for the analysis of source–
sink population dynamics (Pulliam, 1988; Dias, 1996; Gyllenberg et al.,
1996). The term ”source” refers to a habitat in which the local birth rate
(or fecundity in discrete-time models) on average exceeds the death rate
(probability), whereas in a sink population the death rate exceeds the birth
rate. Thus, a sink population may persist only by the means of immigra-
tion from other patches. In metapopulations, abundant migration from
high-quality patches may raise the local population density in low-quality
patches such that, due to the density-dependent effects, the local death
rate exceeds the local birth rate even though the local population would
be viable also alone, with lower local population density however. This
kind of patches were named ”pseudo-sinks” by Watkinson and Sutherland
(1995).
Thus, a structure comprising sources and sinks or pseudo-sinks is nat-
ural to metapopulations. When considering the evolution of resource
specialization, the source–sink structure is not the same for all individ-
uals. Patches that are of high quality to a species that is specialized
to one resource may be low-quality patches to a species specialized to
another resource. In addition, the differences between patches usually
9 2.3 STRUCTURED METAPOPULATION MODELS
appear smaller when they are observed by a generalist compared to the
differences observed by a specialist. Moreover, if periodic or chaotic
population-dynamical attractors are possible, different local populations
of the same species may have qualitatively different population-dynamical
attractors. Simultaneously, it is also possible that, even within a single
patch, species with different specialization strategies encounter qualita-
tively different population-dynamical attractors.
Moreover, dispersal is a key ingredient in spatial population models
with non-equilibrium attractors: intermediate dispersal propensity may
stabilize the local population dynamics in the patches that would, in the
absence of dispersal, exhibit periodic or chaotic population dynamics, but
more abundant dispersal may have a synchronizing effect instead of a sta-
bilizing one. Then again, the type of the population-dynamical attractor
affects the evolution of dispersal propensity: if all the local population
densities are at their equilibrium values, dispersal is selected against, but
when the local population densities fluctuate, dispersal may become ben-
eficial (Hastings, 1983; Parvinen, 1999), and furthermore, the dispersal
propensity may even undergo evolutionary branching where the popu-
lation splits into two morphs; one dispersing abundantly and the other
scarcely (Doebeli and Ruxton, 1997; Parvinen, 1999).
As the evolution of specialization interacts significantly with that of
dispersal, it is necessary to understand both the effects of dispersal on lo-
cal population dynamics and the consequences of source–sink structures
to be able to study the evolution of specialization in spatially heteroge-
neous environments (Ronce and Kirkpatrick, 2001; Nurmi and Parvinen,
2008, 2011, 2013).
When the number of local habitat patches is increased from two in the
metapopulation model (1), the modeler has to consider the details of the
dispersal process, since the dispersal rates (probabilities) and dispersal
survival may be different for each pair of patches. The means of matrix
calculus may enable the mathematical analysis of such models for some
extent (Parvinen, 1999), but usually some mean-field approximation is
necessary when modeling dispersal. For example, the patches may be
assumed to be equally connected by dispersal (Levin et al., 1984; Cohen
and Levin, 1991). Some general conclusions can also be drawn, if it is
assumed that the habitat patches form a grid, and dispersal in this grid
is distance-limited, for example, only to nearest neighbors. In this case,
2 METAPOPULATION MODELS 10
one ends up with coupled map lattices analyzed, e.g., by Kaneko (1992,
1998) and Karonen (2011). However, if one wishes to build a spatially
explicit model, where all the connections between patches are taken into
account, the only way to analyze the resulting model is via simulations
that require careful parameter estimation (see, e.g., Hanski and Thomas
(1994), Hanski et al. (1994) and Hanski and Ovaskainen (2003)).
The two-patch and n-patch models introduced above lack one essen-
tial feature included in the Levins’ metapopulation model: the frequent
but random catastrophes that occasionally wipe out local populations but
leave the patches habitable and recolonizable. If the number of patches
is finite, such catastrophes are liable to drive any population to extinc-
tion, at least in the evolutionary time-scale. There are, however, models
with finite number of patches, where local conditions in patches alternate
randomly, but these temporal variations are mild in the sense that local
populations are not wiped out completely, which enables the viability of
the population in the evolutionary time-scale (McPeek and Holt, 1992;
Kisdi, 2002).
Altogether, in any model intended for evolutionary analysis, the as-
sumption of randomly occurring disasters destroying entire local pop-
ulations must be accompanied with the assumption of infinitely many
patches. If one, in addition, assumes global dispersal ignoring the spa-
tial arrangement of the patches, the model even becomes mathematically
tractable (Hastings and Wolin, 1989; Hastings, 1991; Gyllenberg and Han-
ski, 1992, 1997; Gyllenberg et al., 1997). One can then assume that all
the emigrating dispersers enter a disperser pool, from which they are dis-
tributed evenly to all of the patches. Let now Dn be the size per patch of
the disperser pool at time n. If one now focuses on a single patch with
population density xn at time n, one can determine the local discrete-time
dynamics as
xn+1 = C(n+ 1)(1− e)f(x)xn + piDn(s),
where the function f determines the local growth and survival within the
patch. This function may vary from patch to patch. The parameters pi
and e determine, respectively, dispersal survival and the emigration prob-
ability of an individual during one time step. Furthermore, C(n + 1) is a
random variable drawn from the Bernoulli distribution with parameter c.
11 2.3 STRUCTURED METAPOPULATION MODELS
It determines the occurrence of the local catastrophes, i.e.,
C(n+ 1) =

1, if there is no local catastrophe in the focal
patch after period n (probability 1− c),
0, if a local catastrophe occurs
after period n (probability c).
(2)
When there is only a finite number of different patch types, the dynamics
of the disperser pool size can be heuristically defined as
Dn =
∑
m
pm
(
Expected number of emigrants
from a type m patch at time n
)
,
where pm is the fraction of type m patches. The actual calculation of
Dn from this equation is rather demanding. However, in metapopulations
with globally attracting fixed point equilibrium, one can neglect this cal-
culation and solve Dn from a fixed point equation, since in the fixed point
Dn has a constant value D and this value must be such that once a dis-
perser enters a local population, the local clan it initiates, i.e., itself and
all of its descendants, their descendants, etc, will on average produce ex-
actly one new successful disperser before the clan is destroyed by the next
catastrophe in the patch..
Below in section 4, this metapopulation model is adjusted for resource–
consumer dynamics and the evolution of resource specialization of the
consumers. Derivation of the local dynamics follows the guidelines given
by Geritz and Kisdi (2004), and the calculation of invasion fitness (or
more precisely a proxy for the metapopulation-level invasion-fitness) is
based on the method by Parvinen (2006), who adapted the metapopulation
reproduction ratio concept introduced by Gyllenberg and Metz (2001)
and Metz and Gyllenberg (2001) to discrete-time metapopulation mod-
els. However, before considering fitness in metapopulations, the adap-
tive dynamics framework is introduced as a general toolbox for model-
ing frequency- and density-dependent long-term evolution of continuous
traits in ecologically realistic settings.
12
13 3 ADAPTIVE DYNAMICS
3 Adaptive dynamics
3.1 Historical background
In the days of Darwin (1859), the Mendelian genetics was not widely
known, and thus, it was natural that all the evolutionary considerations
took place at the phenotypic level: the traits that are beneficial for the re-
production and survival of an individual were simply predicted to become
more common in nature. When the results of Mendel were rediscovered at
the beginning of the 20th century (see, e.g., Fischer (1936)), the perma-
nence of genetic material and the consequent discreteness of hereditary
alteration first seemed to conflict with Darwin’s ideas of gradual evolu-
tion (see, e.g., Mayr (1982)). This controversy was solved by the rise
of population genetics conducted by Fischer (1930), Wright (1931) and
Haldane (1932) and the resulting modern synthetic evolutionary theory
(Dobzhansky, 1937; Huxley, 1942).
Mathematical population genetics considers evolution as fluctuations
in the frequencies of different alleles or genotypes in populations. Besides
the randomly occurring mutations and natural selection, these fluctuations
are affected also by random genetic drift (especially in small populations)
and gene flow caused by dispersal. Furthermore, the genetic architecture
of the species affects the fluctuations via, e.g., epistasis, linkage, and re-
combination. Population-genetic models aim to model this genetic com-
plexity in detail. A trade-off that is required to keep the models analyz-
able, is that the species’ ecological framework must be assumed to be rel-
atively simple. Therefore, despite the increasing knowledge on genetics,
phenotypic models of evolution are still useful when pursuing ecological
realism in evolutionary models and predictions.
Classical population genetics usually assume that a unique measure
of fitness is directly attached to each possible trait combination, and fur-
thermore, that this measure is independent of the traits of the rest of the
population (Wright, 1932; Lande, 1976). This means that selection is as-
sumed to be frequency- and density-independent and the fitness values of
the possible trait combinations form a so-called fitness landscape, where
evolution proceeds always uphill: a trait combination with given fitness
can always outcompete the combinations with lower fitness, as well as it
will be outcompeted once a trait combination with higher fitness appears.
3 ADAPTIVE DYNAMICS 14
This results in optimization models where evolution leads to a trait com-
bination whose fitness value is a local maximum of the fitness landscape.
With two-dimensional traits this process corresponds to finding the hill
peaks on a topographical map that describes the fitness landscape. Note
that if the mutations are assumed to be small in effect, evolution does
not necessarily lead to the highest peak, but instead only the nearest local
maximum is achieved.
The incorrectness of the assumption of frequency-independence was
realized already in the early history of population genetics as it was noted
that an allele or trait may benefit from being rare (Haldane, 1932; Lewon-
tin, 1958; Ayala and Cambell, 1974). Most obviously, this assumption
fails in the case where the fitness of an individual depends on pairwise
interactions between conspecifics, such that the strategy (evolving trait)
of the opponent affects the success of an individual. Then, the fitness
value of any trait combination is not constant but depends on the trait fre-
quencies in the population. This means, that the fitness landscape is not
constant, but it depends on the frequencies of the different strategies in
the resident population. This idea is included into the framework of evo-
lutionary game theory introduced by Maynard Smith (1974, 1976, 1982).
In evolutionary game theory (Nowak and Sigmund, 2004), it is as-
sumed that the fitness of an individual is affected by the individual’s suc-
cess in consecutive pairwise interactions with conspecifics. In each en-
counter, an individual may select from several behavioral patterns, e.g.,
escalate a conflict, display, negotiate or withdraw. These patterns are the
traits, the evolution of which evolutionary game theory considers. An
individual may always use the same behavioral pattern. This is called a
pure strategy. However, an individual may also use a mixed strategy, i.e.,
use different behavioral patterns with different probabilities. In this case,
the strategy vector of an individual determines these probabilities. In a
specific encounter, the payoffs that the interacting individuals obtain (or
losses they suffer) are determined by the behavioral patterns (pure strate-
gies) used by the interacting individuals in this encounter.
In the classical evolutionary game theory, it is usually assumed that
the fitness is determined by the average payoff obtained in consecutive
independent encounters. This means that the fitness of an individual be-
comes linear both with respect to the strategy of the resident population
and with respect to the strategy of the individual. This linearity results in
15 3.2 THE ADAPTIVE DYNAMICS APPROACH
some rather unrealistic features as indicated, for example by the Bishop
and Cannings (1978) theorem (see, e.g., Mesze´na et al. (2001)). More-
over, even though evolutionary game theory considers the frequencies of
different strategies, it omits the overall population density. This is a major
drawback, especially when considering long-term evolution (Heino et al.,
1998).
3.2 The adaptive dynamics approach
Adaptive dynamics (Metz et al., 1992, 1996; Dieckmann and Law, 1996;
Geritz et al., 1997, 1998) is a tool for studying the course of frequency-
and density-dependent evolution of continuous traits (strategies) in eco-
logical models. The first step in any application of adaptive dynamics
is the identification of traits, the evolution of which one is interested in.
These traits form the strategy of an individual, and the set of their possible
values is the strategy space. In the simplest case, the strategy is one-di-
mensional, e.g., age at maturation, and the strategy space is some interval
on the real line. Below, adaptive dynamics tools are introduced in the case
of one-dimensional strategies. The generalization to vector-valued strate-
gies is rather straightforward (Dieckmann and Law, 1996; Matessi and
Di Pasquale, 1996; Leimar, 2001, 2005, 2009), but the case of infinite-
dimensional (function-valued) strategies requires more care (Dieckmann
et al., 2006; Parvinen et al., 2006, 2013). The strategies studied in this
thesis are either one- or two-dimensional.
The key idea in adaptive dynamics is to model explicitly the ecological
dynamics and to derive the invasion fitness function mechanistically from
the life-history of the individuals, whereas most of the other approaches
of evolutionary modeling are based on phenomenologically built fitness
functions. For the derivation of the invasion fitness function, it is nec-
essary that invasion fitness itself is exactly mathematically defined. This
definition was given by Metz et al. (1992) who stated that the invasion fit-
ness of a rare mutant with strategy smut is its long-term exponential growth
rate r(smut, Eres) in the environment Eres set by the residents. If r < 0,
the mutation will sooner or later vanish from the population. If r > 0,
the mutant strategy may still be eliminated from the population due to the
demographic stochasticity at the initial phase of the invasion, but it may
also increase in population density and either coexist with the residents or
3 ADAPTIVE DYNAMICS 16
oust some of the resident strategies.
The derivation of the invasion fitness function and the analysis of the
evolutionary dynamics are based on the following three basic assump-
tions:
1. Clonal reproduction.
2. Rarely occurring mutations allowing the separation of ecological
and evolutionary time-scales.
3. Small initial mutant frequency in a large resident population.
In addition, it is usually assumed that:
4. The mutational steps are small, i.e., a new mutant always resembles
one of the existing residents.
5. If a mutant can invade a monomorphic resident population, but in-
vasion under reversed roles is not possible, the mutant will replace
the resident.
6. If a mutant can invade a monomorphic resident population, but the
invasion under reversed roles is also possible, then the resident and
the mutant will coexist.
Detailed discussions on the status of these assumptions are given by Geritz
et al. (2002), Geritz (2005), Geritz and Gyllenberg (2005) and Mesze´na
et al. (2005).
Whereas population genetics considers the short-term evolution of al-
lele distributions, the adaptive dynamics analysis usually involves only a
limited number of different strategies present in the resident population
although the number of possible strategies may be infinite. This limita-
tion allows one, based on the known ecological dynamics, to calculate the
population-dynamical attractor of the resident population.
It is possible to formulate almost any reasonable ecological model
of population dynamics such that it contains an environmental interaction
variable, say E, such that the population dynamics affect this variable, but
once its value is known, the equations describing population dynamics are
linear (Diekmann et al., 1998, 2001, 2003, 2007). Due to the assumption
(2.) of rarely occurring mutations, the resident population is always on
17 3.2 THE ADAPTIVE DYNAMICS APPROACH
the population-dynamical attractor when a new mutant strategy enters the
population. On the population-dynamical attractor, the resident sets, to-
gether with the abiotic factors, the value of the environmental interaction
variable. Let this environment set by the resident population be Eres. This
variable, Eres, may be a scalar, a vector, or even an infinite-dimensional
variable. This thesis focuses on discrete-time population models. In such
models, it is natural that the biotic factors affecting the environment Eres
are different for each time unit. The invasion fitness r(smut, Eres), how-
ever, is not determined for any single time unit, but it is the long-term av-
erage of the exponential growth rate. Therefore, it is obvious that the vari-
able Eres must be of the form Eres = (Eres(1), Eres(2), . . . , Eres(n), . . .),
whereEres(n) is the environment that determines the growth of the mutant
population at time-unit n.
According to the assumption (3.), the mutant population is initially
small, and thus, its effect on the environment is negligible. Therefore,
at the initial phase of invasion, its population dynamics may be approx-
imated by a linear differential (or difference) equation, where the per
capita growth rate of the mutant population determines the invasion fit-
ness of the mutant strategy (Metz et al., 1992). Let now smut denote the
mutant strategy and let r(smut, Eres) denote the invasion fitness of the mu-
tant in the environment set by the resident.
Assumption (4) is necessary when one wants to deduce the expected
direction of evolution based on the local properties of the invasion fitness
and local fitness gradient that will be derived below derived on the basis
of invasion fitness.
Assumptions (5.) and (6.) allow majority of the evolutionary analysis
of ecological models to be built on invasion fitness (Geritz et al., 1998,
2002). In most ecological scenarios, these assumptions follow directly
from the previous assumptions when the population-dynamical attractor
of the resident population is unique. When there are several possible eco-
logical attractors for the resident population dynamics, the situation is
more complicated. Consider, for example, the case in which the resi-
dent population has two alternative stable attractors, say A and B. Then
the environment set by the resident is not unique, but it is different for
each attractor. Denote now the environment set by the resident while on
the attractor A by EresA and the environment set by the resident while on
the attractor B by EresB . Furthermore, assume that r(smut, EresA ) > 0 and
3 ADAPTIVE DYNAMICS 18
r(smut, EresB ) < 0. Now, a mutant that enters while the resident is on the
attractor A starts to increase in population density. In most cases, the
appearance of the mutant does not cause significant changes in the at-
tractors of the population dynamics even if the mutant ousts the resident
(Geritz et al., 2002). Sometimes, however, it is possible that, due to the
appearance of the mutant, the population dynamical attractor A becomes
unstable, and the population (mutant–resident dynamics) evolves to the
alternative attractor B, on which the mutant population starts to diminish
and finally dies out. Therefore, the prevalent strategy of the population
remains unchanged but the population-dynamical attractor changes qual-
itatively. This is the so called ”resident strikes back” scenario (Doebeli,
1998; Mylius and Diekmann, 2001; Dercole et al., 2002).
A special extreme case of this scenario is the evolutionary suicide,
where the alternative resident attractor B is the trivial attractor that cor-
responds to extinction. Under certain ecological conditions, it is possible
that an invading mutant can oust the resident, even though it is not viable
alone. In this case, it is possible that evolution drives the species to ex-
tinction, i.e., evolutionary suicide occurs (Matsuda and Abrams, 1994a,b;
Ferrie`re, 2000; Rankin and Lopez-Sepulcre, 2005; Parvinen, 2005, 2007).
In the case of a polymorphic population, it is also possible that only
one morph is driven to extinction, which may even result in evolution-
ary branching–extinction cycles (Kisdi et al., 2001; Dercole, 2003; Nurmi
and Parvinen, 2013). Evolutionary suicide (evolutionary self-extinction,
Darwinian extinction) is possible, since traits that are harmful to the via-
bility of the species may still be beneficial at the individual level, which
allows them to become more common in the population (Webb, 2003).
This may be related, e.g., to the ”tragedy of commons” (Hardin, 1968).
There are two different types of evolutionary suicide. Mutations that
are harmful at the population-level may cause the population size to be-
come extremely small such that the population is finally wiped out by
demographic stochasticity (Matsuda and Abrams, 1994a), but it is also
possible that the evolutionary suicide occurs fully deterministically (Gyl-
lenberg and Parvinen, 2001; Gyllenberg et al., 2002). Typically, scenar-
ios resulting in deterministic evolutionary suicide involve Allee-effects
(Stephens et al., 1999). However, Allee-effects are not the only route
to deterministic evolutionary suicide, because it may be enabled also by,
e.g., non-equilibrium ecological dynamics (Parvinen, 2005; Nurmi and
19 3.3 SCALAR-VALUED STRATEGIES
Parvinen, 2013).
It is noteworthy that the condition r(smut, Eres) < 0 implies that the
mutant is liable to become ousted from the population, whereas the con-
dition r(smut, Eres) > 0 only implies that the mutant population is capable
to invade the mutant population. However, at the initial phase of an inva-
sion, the invading mutant population only comprises one (or a few) indi-
vidual(s). Therefore, a mutant, however fit it may be, can vanish from the
resident population due to demographic stochasticity. In this case how-
ever, the resident population remains unchanged. Thus, a corresponding
mutant is liable to later again repeatedly appear in the resident population
until it survives the stochastic initial phase of the invasion, and finally
invades the resident population.
3.3 The evolutionary analysis of scalar-valued strategies
in monomorphic populations
Below, it is assumed that the strategy under consideration is scalar-valued,
i.e., one-dimensional. It is also assumed that the resident population is ini-
tially monomorphic, i.e., all the resident individuals have the same strat-
egy. However, the generalization of the presented results to polymorphic
resident populations is rather straightforward. Furthermore, it is assumed
that the population-dynamical attractor of the resident strategy is unique.
As mentioned above, this simplifies the evolutionary analysis, since then
also the environment Eres set by the resident is uniquely determined for
each resident strategy, and thus, it is possible to base the evolutionary
analysis solely on the invasion fitness, which can be considered as a func-
tion of two variables; strategies smut and sres, of which the latter one acts
through the environmental interaction variable Eres.
Since the mutational steps are assumed to be small (assumption (4.)),
the expected direction of evolution in this monomorphic population is
given by the local fitness gradient
D(sres) =
[
∂r(smut, Eres)
∂smut
]
smut=sres
. (3)
Of special interest are the so called singular strategies s∗ for whichD(s∗) =
0, i.e., directional selection vanishes in the monomorphic population. A
3 ADAPTIVE DYNAMICS 20
classification of all possible generic types of singular strategies and their
interpretation is given by Metz et al. (1996) and Geritz et al. (1997, 1998).
Properties of singular strategies and directions of evolution in a monomor-
phic population may be analyzed graphically by a pairwise invadability
plot, or PIP, (Matsuda, 1985; van Tienderen and de Jong, 1986; Metz
et al., 1996; Geritz et al., 1998). PIPs representing the four most impor-
tant classes of singular strategies are illustrated in Figure 1.
M
u
ta
n
ts
tr
at
eg
y
A B C D
++
-
-
+
+
-
-
+
+
-
-
+
+
-
-
Resident strategy
Figure 1: Examples of pairwise invadability plots and qualitatively differ-
ent singular strategies. In the white areas a mutant strategy may invade
the resident population. In the gray areas the invasion is not possible.
Panel A: Evolutionarily attracting and uninvadable singular strategy.
Panel B: Evolutionarily repelling and invadable singular strategy.
Panel C: ”Garden of Eden” evolutionarily repelling singular strategy.
Panel D: Evolutionary branching point.
A pairwise invadability plot is the sign plot of the invasion fitness
r(smut, Eres) such that the horizontal axis corresponds to the set of all pos-
sible resident strategies and the vertical axis to the set of all possible mu-
tant strategies. A white point in the PIP indicates that the corresponding
mutant strategy can invade a population with the corresponding resident
strategy, i.e., r(smut, Eres) > 0. Correspondingly, a black point indicates
that the mutant cannot invade, i.e., r(smut, Eres) < 0. The curves separat-
ing white and black regions in the PIP are the fitness isoclines given by
the trait combinations for which r(smut, Eres) = 0. The main diagonal is
trivially such an isocline, since r(sres, Eres) = 0 due to the assumption (2)
that ensures that the resident is always on a population-dynamical attrac-
tor, and on a population-dynamical attractor, the population does neither
21 3.3 SCALAR-VALUED STRATEGIES
grow nor decrease in population size. The configuration of the other, non-
trivial, isocline(s) determines the singular strategies and their properties.
Singular strategies lie at those points where a nontrivial fitness isocline
crosses the diagonal. Even though each PIP in figure 1 has only one sin-
gular strategy (s∗), it is possible that the strategy space contains arbitrarily
many singular strategies.
Assuming that only mutants slightly different from the resident can
occur (assumption (4.)), one can confine the analysis of each PIP to a nar-
row strip along the diagonal where the mutant and resident strategies are
identical. For example, consider the PIP in Figure 1A. From the black-
and-white pattern it can be seen that a resident population with an arbi-
trary strategy s such that s < s∗ can be invaded by mutants with a slightly
larger strategy but not by mutants with a slightly smaller strategy. The
opposite is true for a resident population with any strategy s > s∗. In this
sense, the strategy s∗ is evolutionarily attracting. Moreover, it can also
be seen that a resident population with strategy s = s∗ cannot be invaded
by any nearby mutant, and therefore it is uninvadable, i.e., evolutionarily
stable strategy (ESS, Maynard Smith and Price (1973)).
The singular strategy in the figure 1B has the opposite properties. It
is evolutionarily repelling and, moreover, can be invaded by any nearby
mutant. The singular strategy in figure 1C represents so called ”Garden of
Eden” configuration: It is evolutionarily stable in the sense, that once the
resident population has exactly the singular strategy s∗, it is uninvadable
by any nearby mutant. However, the singular strategy is not evolutionar-
ily attracting, and therefore, any slightest deviation makes the population
to evolve away from the neighborhood of the singular strategy. In natural
systems, such deviations are unavoidable, and thus in practice, there is no
need to distinguish invadable and uninvadable singular strategies when-
ever they are evolutionarily repelling.
The singular strategy in figure 1D is evolutionarily attracting but in-
vadable. A singular strategy of this type is called an evolutionary branch-
ing point. In the neighborhood of an evolutionary branching point, there
exists a domain of strategies that can coexist in a protected dimorphism
in the ecological time-scale. Consider now two strategies, say x and y.
Let Ex (or Ey) be the environment determined by a monomorphic resi-
dent population with strategy x (or y). Strategies x and y can coexist in
a protected dimorphism if both r(x,Ey) and r(y, Ex) are positive. This
3 ADAPTIVE DYNAMICS 22
means that if, in this coexistence, one of the two strategies would be rare,
it would grow in population size since the environment would be, practi-
cally, set by the competing strategy.
The existence of strategy pairs capable for such protected coexistence
can be identified from pairwise invadability plots by switching the roles
of the mutant and resident strategy (mirroring with respect to the diago-
nal) and placing the resulting PIP on top of each other with the original
PIP. Altogether, close to the branching point, the population becomes di-
morphic. When the population is dimorphic in the neighborhood of an
evolutionary branching point, it can be invaded only by mutants that are
further away from the branching point. Thus, the population encoun-
ters divergent selection and, on each successive invasion, the two resident
strategies become, at least initially, more and more distinct from each
other (Metz et al., 1996; Geritz et al., 1997, 1998, 2004).
Whenever evolutionary branching is considered, the basic assump-
tion (1) of clonal reproduction becomes crucial. Kisdi and Geritz (1999)
have shown that clonal adaptive dynamics can for large extent predict the
course of evolution in monomorphic diploid sexually reproducing pop-
ulations as well. In the case of branching points, however, the clonal
adaptive dynamics predicts that the strategy of a monomorphic popula-
tion evolves towards the neighborhood of the branching point where dis-
ruptive selection promotes ecological diversification. The same is true
also for monomorphic sexually reproducing populations. What happens
under the influence of such disruptive selection, depends on the genet-
ical architecture and the mating system of the species (Dieckmann and
Doebeli, 1999; Geritz and Kisdi, 2000; van Doorn and Weissing, 2001).
In clonally reproducing populations, diversification splits the population
to two distinct lineages that encounter divergent evolution, which makes
their strategies to evolve further away from each other. In diploid popu-
lations, however, this split is prevented by the averaging effect of sexual
reproduction, unless some form of assortative mating evolves (see, e.g.,
van Doorn and Dieckmann (2006); van Doorn et al. (2009); Ripa (2009);
Kisdi and Priklopil (2011)). Altogether, the mere existence of an evolu-
tionary branching point does not lead to ecological speciation. Branching
points can only indicate ecological circumstances that may promote di-
versification which may, if mating barriers evolve, result in speciation.
Besides the properties introduced in Figure 1, the isocline configu-
23 3.3 SCALAR-VALUED STRATEGIES
rations in pairwise invadability plots may differ qualitatively in the abil-
ity of the singular strategy to invade other strategies in its neighborhood.
However, this property is of interest only in the case of an evolutionar-
ily attracting ESS, and even then the interest is minor, since this property
only determines the way the ESS is approached. If the ESS can invade
neighborhood strategies, it is possible, that the population ends up exactly
to the ESS in a discrete step. In the opposite case, population can only
approach the ESS as a limit process that may be restricted by the mini-
mum size of possible mutations, which is usually assumed to exist in the
adaptive dynamics analysis.
If the mutations were infinitesimally small, evolutionary analysis based
on dynamical systems theory would be possible using the canonical equa-
tion of adaptive dynamics (Dieckmann and Law, 1996; Champagnat et al.,
2001, 2006, 2008; Durinx et al., 2008). Thus, the existence of the min-
imum size of possible mutations together with mutational stochasticity
separates adaptive dynamics approach from standart dynamical systems
theory and enables, e.g., the analysis of evolutionary branching, which in-
creases the dimensionality of the evolving system and is therefore outside
the scope of the dynamical systems theory as such.
When selection is both frequency- and density-dependent, the invad-
ability and the evolutionary attractivity of a singular strategy are indepen-
dent of each other, whereas in optimization models (fitness landscapes)
and game-theoretical models they are contingent on each other (Mesze´na
et al., 2001; Dieckmann and Metz, 2006). This, together with the game-
theoretical history of adaptive dynamics, has caused some variation and
inconsistency in the terminology used by different authors. The term
ESS (evolutionarily stable strategy) (Maynard Smith and Price, 1973),
that refers to strategies that cannot be invaded by any nearby strategy, is
nowadays well established, even though the established interpretation is
rather misleading from the point of view of the traditional theory of dy-
namical systems, where an equilibrium point in a state-space is stable if
the state of the system converges to this point whenever the initial state is
close enough to this point (Devaney, 1989; Verhulst, 1996). However in
adaptive dynamics, evolution starting from a neighborhood of an ESS that
is not evolutionarily attracting will direct away from the ESS. In Figure
1, both cases A and C illustrate evolutionarily stable (uninvadable) strate-
gies even though only the singular strategy illustrated in panel A would
3 ADAPTIVE DYNAMICS 24
be stable in the terminology of dynamical systems. Moreover, evolu-
tionarily attracting strategies are also called convergence stable strategies
(Christiansen, 1991). Furthermore, Eshel and coworkers use the term con-
tinuously stable strategy (CSS) for a convergence stable ESS (Eshel and
Motro, 1981; Eshel, 1983; Eshel et al., 1997).
The pairwise invadability plots (figure 1) allow the graphical analysis
of the global evolutionary attractivity and global invadability of singu-
lar strategies. However, due to assumption 4 of small mutational steps,
even local evolutionary attractivity and invadability are sufficient for evo-
lutionary analysis. The local properties of the singular strategies may be
analyzed also algebraically based on the values of the second order partial
derivatives of the function r(smut, Eres) (Geritz et al., 1998). Let now s∗
be a singular strategy, i.e.,
D(s∗) =
[
∂r(smut, Eres)
∂smut
]
smut=sres=s∗
= 0.
If s∗ is a local fitness maximum in the environment set by the strategy s∗,
i.e., [
∂2r(smut, Eres)
(∂smut)2
]
smut=sres=s∗
< 0,
then s∗ is a locally uninvadable strategy (compare to Figure 1A). Simi-
larly, if this second order partial derivative is negative, then s∗ is a fitness
minimum in the environment set by the strategy s∗ Thus, it can be invaded
by any nearby strategy, which means that it is a branching point (compare
to Figure 1D). Monomorphic evolution to such fitness minimums is pos-
sible since, under frequency-dependent selection, each resident strategy
sres determines different environment Eres where fitness landscape expe-
rienced by a mutant with strategy smut, i.e, r(smut, Eres) considered as a
function of the mutant strategy smut, determines which mutants may in-
vade the resident population. However, once a mutant invades and re-
places the resident, it determines a new, different, fitness landscape. Fig-
ure 2 illustrates the way this process may lead either to a (local) fitness
maximum or to a (local) fitness minimum.
In monomorphic populations, the expected direction of evolution is
given by the sign of the local fitness gradient D(s) (see equation 3). For
singular strategies s∗ , the fitness gradient D(s∗) = 0. Furthermore, the
25 3.3 SCALAR-VALUED STRATEGIES
Evolution to an evolutionary stable strategy (ESS)
Evolution to a branching point
=⇒ Evolutionary steps =⇒
Figure 2: In each panel, the invasion fitness r(smut, Eres) (vertical axis)
is plotted with respect to the mutant strategy smut (horizontal axis) in the
environment set by a monomorphic resident population with the strategy
sres indicated by the vertical dashed line. In each panel, next evolutionary
step will be towards right, i.e., the resident strategy sres is replaced with
a mutant strategy smut such that smut > sres, until, in the rightmost panel,
a singular strategy is reached. On the upper row, this singular strategy is
a local fitness maximum, i.e., an ESS, and on the lower row, the singular
strategy is a local fitness minimum, i.e., a branching point.
3 ADAPTIVE DYNAMICS 26
sign of the fitness gradient in the neighborhood of s∗ can be deduced
from D′(s∗). Thus, the singular strategy is evolutionarily attracting, if
D′(s∗) < 0, and repelling if D′(s∗) > 0. Furthermore, the value of
D′(s∗) can be calculated as
D′(s∗) =
[
∂2r(smut, Eres)
(∂smut)2
−
∂2r(smut, Eres)
(∂sres)2
]
smut=sres=s∗
.
Moreover, if [
∂2r(smut, Eres)
(∂sres)2
]
smut=sres=s∗
is positive, then s∗ can invade neighborhood strategies. If any of these
expressions vanishes, the properties of the singular strategies must be de-
duced from higher order partial derivatives (based on Taylor-series ex-
pression of the invasion fitness function).
3.4 Evolutionary analysis of scalar-valued strategies in
polymorphic populations
So far, only monomorphic populations have been considered. The adap-
tive dynamics approach, however, applies to di- or polymorphic pop-
ulations as well. The algebraic tools provided by adaptive dynamics
are applicable, given that it is possible to find the attractor of the eco-
logical dynamics, be it an equilibrium or a periodic orbit. Even when
the population-dynamical attractor cannot be found algebraically, adap-
tive dynamics tools may still enable evolutionary analysis. If a stable
population-dynamical attractor exists, it can often be found by iterating
the ecological population dynamics sufficiently long. Once the attractor
has been found with numerical methods, the theoretical methods provided
by adaptive dynamics apply for evolutionary analysis.
Furthermore, the adaptive dynamics approach allows efficient evolu-
tionary simulations since the ecological model for the population dynam-
ics is specified, and thus, resource-consuming individual-based simula-
tions can be replaced with simulations that are built on the iteration of
the ecological dynamics of a polymorphic population together with in-
frequent insertions of new mutants, and removals of strategies that have
become rare enough to be considered extinct. In this thesis, all these tools
27 3.5 VECTOR-VALUED STRATEGIES
are being used: algebraic analysis, numerical analysis and simulations
based on iteration of ecological dynamics with rare randomly occurring
mutations.
3.5 Joint evolution of several traits (vector-valued
strategies)
One of the main topics of this thesis is to show that the joint evolution of
specialization and dispersal propensity may allow an initially monomor-
phic population to become trimorphic such that a generalist morph co-
exists with two specialist morphs. Studying the joint evolution of two
traits means that one has to consider vector-valued traits. Leimar (2001,
2005, 2009) has shown that, in this case, different mutational variance–
covariance structures and fitness interactions may crucially affect the evo-
lutionary dynamics.
In the case of one-dimensional traits and small mutations, the evolu-
tionary dynamics are rather simple: if the fitness gradient D(sres) is posi-
tive, only mutants with higher trait value may invade the resident strategy
sres, and the evolutionary path is qualitatively similar for any sequence of
successive mutations. In the case of two co-evolving traits, there are usu-
ally at least two qualitatively different types of mutants that may invade
the resident strategy.
Consider, for example, the joint evolution of dispersal propensity and
specialization. Then, in the absence of pleiotropy, the resident popula-
tion may be invaded either by mutants that differ only in the dispersal
propensity or by the mutants that differ only in the specialization strategy.
Even in this non-pleiotropic case the order of stochastic mutation events
may significantly affect the outcomes of evolution, and the evolutionary
dynamics are not, even qualitatively, independent of the mutation process
(Nurmi and Parvinen, 2011).
If pleiotropic mutations affecting simultaneously both the dispersal
propensity and specialization are possible, the set of mutant strategies
capable of invading the resident is notably larger. Furthermore, there
may be fitness interactions such that the sign of the invasion fitness of
a pleiotropic mutant cannot be deduced from the invasion fitnesses of the
non-pleiotropic mutants. For example in the case of joint evolution of dis-
persal propensity and specialization, biological intuition might let one to
3 ADAPTIVE DYNAMICS 28
expect that a mutant that is simultaneously both more specialized and less
dispersive might be able to invade a resident strategy that is uninvadable
against both strategies that differ only in the specialization strategy and
strategies that differ only in the dispersal propensity.
In this thesis, as well as in the analysis committed by Nurmi and Parvi-
nen (2011) pleiotropic mutations are assumed to be impossible. Since
even non-pleiotropic mutations are sufficient to enable the evolution to
the trimorphic coexistence of specialists and generalists, it is not neces-
sary to add in the full complexity of pleiotropic mutations even though
they may sometimes enable the emergence of additional biodiversity.
29 4 MECHANISTIC MODEL DERIVATION
4 Mechanistic derivation of ecological models
for the adaptive dynamics of resource use
The agenda of this section is to show how to derive metapopulation mod-
els that are suitable for the evolutionary analysis of resource usage. As
mentioned above, this process has to start from the individual level; here
the starting point is a continuous-time resource-consumer model with two
alternative resources. Geritz and Kisdi (2004) have shown that a simple
argument of time-scale separation allows one to derive from this model a
discrete-time model for the consumer population. Once the discrete-time
consumer population dynamics have been specified in a single well-mixed
population, lifting this model to the metapopulation level is just a ques-
tion of book-keeping, as has been shown by, e.g., Gyllenberg et al. (1997)
and Parvinen (2006).
The model derivation is followed by the derivation of the invasion
fitness function in these models. In order to calculate invasion fitness in
metapopulations, it is necessary to understand the calculation of invasion
fitness for well-mixed populations. Therefore, both of the calculations
will be presented here.
4.1 Discrete-time model for local population dynamics
The derivation of a discrete-time model for the well-mixed population is
based on the guidelines given by Geritz and Kisdi (2004). Their approach
applies to species that hatch at the beginning of season, use resources
from the environment to produce new eggs that also encounter mortality
during the breeding season. At the end of the breeding season, all of the
adults die and only a fraction of the eggs survives to the following season.
The other eggs are lost. For simplicity, it is also assumed that there is no
within-season mortality among the adults.
In the modeling technique of Geritz and Kisdi (2004), the details of
the continuous-time resource-dynamics determine the type of the discrete-
time consumer-dynamics. Below, the model derivation is presented in the
case of general resource-growth functions and a monomorphic consumer
population (all the consumers are identical). Later, the model is general-
ized to the case of several consumer types, and specific resource-growth
4 MECHANISTIC MODEL DERIVATION 30
functions are introduced in order to derive some well-known discrete-time
population models.
First, let the variables n ∈ N and t ∈ [0, 1] denote two different mea-
sures of time such that the discrete variable n determines the number of
year (or breeding season) whereas the continuous variable t determines
time within that season. Let nowA(i)n (t) be the availability of the resource
i at time t during season n, and let αiGi(Ai) be the density-dependent per
capita growth rate of the resource i, where Gi is assumed to be a decreas-
ing function.
Then the within-season continuous-time resource dynamics, in the ab-
sence of consumers, are
dA
(i)
n
dt
(t) = αiGi
(
A(i)n (t)
)
A(i)n (t). (4)
Assume now, that the resources are used by a monomorphic consumer
population with population density xn during the breeding season n. The
consumer population size xn is constant since it is assumed that the con-
sumers do not encounter within-season mortality, but they all perish at
the end of the breeding season. Assume further that consumers use the
resource i according to the law of mass-action with the rate βi, and the
consumed resources are converted to new eggs with efficiency γi. Now,
let the density of the eggs of the consumer at time t during season n be
Un(t) and assume that, during breeding season, the already oviposited
eggs are destroyed with rate δ. The eggs are identical, independent of the
resource usage of the consumer who produced the egg.
With these assumptions, it is possible to formulate the within-season
dynamics for a monomorphic consumer population as
ε
dA
(i)
n
dt
(t) = αiGi
(
A(i)n (t)
)
A(i)n (t)− βiA
(i)
n (t)xn
dUn
dt
(t) =
(
γ1β1A
(1)
n (t) + γ2β2A
(2)
n (t)
)
xn − δUn(t),
(5)
where ε is a small dimensionless scalar that allows one to assume that the
resource dynamics are fast enough (compared to consumer egg dynamics)
in order to assume that the resource densities are always at the stable
31 4.1 LOCAL POPULATION DYNAMICS
quasi-equilibrium value set by the current consumer population density
xn. This value,
Â(i)n = max
{
0, G−1i
(
βi
αi
xn
)}
, (6)
can be interpreted as the availability of the resource i during season n.
For some resource-growth functions, high consumer density may result
in negative values of G−1i
(
βi
αi
xn
)
. In these cases, the resource availabil-
ity diminishes (rapidly) until the resource has become completely absent
(exhausted), which means that this resource cannot be used for egg pro-
duction. Once a resource is exhausted, devoted specialists, utilizing solely
this resource, cannot produce any eggs, and thus perish over the winter.
If both resources are exhausted simultaneously, none of the consumers
can produce any eggs, which means that once the adult consumers die at
the end of the season, the entire population has perished. The exhausted
resource recovers at the beginning of the next breeding season given that
the consumer population has diminished sufficiently.
Now, the egg density obeys the linear differential equation,
dUn
dt
(t) =
(
γ1β1Â
(1)
n + γ2β2Â
(2)
n
)
xn − δUn(t). (7)
It is easy to find the solution of this equation:
Un(1) =
1− e−δ
δ
(
γ1β1Â
(1)
n + γ2β2Â
(2)
n
)
xn.
Now, assuming further that fraction σi of these eggs survives to next
season and hatches successfully, one can calculate
xn+1 = σiUn(1).
It is now possible to simplify the notation by defining a new compound
parameter
λi =
σiγi
δ
(1− exp(−δ)).
With this notation, one can write down the discrete-time model for the
consumer population as
xn+1 =
2∑
i=1
λixnβiÂ
(i)
n . (8)
4 MECHANISTIC MODEL DERIVATION 32
Next, consider the case of several consumers that are identical except
for the resource consumption rates β. Let j denote the consumer type and
let x(j)n be the type j consumer population density during the breeding
season n. Assume also that the type j consumers use the resource i ac-
cording to the law of mass action with rate βij. Furthermore, assume that
the other parameters in the resource–consumer model (5) are independent
of the consumer type. Then the resource dynamics for type i resource
become
ε
dA
(i)
n
dt
(t) = αiGi
(
A(i)n (t)
)
A(i)n (t)− A
(i)
n (t)
∑
m
βimx
(m)
n .
As above, it is possible to solve the quasi-equilibrium resource density
Â(i)n = max
{
0, G−1i
(∑
m βimx
(m)
n
αi
)}
. (9)
Once this value is known, the differential equation determining the egg
dynamics is the same as above (equation 7) and one obtains a general
discrete-time model with two resources for several consumers:
x
(j)
n+1 =
2∑
i=1
λiβijÂ
(i)
n x
(j)
n . (10)
In this equation, λi is a resource-specific parameter, and βij depends on
the consumer strategies but not on the consumer population sizes. Thus at
the level of ecological dynamics, they are constant parameters. Therefore,
if the resource availabilities Â(1)n and Â(2)n are known, the ecological dy-
namics (equation 10) are linear. Thus at time unit n, the environment set
by competing residents is determined by the resource availabilities, i.e.,
Eres(n) =
(
Â
(1)
n
Â
(2)
n
)
. (11)
If one now defines the fecundity function of type j consumers with strat-
egy sj as
f(sj, Eres(n)) =
(
λ1β1jÂ
(1)
n + λ2β2jÂ
(2)
n
)
, (12)
33 4.1 LOCAL POPULATION DYNAMICS
then the population model (10) can be written in the form
x
(j)
n+1 = f(s
j, Eres(n))x(j)n . (13)
In order to illuminate the differences between generalists and special-
ists, the resources are, in this thesis, assumed to be equivalent both in
nutritional values and in renewal rates, but possibly different in availabil-
ities, i.e., α1 = α2 = α, λ1 = λ2 = λ, but K1 can be different from
K2.
This mechanistically underpinned population model is, in slightly dif-
ferent forms (based on different resource growth functions), widely uti-
lized and analyzed in the articles 2, 3 and 4, in which it is generally as-
sumed, that the resource growth rate has been scaled such that α = 1.
However, in article 1 a different modeling approach is assumed in order
to create a model which is equivalent to the models of habitat specializa-
tion and habitat selection, but which underpins the differences between
habitats by varying resource availabilities.
Below, three different resource-growth functions and the three differ-
ent resulting discrete-time consumer population models are introduced.
The Beverton–Holt model
Let now the resources to have chemostat dynamics such that the internal
within-season growth rate of the resource population i equals α and car-
rying capacity of the resource equals Ki. Then the resource dynamics in
the absence of consumers (equation 4) are
dA
(i)
n
dt
(t) = αG
(
A(i)n (t)
)
A(i)n (t) = α
(
1−
A
(i)
n (t)
Ki
)
, (14)
This equation can be equally interpreted such that there is a constant in-
flux of the resource to the system with rate α and the resources decay
exponentially with rate α
Ki
. In this case (see equation 6),
Gi(A) =
(
1
A
−
1
Ki
)
and G−1i (x) =
1
1
Ki
+ x
.
The inverse function G−1i is always positive, which means that the quasi-
equilibrium resource density in the case of several consumers (equation
4 MECHANISTIC MODEL DERIVATION 34
9) can be written as
Â(i)n =
α
α
Ki
+
∑
m βimx
(m)
n
, (15)
and the between-season consumer dynamics (equation 10), with α = 1,
obey the difference equation
x
(j)
n+1 = λx
(j)
n
(
β1jK1
1 +
∑
mK1β1mx
(m)
n
)
+ λx(j)n
(
β2jK2
1 +
∑
mK2β2mx
(m)
n
)
,
(16)
which, in the case of one resource and one consumer, is the famous Bev-
erton and Holt (1957) model
xn+1 =
λβKxn
1 + βKxn
.
The discrete-time logistic model
Assume that the resources have logistic dynamics in the absence of con-
sumers (see equation 4), i.e.,
dA
(i)
n
dt
(t) = α
(
1−
A
(i)
n (t)
Ki
)
A(i)n (t) = αGi
(
A(i)n (t)
)
A(i)n (t),
which means that (see equation 6)
Gi(A) =
(
1−
A
Ki
)
and G−1i (x) = Ki(1− x),
of which the latter one is negative for large values of x.
Then, the quasi-equilibrium resource densities (equation 9) are
Â(i)n = max
{
0, Ki
(
1−
1
α
∑
m
βimx
(m)
n
)}
. (17)
This means that, if the consumer population becomes overly large, a re-
source may be exhausted. An exhausted resource cannot be used to the
35 4.1 LOCAL POPULATION DYNAMICS
production of new eggs. If both of the resources are exhausted simul-
taneously, the consumer population cannot produce any eggs, and thus
perishes over the winter. Once exhausted, the resource population is as-
sumed to recover immediately at the beginning of the following season.
Altogether, one now obtains a version of the truncated discrete-time
logistic model (May, 1976) for the consumer population (with α = 1):
x
(j)
n+1 = λK1x
(j)
n β1j max
{
0,
(
1−
∑
m
β1mx
(m)
n
)}
+ λK2x
(j)
n β2j max
{
0,
(
1−
∑
m
β2mx
(m)
n
)}
.
(18)
The Ricker model
Assume that the resource dynamics (equation 4) are, in the absence of
consumers, given by the Gompertz (1825) equation
dA
(i)
n
dt
= αLn
(
Ki
A
(i)
n (t)
)
A(i)n (t) = αGi
(
A(i)n (t)
)
A(i)n (t),
which means that
Gi(A) = Ln
(
Ki
A
)
and G−1i (x) = Ki exp(−x).
As in the case with the Beverton–Holt model (equation 16), the inverse
function is again always positive, and the quasi-equilibrium resource den-
sities (equation 9) can be written as
Â(i)n = Ki exp
(
−
∑
m βimx
(m)
n
α
)
.
Thus, one obtains the famous Ricker (1954) model that, in the case of two
resources and several consumer types (and α = 1), has the form
x
(j)
n+1 =λK1β1jx
(j)
n exp
(
−
∑
m
β1mx
(m)
n
)
+λK2β2jx
(j)
n exp
(
−
∑
m
β2mx
(m)
n
)
.
(19)
4 MECHANISTIC MODEL DERIVATION 36
4.2 Trade-off between the resource consumption rates
In the population models derived above, the resource usage of type j con-
sumers is determined by two consumption rates: β1j and β2j . It is char-
acteristic to the above models that the dynamics of the resources interact
only via shared consumers and, above all, consumers interact only via re-
source availabilities: the more there are consumers around, and the more
efficiently they use resources, the lower are the quasi-equilibrium values
Â
(i)
n of the resource densities, and the more efficient the consumers have
to be in using these resources in order to maintain viability. Moreover in
these population models, increasing usage rates of the resources do not
involve any additional costs, such as increasing exposure to predation.
Thus, if resource consumption rates β1j and β2j were to evolve freely
without any limitation, these rates would most likely encounter evolution
towards ever increasing values. Therefore, there is an obvious need for ex-
ternally determined limits for these rates, which is typical for all kinds of
specialization evolution, whereas for example, the evolution of dispersal
(as well as that of reproduction timing) takes place in the balance of in-
herent costs and benefits of dispersal: dispersal is necessary for long-term
survival of the species, but overly abundant dispersal causes unnecessary
risks and consumes resources.
Usually in evolution of specialization literature, as well as in this the-
sis, it is assumed that the growth of the resource consumption rates is
limited by a trade-off curve. Below this curve, mutations increasing both
of the consumption rates are possible, which means that, in the evolution-
ary process, the trade-off curve is reached rapidly. Thus on evolutionary
analysis, one can focus solely on the evolution along the trade-off curve.
(See figure 3B for examples of trade-off curves). On this curve, the better
an individual is in utilizing one resource, the worse it is in utilizing the
other, and any mutation increasing the consumption rate of one resource
must cause a decrease in the consumption rate of the other.
Assume now that the resource consumption rates are determined by
the strategy s ∈ [0, 1] of an individual. Assume also that the resource
consumption is symmetric in the sense that there exists a function β such
that, for type j individuals with strategy s, one can determine β1j = β(s)
and β2j = β(1− s). In other words, strategy s individuals use resource 1
with rate β(s) and resource 2 with rate β(1− s) (according to the law of
37 4.2 TRADE-OFF
mass-action). In algebraic analysis, the following assumptions are made:
1. Function β is strictly increasing, i.e., the more specialized an indi-
vidual is on a specific resource, the more efficiently the individual
can use this resource.
2. If nothing is invested to the use of a certain resource, nothing is
obtained from this resource, i.e., β(0) = 0.
Since in population-dynamical equations, the function β occurs al-
ways as a product with resource carrying capacities, fixing the maximum
value of β is just a matter of scaling these parameters appropriately, and
one can without loss of generality assume β(1) = 1. Now, the case s = 0
corresponds to a devoted specialist using only resource 2, and the case
s = 1 to a devoted specialist using only resource 1. The case s = 0.5
corresponds to an unbiased generalist.
In numerical explorations, it is necessary to fix the functional form of
the resource consumption function (the trade-off curve). In these cases, it
is assumed that
β(s) =
1− e−θs
1− e−θ
, θ 6= 0. (20)
This formula is not defined for θ = 0, but since limθ→0 β(s) = s it is nat-
ural to define β(s) = s when θ = 0. This trade-off function in illustrated
in Figure 3.
The trade-off parameter θ determines whether the resource consump-
tion function β is convex (θ < 0), concave (θ > 0), or linear (θ = 0).
In the case of concave resource consumption function, the resource con-
sumption function increases deceleratingly. This case is sometimes re-
ferred as the case of weak trade-off since a generalist can use resources
more efficiently than a linear combination of the two specialists, i.e.,
β(0.5) >
β(0) + β(1)
2
.
Analogously, in the case of convex resource consumption function, the
resource consumption function increases acceleratingly, i.e.,
β(0.5) <
β(0) + β(1)
2
,
4 MECHANISTIC MODEL DERIVATION 38
A) B)
0.2 0.4 0.6 0.8 1
0.5
1
θ=
0
θ=
3
θ=
−
3
s
β(s)
θ=
−
7
θ=
7
0.2 0.4 0.6 0.8 1
0.5
1
θ=
−
2
θ=
−
1
θ=
0
θ=
1
θ=
2
β(s)
β(1− s)
Figure 3: Trade-off curves.
Panel A: Resource consumption rate β(s) as a function of the specializa-
tion strategy s for different values of the trade-off parameter θ.
Panel B: Consumption rate of resource 2
(
β(1− s)
)
as a function of the
consumption rate of resource 1
(
β(s)
)
, i.e., the curves delimiting fitness
sets in the spirit of Levins (1962, 1963).
39 4.2 TRADE-OFF
which can be interpreted as strong trade-off. In the terminology used
by, e.g., White et al. (2006) and Hoyle et al. (2008, 2011), the case of
concave resource consumption function corresponds to a trade-off with
accelerating costs, and the case of convex resource consumption function
corresponds to a trade-off with decelerating costs.
The resource consumption function is the only ingredient in the model
presented here that has no mechanistic interpretation. Negative values of
θ are used to model phenomenologically the situations in which there
is an additional cost of generalism (or switching cost), whereas positive
values of θ correspond to cases in which there is an additional benefit of
generalism (switching benefit). The linear resource consumption function
(β(s) = s, θ = 0) is an important special case since it can be interpreted,
for example, as the search time allocation between the two resources.
In the literature considering the evolution of specialization, an as-
sumption that corresponds to assuming β(s) = sν , where ν > 0, is rather
usual (see, e.g., Egas et al. (2004); Rueffler et al. (2007); De´barre and
Gandon (2010) and Zu et al., (2011a)). However in the model described
above, this formulation would result in
lim
ŝ→0
[
dβ(s)
ds
]
s=ŝ
= ∞ and lim
ŝ→1
[
dβ(1− s)
ds
]
s=ŝ
= ∞, if 0 < ν < 1,
lim
ŝ→0
[
dβ(s)
ds
]
s=ŝ
= 0 and lim
ŝ→1
[
dβ(1− s)
ds
]
s=ŝ
= 0, if ν > 1,
which may generate artificial singularities extremely near to the borders
of the strategy space. With the formulation (20), one obtains resource
consumption functions that resemble the case of β(s) = sν , but avoids
these artificial singularities.
The family of resource consumption functions given by equation (20)
covers a wide range of qualitatively different ecological scenarios. How-
ever, the functions in this family are always either everywhere concave or
everywhere convex. Hence, e.g., the cases with sigmoidal trade-offs can-
not be covered. There are, however, methods in the adaptive dynamics
toolbox that are independent of the particular shape of the trade-off func-
tion (de Mazancourt and Dieckmann, 2004; Bowers et al., 2005; Kisdi,
2006; Geritz et al., 2007; Kisdi, 2014). These methods can, for exam-
ple, reveal ecological scenarios where evolutionary branching may occur.
From the point of view of evolution of specialization, it would be useful
4 MECHANISTIC MODEL DERIVATION 40
to further develop these methods such that they could be used to reveal
or exclude the possibility of the trimorphic coexistence of one general-
ist strategy with two different specialist strategies in the case of two re-
sources.
4.3 Metapopulation dynamics
Once the local dynamics are derived from the individual level processes
(section 4.1), one can build a structured metapopulation model based on
this local population model. Below the required assumptions and model
derivation are introduced in detail.
It is assumed that the landscape consists of an infinite number of large
local habitat patches that are prone to local catastrophes. However, there
is only a finite number of different patch types. Each patch can support a
local population. Patch types differ from each other only in the carrying
capacities of the two resources. In an individual patch, the local popula-
tion growth rate (fecundity) at time n is set by the resource availabilities
at that time. These availabilities are determined by the local population
sizes, specialization strategies of the consumers and the local resource
carrying capacities as explained in equation (9).
In equation (11) these availabilities were used to determine the envi-
ronment Eres(n) set by the resident at time-unit n. In metapopulations,
however, the environment set by the resident population is determined
at the metapopulation level. Instead, the local resource availabilities de-
termine Eresloc(n) the local environment set by the current local resident
population at time n, equally with the definition 11.
Now, in the absence of dispersal and catastrophes, one can write the
local dynamics of type j consumers with strategy s(j) in a patch of type
m in the form (compare with equation (13))
x
(j)
n+1 = f
m
(
s(j), Eresloc(n)
)
x(j)n .
In fact, once the resource availabilities are known, the fecundity function
fm is identical in all patch types m. However, the notations in the fitness
calculations are simpler when differences in the local growth, caused by
different resource carrying capacities, are denoted also explicitly in the
fecundity function.
41 4.3 METAPOPULATION DYNAMICS
Once an individual decides to emigrate, it is assumed to enter the pool
of dispersers. The dispersers that survive migration are distributed evenly
to all of the patches, independent of their origins. Each individual emi-
grates with probability e and survives migration with probability pi (inde-
pendent of e). Furthermore,Dn(s) denotes the average number of strategy
s dispersers per patch emigrated from the patches at period n (disperser
pool size of the strategy s dispersers).
During one time step, a single patch encounters a catastrophe with
probability c. These catastrophes occur independently in different patches.
When a catastrophe takes place, it wipes out the entire local population.
After a catastrophe a new local population is founded by dispersers from
the disperser pool. The order of events during a season is assumed to
be: Potential catastrophe destroying all the eggs in a patch – hatching –
emigration to the disperser pool – immigration from the disperser pool –
census – production of the new eggs in the patches.
With these assumptions and notations, the local dynamics of a type j
consumer population with strategy s(j) are
x
(j)
n+1 = C(n+ 1)(1− e)f
m
(
s(j), Eresloc(n)
)
x(j)n + piD
(j)
n (s), (21)
where C(n + 1) is a random variable determining the occurrence of the
catastrophes (see equation (2)), and
D(j)n (s) =
∑
m
pm
(
Expected number of strategy s(j) emigrants
from a type m patch at time n
)
.
(22)
Equations (21) and (22) form the metapopulation model that is analyzed
in this thesis.
42
43 5 INVASION FITNESS
5 Invasion fitness and environmental interac-
tion variable
5.1 Invasion fitness in well-mixed populations
Consider a k-morphic resident population where the resident strategies
are (s(1), s(2), . . . , s(k)). Suppose that the resident population has settled
to an attractor
(
X res1 , X
res
2 , X
res
3 , . . .X
res
n , . . .
)
, where each X resn comprises
the resident population sizes (x(1)n , x(2)n , . . . , x(k)n ) at time n. For each n,
the local population sizes together with resident strategies and resource
carrying capacities determine the environment Eres(n) set by the resident
population at time n. Consider now a negligibly small mutant popula-
tion with strategy smut. The small mutant population does not affect the
environment, and thus based on equation (13), the mutant population dy-
namics obey the linear difference equation
xmutn+1 = f(s
mut, Eres(n))xmutn . (23)
If, furthermore, the resident population dynamics have settled to a fixed
point attractor, then the environment set by the resident remains con-
stant (Eres(n) = Eres for each n), and the equation (23) becomes an
autonomous linear difference equation. This means that f(smut, Eres) is
equivalent to the basic reproduction ratio of the mutant population (Diek-
mann et al., 1990; Heffernan et al., 2005), and one can, in the spirit of
Metz et al. (1992), determine the invasion fitness of a rare mutant in
the environment set by the residents as (see, e.g., Mylius and Diekmann
(1995)).
r(smut, Eres) = ln
(
f(smut, Eres)
)
.
In principle, the generalization of this invasion fitness function to the
case of non-equilibrium dynamics is simple:
r(smut, Eres) = lim
t→∞
ln
 t
√√√√ t∏
i=1
f(smut, Eres(n))

= lim
t→∞
1
t
t∑
i=1
ln
(
f(smut, Eres(n))
)
.
(24)
5 INVASION FITNESS 44
In practice however, it is possible to calculate invasion fitness only in
the case of p-periodic resident population dynamics when p ∈ N. In this
case,
r(smut, Eres) =
1
p
p∑
i=1
ln
(
f(smut, Eres(n))
)
.
5.2 Invasion fitness in metapopulations
Defining fitness in metapopulations is not straightforward, since individ-
uals compete with each other in the local patches, but a trait combina-
tion that is extremely profilic locally, may be completely destroyed by a
catastrophe, if it fails to send out successful dispersers. Below, the cal-
culation of invasion fitness in metapopulations is presented in the case of
a monomorphic resident population. The generalization to polymorphic
residents is rather straightforward, but notationally more complex.
In metapopulation models, fitness must be determined at the level
of dispersers and local clans initiated by the dispersers (Gyllenberg and
Metz, 2001; Metz and Gyllenberg, 2001; Parvinen, 2006). Once a dis-
perser enters a local patch, it starts a new clan. This clan consists of the
disperser itself, its descendants, their descendants, etc. Due to clonal re-
production, all the individuals in the clan have the same strategy as the
initiating disperser. Depending on this strategy and the local conditions
in the patch, the clan may either die out due to the local competition in
the patch, or increase in population size until it is destroyed by the next
local catastrophe in the patch.
Each generation in the clan sends out new dispersers until the whole
clan, as well as the whole local population, is destroyed by a local catas-
trophe. The expected number of successful dispersers (i.e., initiated new
local clans) produced by a local clan can be interpreted as the basic re-
production number of the dispersers (or, equally, of the local clans), and
it can be used as a proxy for the invasion fitness as above in well-mixed
populations.
In metapopulation models, the environment set by the resident must
be determined at the metapopulation level. Throughout this thesis, the
ecological parameters are always chosen such that a metapopulation-level
quasi-steady state exists (Article 4 that considers non-equilibrium dynam-
45 5.2 FITNESS IN METAPOPULATIONS
ics focuses solely on the well-mixed populations). This means that, at
this quasi-equilibrium, the size D of the resident disperser pool is con-
stant. However, as long as local disasters may occur, the local population
sizes still vary. After a catastrophe a new local population is immedi-
ately founded by the immigrants from the disperser pool. This new lo-
cal population is, however, usually very small compared to the resident-
population’s fixed point size set by the resource carrying capacities, the
resident strategy and disperser pool size D. The local population size then
approaches a stable attractor until the next local disaster occurs. It is even
possible that the metapopulation level dynamics have a fixed-point attrac-
tor (constant D) even though the local dynamics have cyclic attractors
(Gyllenberg et al., 1993). However, in this thesis the main focus is on the
metapopulations with local dynamics of the Beverton–Holt type, where
population size always approaches a fixed point value monotonically.
In a metapopulation-dynamical equilibrium, the disperser pool size
and the distribution of local population sizes remain constant, although
the size of the local population in each patch varies. It may be crucial
for a mutant, whether it enters a patch that is almost empty after a recent
local disaster, or a patch where the local population size has already grown
large.
Let now R(smut, Eres) denote the expected number of new successful
dispersers sent out by an average local clan initiated by a strategy smut
disperser in an environment Eres set by the strategy sres resident.
Consider a monomorphic resident metapopulation that has settled to a
metapopulation-dynamical equilibrium with constant disperser pool size
D. Then, all patches of type m and age n (time elapsed since the latest
catastrophe in the patch) have the same population density xmn . It is easy
to iteratively calculate these densities from the equation
xmn+1 = (1− e)f
m(sres, Eresloc(n))x
m
n + piD, x
m
0 = piD, (25)
where Eresloc(n) is the local environment (resource availabilities) in the
patch under consideration determined by the local resident population
n time-units after the latest local catastrophe. Once the successive res-
ident population densities have been calculated using equation (25), it is
possible to further calculate the vector of successive local environmental
5 INVASION FITNESS 46
conditions set by this resident, i.e.,
Eresloc = (E
res
loc(1), E
res
loc(2), . . . , E
res
loc(n), . . .).
These local resource availabilities allow one to calculate iteratively
the dynamics of the mutant clan in this patch as a function of the time
elapsed since the latest catastrophe and the time elapsed since the foun-
dation of this clan. The local populations are assumed to be large (math-
ematically speaking infinite), which allows one, e.g., to neglect demo-
graphic stochasticity. Therefore, numbers xmn do not represent individuals
but some abstract units of population density. It is clearly impossible to
determine the size of a mutant clan consisting of only a single mutant in-
dividual (or a few mutant individuals) using these units. Fortunately, this
is not even necessary, since the mutant population is assumed to be small.
Thus, the resident population determines the density-dependent factors in
the mutant dynamics, i.e., one can neglect the changes in the resource
availabilities caused by the mutants as well as the effects of immigration
on the local population dynamics of the mutants. This means that the dy-
namics of a mutant clan are linear with growth set by the properties of the
patch and the resident population densities. Hence, one can use the rel-
ative size (actual size divided by the initial size) of the clan to determine
how many new successful dispersers a clan is expected to produce.
Let now a local mutant clan with strategy smut be founded in a type
m patch that has encountered its latest local catastrophe η0 time units
ago. Denote the relative size of this clan when η time units have elapsed
since the latest catastrophe by ymη0(η, s
mut, Eresloc), where Eresloc refers to the
local environment set by the resident population. Now η − η0 is the time
elapsed since the foundation of this clan.
It is now possible to solve ymη0(η, s
mut, Eresloc) from a linear difference
equation
{
ymη0(η + 1, s
mut, Eresloc) = (1− e)f
m(smut, Eresloc(η))y
m
η0
(η, smut, Eresloc),
ymη0(η0, s
mut, Eresloc) = 1.
Therefore
ymη0(η, s
mut, Eresloc) =
η−1∏
i=η0
(1− e)fm
(
smut, Eresloc(i)
)
. (26)
47 5.2 FITNESS IN METAPOPULATIONS
Now, ymη0(η, s
mut, Eresloc) is the size of the mutant clan given that there
are no local catastrophes. When calculating the expected number of dis-
persers produced by this clan, however, the catastrophes have to be taken
into account. Furthermore, the exact ordering of the events during season
has to be considered also. A local catastrophe destroys the clan along
with the entire local population. The clan founded η0 time units after the
latest local catastrophe is still alive η0 + 1 time units after the catastrophe
with probability 1− c, and at the census of that season it has relative size
ymη0(η0 + 1, s
mut, Eresloc) = (1− e)f
m(smut, Eresloc(η0)).
However, emigration takes place before census, especially before the size
of the clan has been diminished by the factor 1 − e. Therefore, the ex-
pected number of successful dispersers produced by the clan in the first
season after its foundation is
pie(1− c)fm(smut, Eresloc(η0)) = pie(1− c)
ymη0(η0 + 1, s
mut, Eresloc)
1− e
.
This reasoning can be generalized forward, and altogether, the mutant
clan is expected to produce
pie
1− e
∞∑
η=η0
(1− c)1+η−η0ymη0(η, s
mut, Eresloc)
new successful dispersers.
The probability that the clan is founded in a patch, where η0 time
units have elapsed since the latest local catastrophe, is (1− c)η0c, and the
probability that the clan is founded in a type m patch is pm, the fraction
of type m patches. Thus, one can calculate R(smut, Eres), the expected
number of new mutant clans initiated by an average strategy smut mutant
clan in an environment Eres set by the resident population, as
R(smut, Eres) =∑
m
pm
∞∑
η0=0
(1− c)η0c
(
epi
1− e
∞∑
η=η0
(1− c)1+η−η0ymη0(η, s
mut, Eresloc)
)
,
(27)
5 INVASION FITNESS 48
which simplifies into
R(smut, Eres) =
epic
1− e
∑
m
pm
∞∑
η0=0
∞∑
η=η0
(1− c)1+ηymη0(η, s
mut, Eresloc). (28)
Until now, it has been assumed that the sizeD of the resident disperser
pool is a known constant. Now, one can finally solve the actual value of
D from a fixed point equation
R(sres, Eres) = 1, (29)
since in the metapopulation-dynamical quasi-equilibrium, each success-
ful disperser has to produce on average exactly one new successful dis-
perser. In the case of a polymorphic resident population, one obtains one
fixed-point equation for each resident strategy, and the sizes of the disper-
sal pools of each strategy may, in principle, be solved from this equation,
but in practice, the actual calculation becomes rapidly overly cumbersome
as the amount of resident strategies increases.
Note, that even though the disperser pool size D is not explicitly in-
volved in the equation (29), it affects the values of the environmental
interaction variables Eresloc as it determines the local population density
distributions of the residents in each patch. As mentioned before, the in-
teraction variable contains the information about the nonlinear feedback,
and thus, the model becomes linear if its value is assumed to be known.
5.3 Environmental interaction variable and the
principle of competitive exclusion
The traditional interpretation of the principle of competitive exclusion
(Gause, 1934; Hardin, 1960; Levin, 1970; Armstrong and McGehee, 1980)
states that at steady state there cannot be more coexisting species (strate-
gies) than there are resources (or other limiting factors). In any model
with two distinct resources, including the current model, this would pre-
vent the coexistence of more than two different strategies. However, this
statement has already been shown incorrect in several occasions (see,
e.g., Wilson and Yoshimura (1994)). The modern version of this prin-
ciple (Diekmann et al., 2003; Mesze´na et al., 2006), however, states that
49 5.3 THE PRINCIPLE OF COMPETITIVE EXCLUSION
the maximum number of species (strategies) that can robustly coexist at
steady state is less than or equal to the dimension of the interaction vari-
able.
In a well-mixed population with a fixed-point attractor, the environ-
mental interaction variable only includes the equilibrium availabilities of
the two resources, and hence the maximum number of coexisting strate-
gies is limited to two. However, already in the case of a two-periodic
attractor in a well-mixed population, the interaction variable includes two
successive availabilities for each resource, and hence its dimension is
four. In metapopulation models with quasi-equilibrium dynamics, i.e.,
fixed disperser pool size even though local population sizes vary due to
catastrophes, the dimension of the interaction variable is, in principle, in-
finite. Thus, the principle of competitive exclusion sets no limits for the
coexistence of different strategies, even though the robust coexistence of
a continuum of strategies is still not possible (Gyllenberg and Mesze´na,
2005).
50
51 6 EVOLUTION OF RESOURCE SPECIALIZATION
6 Evolution of resource specialization
6.1 Ways to model specialization
As mentioned at the beginning of this thesis, evolution of specialization
affects the dynamics of virtually any other life-history trait. Therefore,
it has been studied within numerous frameworks. In this thesis, the fo-
cus is on the case of usage of two distinct resources that has been also
considered, e.g., by MacArthur and Levins (1964); Lawlor and May-
nard Smith (1976); Schreiber and Tobiason (2003); Ma and Levin (2006);
Rueffler et al. (2006, 2007) and Abrams (2012). However, evolution of
resource utilization has also been widely studied in the case of a single
resource with a continuously varying character (see, e.g., MacArthur and
Levins (1967); Dieckmann and Doebeli (1999); Kisdi and Geritz (1999);
Egas et al. (2005) and, for the case with several resources (Bu¨chi and
Vuilleumier, 2014)). This approach relates closely to the studies of niche
evolution (see, e.g., Roughgarden (1972, 1976); Abrams (1986); Kassen
(2002); Ackermann and Doebeli (2004); Holt (2009)).
Resource continuums have also been studied in the context of ecolog-
ical character displacement (see, e.g., Brown and Wilson (1956); Slatkin
(1980); Grant (1994); Doebeli (1996); Kawecki and Abrams (1999); Miz-
era and Mesze´na (2003)), where the main interest is the co-evolution of
two competing species or strategies. In the studies of character displace-
ment, it is usually assumed that there exists, in the environment under
consideration, a single optimal phenotype towards which the monomor-
phic population evolves. However, monomorphic populations or evolu-
tionary branching are not usually considered, but the main focus is on
the effects of interspecific competition to the evolution of two compet-
ing species, or strategies: How far from the optimal phenotype can the
phenotypes of the competing species be driven by the tendency to avoid
competition. The ecological character-displacement approach is rather
closely related to the theories of optimal foraging (see, e.g., MacArthur
and Pianka (1966); Schoener (1971); Charnov (1976); Oaten (1977); Pyke
(1984); Stephens and Krebs (1986)).
The evolution of specialization may also be approached from the point
of view of phenotypic plasticity (see, e.g., Via and Lande (1985); Moran
(1992); Scheiner (1993); van Tienderen (1997); Sultan and Hamish (2002)).
6 EVOLUTION OF RESOURCE SPECIALIZATION 52
In this case, the specialist strategies correspond to non-plastic phenotypes
utilizing one resource, whereas the generalist strategy exhibits phenotypic
plasticity being able to utilize both of the resource but less efficiently than
the specialists on these resources. The trade-off parameter θ, in this case,
measures how limited are the resource consumption abilities of the plastic
phenotype compared to the specialist phenotypes (DeWitt et al., 1998).
Biologically more specific models of the evolution of resource spe-
cialization have been constructed, e.g., for the analysis of evolution of
host specialization of parasites and phytophagous insects, where it is nat-
ural to interpret alternative hosts as different resources for the parasite
or phytophagous insect (see, e.g., Jaenike (1990); Joshi and Thompson
(1995); Fry (1996); Abrams and Kawecki (1999); Nosil (2002); Poulin
et al. (2006)). Moreover, in spatially heterogeneous models, different
types of suitable habitats may also be considered as resources (habitat spe-
cialization, see, e.g., Levins (1962, 1963); van Tienderen (1991); Fryxell
(1997); Kisdi (2002); Morris (2003))
Also, diverse modeling approaches have been used. Genetic models
(see, e.g., Taper and Chase (1985); Drossel and McKane (1999, 2000);
Bu¨rger (2002, 2005); Via (2002)) are able to treat different genetic archi-
tectures in detail but usually require one to use rather simple models for
the ecological dynamics. Phenotypic models of evolution (Lande, 1976;
Emlen, 1980) sometimes lack immediate genetic underpinnings but on
the other hand let one to study ecologically more complex systems. The
traditional approaches on the phenotypic modeling of evolution of spe-
cialization have included for example game theoretic models (see, e.g.,
Brown (1990); Parker and Maynard Smith (1990); Brown and Vincent
(1992); Hofbauer and Sigmund (1998)) and models using the adaptive
dynamics approach (see, e.g., Mesze´na et al. (1997); Parvinen and Egas
(2004); Ma and Levin (2006)).
6.2 Evolution of specialization in well-mixed populations
under equilibrium dynamics
Majority of mathematical models focusing on the evolution of specializa-
tion assumes a well-mixed population. As a consequence, the possible
evolutionary scenarios are, especially in the case of equilibrium popula-
tion dynamics, rather well-known. In the case of two different resources,
53 6.2 WELL-MIXED POPULATIONS
the most striking common feature of the evolutionary dynamics is the
importance of the trade-off: A strong trade-off between the abilities to
utilize the resources leads to specialist populations whereas weak trade-
off results in generalist populations. These are the evolutionary scenarios
observed when evolution is frequency-independent (Levins (1962, 1963),
but see also Rueffler et al. (2004)). However, when selection is frequency-
dependent and trade-off is moderately strong, it is possible that evolution
of a monomorphic population directs to increased generalism, but the gen-
eralist population undergoes evolutionary branching, and finally, the pop-
ulation comprises two different specialist strategies (see, e.g., Mesze´na
et al. (1997)).
The evolutionary scenarios observed in this thesis correspond to this
general overview. They are illustrated in Figure 4. Note that evolution-
ary branching illustrated in panel B requires that the ecological dynam-
ics are modeled such that the dimension of the environmental interaction
variable is at least two (so that selection is frequency-dependent). When
there are two resources and the ecological dynamics have an equilibrium
attractor, it is rather natural to the environmental interaction variable to
have two dimensions (two scalars, each describing the equilibrium avail-
ability of one resource). This is the case also in the models analyzed in
this thesis. Thus, the modern competitive exclusion principle by Mesze´na
et al. (2006) prevents the robust coexistence of more that two strategies.
However, Rueffler et al. (2006) have shown that there are natural ways
to model specialization also such that the interaction variable has only
one dimension, which excludes evolutionary branching and robust coex-
istence of any pair of strategies.
The effect of the trade-off strength (trade-off parameter θ) on the evo-
lutionary dynamics is summarized in Figure 5 that illustrates evolution-
ary bifurcation diagrams, where the evolutionary singular strategies of a
monomorphic population are plotted as a function of θ. It is noteworthy
that, the devoted specialist strategies may still maintain their evolution-
ary attractivity for a while, even though the unbiased generalist strategy
becomes evolutionarily attracting as the value of θ increases.
After evolutionary branching, the evolution of the dimorphic popula-
tion usually directs towards the combination of the two devoted specialist
strategies. This is the always case in this thesis when the ecological dy-
namics have fixed point attractors in a well-mixed population. However,
6 EVOLUTION OF RESOURCE SPECIALIZATION 54
St
ra
te
gi
es A B C D
0
0.5
1
150 300 0
0.5
1
400 800 0
0.5
1
150 300 0
0.5
1
150 300
Evolutionary time
Figure 4: Evolutionary scenarios under equilibrium population
dynamics.
Strategies present in the population as a function of the evolutionary time.
One unit of evolutionary time corresponds to one loop of the simulation
procedure depicted in the Appendix of Nurmi and Parvinen (2013). Thus,
it is only applicable for comparison between different simulations using
the same procedure.
Panel A: Concave resource consumption function (weak trade-off) –
Evolution leads to generalism.
Panel B: Weakly convex resource consumption function (moderately
strong trade-off) – Evolution of a monomorphic population leads to
generalism, where evolutionary branching takes place. The evolution
of a dimorphic population leads to the combination of the two devoted
specialist strategies.
Panels C and D: Strongly convex resource consumption function (strong
trade-off)– Evolution leads to the nearest devoted specialist strategy.
55 6.2 WELL-MIXED POPULATIONS
K1 = K2 = 1 K1 = 1.1, K2 = 1 K1 = 3, K2 = 1
−2 −1 0 1
0
0.5
1
θ
−2 −1 0−3 1
0
0.5
1
θ
−2 −1 0−3−4 1
0
0.5
1
θ
Figure 5: Singular strategies as a function of the trade-off parameter θ
when λ = 3 in the Beverton–Holt model (equation 16). Thin black curve
indicates evolutionary repellors, thick grey curve branching points and
thick black curve evolutionary endpoints. The arrows in the panels in-
dicate the expected direction of evolution in a monomorphic population.
The evolutionary bifurcation diagrams for other corresponding models,
e.g., discrete logistic model (equation 18) and Ricker model (equation
19) are qualitatively similar.
Zu et al. (2011a, 2011b) have shown that, for complicated trade-off struc-
tures, the evolution of the dimorphic population may lead to a dimorphic
singular strategy combination, in which the coexisting strategies are not
devoted specialists. Further branching, however, is not possible.
Altogether, there are four different endpoints for the specialization
evolution in the models studied in this thesis in well-mixed populations
under equilibrium dynamics. None of them, however, involves evolution
to the trimorphic coexistence of specialists and generalists. This in in
accordance with the majority of previous results, (see, e.g., Brown (1990);
Mesze´na et al. (1997); Parvinen and Egas (2004); Ma and Levin (2006);
Ravigne´ et al. (2009)). In this thesis, it is explored how this modeling
approach could be extended in order to allow evolution starting from a
monomorphic population to lead to trimorphic coexistence.
There are several models where the ecological coexistence of a gener-
alist strategy and two specialist strategies is possible (Wilson and Yoshimura,
1994; Kisdi, 2002; Abrams, 2006b). However, such coexistence may of-
ten be evolutionarily unstable. Even more rarely is such trimorphic coex-
istence evolutionarily attainable, i.e., reachable from an initially monomor-
phic population when mutations are assumed small and infrequent.
The possibility of ecological trimorphic coexistence was first demon-
strated in a model compiled by Wilson and Yoshimura (1994). However,
6 EVOLUTION OF RESOURCE SPECIALIZATION 56
Egas et al. (2004) showed that this coexistence is not evolutionary at-
tainable, and furthermore, evolution even destroys the coexistence. Later
on, trimorphic coexistence has been shown evolutionarily attainable un-
der cyclic resource dynamics (Abrams, 2006a,b), or when the assump-
tions concerning the consumer behavior are relatively restrictive (Egas
et al., 2004), or only in a narrow parameter domain (Kisdi, 2002). In spa-
tially heterogeneous model with spatially aggregated resources, distance-
limited dispersal may also allow evolutionarily attainable trimorphic co-
existence such that generalists live in the habitat boundaries (De´barre and
Lenormand, 2011; Karonen, 2011).
Below, different extensions of the well-mixed consumer population
model with two resources are introduced, that allow the evolution to the
trimorphic coexistence of specialists and generalists.
6.3 Evolution of specialization in the case of well-mixed
populations with non-equilibrium dynamics
In the context of dispersal evolution, the importance of non-equilibrium
ecological dynamics has been recognized for a long time. On one hand,
non-equilibrium population dynamics may forge dispersal and even en-
able evolutionary branching of dispersal strategies, but, on the other hand,
dispersal may stabilize population dynamics (Gyllenberg et al., 1993;
Holt and McPeek, 1996; Parvinen, 1999; Ronce, 2007). However, re-
cent results indicate that the type of population-dynamical attractor may
affect the evolution of other life history traits as well (White et al., 2006;
Geritz et al., 2007; Hoyle et al., 2011). Previous work on other traits
has shown that, under non-equilibrium population dynamics, evolution-
ary branching may be possible also in such ecological scenarios that do
not allow branching under equilibrium dynamics (Parvinen, 1999; White
et al., 2006; Hoyle et al., 2011). Thus, non-equilibrium dynamics may
result in enhanced biodiversity.
In the model analyzed in this thesis, evolutionary branching is possi-
ble already under equilibrium dynamics. However, non-equilibrium dy-
namics may still add in diversity by allowing a secondary evolutionary
branching to occur, which results in the trimorphic coexistence of gener-
alists and specialists. Furthermore, non-equilibrium dynamics may result
in evolutionary suicide. Below, these evolutionary scenarios enabled by
57 6.3 NON-EQUILIBRIUM DYNAMICS
the non-equilibrium dynamics are presented by illustrating results of evo-
lutionary simulations.
In order to illuminate how the population dynamics affect the evo-
lutionary dynamics, one needs to illustrate the population-dynamical at-
tractors during the evolutionary time together with the evolutionary tree
in the strategy space. However, the evolutionary simulations are never
completely mutation limited. Instead, the population is, in practice, al-
ways polymorphic during the simulation. Therefore, in order to illustrate
the population-dynamical attractor of the entire population, one needs to
calculate how much extant strategies use resources, which in turn al-
lows one to calculate the availabilities of the resources. If strategies
(s(1), s(2), . . . , s(k)) are present at time unit n with corresponding popu-
lation sizes (x(1)n , x(2)n , . . . , x(k)n ), then the availabilities of the resources
1 and 2 are for the case with logistic ecological dynamics, according to
equations 9 and 17, respectively
Â
(1)
n = K1 max
(
0, 1−
∑k
i=1 β(s
(i))x
(i)
n
)
Â
(2)
n = K2 max
(
0, 1−
∑k
i=1 β(1− s
(i))x
(i)
n
)
.
(30)
When the population is on a non-equilibrium attractor, these availabil-
ities fluctuate as the consumer population sizes fluctuate. Based on these
availabilities, it is often possible to deduce the type of the population-
dynamical attractor of the consumer population as a whole. For example
in the case with two equally abundant resources, if the population is on
a two-periodic in-phase orbit, the sum of the resource availabilities takes
two different values on the population-dynamical attractor whereas their
difference is close to zero. If the population is on a two-periodic out-
of-phase orbit (asymmetric attractor), the differences alternate between
a positive and a negative value on the population-dynamical attractor
whereas the sum remains virtually constant. More generally: the more
asynchronous are the resource fluctuations, the larger are the absolute val-
ues of the differences in the resource availabilities.
Evolution to singular dimorphic strategy pairs
Under non-equilibrium dynamics, dimorphic evolution may lead to sin-
gular strategy pairs instead of pairs of devoted specialist strategies even
6 EVOLUTION OF RESOURCE SPECIALIZATION 58
when the trade-off function is everywhere convex (concave). On one
hand, Nurmi and Parvinen (2013) showed that, under in-phase oscilla-
tions of the resource availabilities, a dimorphic population usually evolves
towards the combination of the two devoted specialist strategies when pa-
rameter values are such that a monomorhic population evolves to gen-
eralism where evolutionary branching takes place. On the other hand,
asynchronous oscillations of the resource availabilities may benefit gen-
eralists, since the generalists experience less variance in the resource in-
take.
Evolutionary dynamics, in this case increasing specialism, may cause
attractor switches to the ecological dynamics. Due to the effects intro-
duced above, these attractor switches may stop the dimorphic evolution
to a singular dimorphic strategy pair as illustrated in the Figure 6. Fur-
thermore, under chaotic population dynamics, it is even possible that
the stochastic mutations, even though they are small in effect, induce
attractor-switches in the ecological dynamics. These attractor switches
may sometimes generate evolutionary fluctuations (illustrated in Nurmi
and Parvinen (2013)).
Evolution to the trimorphic coexistence of a generalist with two
specialist
Evolution starting from a monomorphic population may, under non-equi-
librium population dynamics, lead to the trimorphic coexistence of a gen-
eralist and two specialists strategies. In such coexistence, each of the spe-
cialists uses the corresponding resource more efficiently than the compet-
ing strategies. The viability of the generalist strategy, on the other hand,
is based on the asynchronous non-equilibrium population dynamics of the
specialists. The population sizes of the specialist strategies fluctuate, and
hence they are repeatedly rather low, which means that the correspond-
ing resource is abundantly available allowing the generalist to increase
in population size. This phenomenon was originally observed by Abrams
(2006b,a) in a continuous-time model involving Holling type II functional
response in the case where the dynamics of the two resources are differ-
ent, which creates sufficient asynchrony to the resource dynamics. How-
ever, non-linear functional response is known to have an essential part in
allowing species coexistence, e.g., several species can coexist even on a
59 6.3 NON-EQUILIBRIUM DYNAMICS
A) Strategies present during the evolutionary time
St
ra
te
gi
es
replacements
0.5
1
1000 2000
R
es
o
u
rc
e
av
ai
la
bi
lit
ie
s
B) Sum of the resource availabilities
2
3
1000 2000
C) Difference between the resource availabilities
0.3
1000 2000
D) Availability Â(1)n of resource 1
2
1000 2000
Evolutionary time
Figure 6: The result of an evolutionary simulation leading to a di-
morphic singular strategy pair under periodic population dynamics
in the logistic model (equation 18).
Panel A: Strategies present in the population as a function of the evolu-
tionary time. One unit of evolutionary time corresponds to one loop of
the simulation procedure depicted in the Appendix of Nurmi and Parvi-
nen (2013). Thus, it is only applicable for comparison between different
simulations using the same procedure.
Panels B, C, and D: Resource availabilities Â(1)n and Â(2)n as defined in
equation (30) as a function of the evolutionary time. For each evolution-
ary time unit, Panel B illustrates the sum of the resources availabilities
during each step on the population-dynamical attractor. Panel C illustrates
the differences of the resource availabilities and panel D the availability
of resource 1.
Parameters: K1 = K2 = 3.5, θ = −0.1, α1 = α2 = 1, λ1 = λ2 = 1.
6 EVOLUTION OF RESOURCE SPECIALIZATION 60
single resource under non-equilibrium dynamics (Armstrong and McGe-
hee, 1980; Kisdi and Liu, 2006; Geritz et al., 2007; Tachikawa, 2008).
In the models analyzed in this thesis, consumers use resources accord-
ing to the law of mass-action with a linear functional response (Holling
type I functional response). Furthermore, evolution to trimorphic coex-
istence is possible also in the case of similar resources since the asyn-
chronous fluctuations in the resource availabilities may be generated solely
by the over-compensatory consumer population dynamics. Thus, the re-
sults indicate that non-equilibrium population dynamics really is the main
factor enabling evolution to trimorphic coexistence.
Figure 7 illustrates an example of an evolutionary simulation lead-
ing to trimorphic coexistence. There, the population first evolves to gen-
eralism, where evolutionary branching occurs. After branching, the di-
morphic population ”inherits” its population-dynamical attractor from the
preceding monomorphic population Geritz et al. (2002). Thus, the di-
morphic population is initially on an in-phase two-periodic orbit. There-
fore, the dimorphic population evolves initially towards the coexistence
of the two devoted specialists. However, as the branches specialize fur-
ther, their ecological dynamics undergoes a series of period-doubling bi-
furcation which leads to chaotic ecological dynamics, which breaks the
synchronism in the dynamics of the two morphs. Finally, the population-
dynamical attractor becomes an out-of-phase two-periodic orbit, where
the evolutionary dynamics lead to a singular dimorphic strategy pair, which
is not uninvadable, and thus, a secondary evolutionary branching occurs
and leads to trimorphic coexistence.
Evolutionary suicide and branching–extinction cycles
The possibility of evolutionary suicide relates to one peculiarity of the
discrete-time version of the logistic population model: if the resources
are abundant and the consumers use them efficiently, it is possible that
one or both of the resources become exhausted. It is not possible to pro-
duce new eggs by utilizing an exhausted resource. Thus, if both resources
are exhausted simultaneously, all the consumers die out even though the
resources recover later. If one of the resources becomes exhausted, all the
devoted specialists utilizing this resource die out. Both of these scenarios
may lead to evolutionary suicide and the latter one even to evolutionary
61 6.3 NON-EQUILIBRIUM DYNAMICS
A) Strategies present during the evolutionary time
St
ra
te
gi
es
replacements
0.5
1
1000 3000
R
es
o
u
rc
e
av
ai
la
bi
lit
ie
s
B) Sum of the resource availabilities
1
3
5
1000 3000
C) Difference between the resource availabilities
-
1
1
3
1000 3000
Evolutionary time
Figure 7: The result of an evolutionary simulation leading to the co-
existence of generalist and specialists in the logistic model (equation
18).
Panel A: Strategies present in the population as a function of the evolu-
tionary time. One unit of evolutionary time corresponds to one loop of
the simulation procedure depicted in the Appendix of Nurmi and Parvi-
nen (2013). Thus, it is only applicable for comparison between different
simulations using the same procedure. Initial population is monomorphic
practicing strategy s = 0.1. Simulation ended in a trimorphic population
practicing strategies s1 = 0, s2 = 0.5, and s3 = 1.
Panels B and C: Resource availabilities Â(1)n and Â(2)n as defined in equa-
tion (30) as a function of the evolutionary time. For each evolutionary
time unit, Panel B illustrates the sum of the resources availabilities during
each step on the population-dynamical attractor. Panel C illustrates the
differences of the resource availabilities.
Parameters: K1 = K2 = 3.8, θ = −0.72, α1 = α2 = 1, λ1 = λ2 = 1.
6 EVOLUTION OF RESOURCE SPECIALIZATION 62
branching–extinction cycles (see Nurmi and Parvinen (2013) for illustra-
tions). However, in the latter scenario, the possibility of evolutionary sui-
cide in evolutionary simulations depends on the details of the simulation
procedure (see the Appendix of Nurmi and Parvinen (2013)).
An overview of the evolutionary dynamics under non-equilibrium
population dynamics
A concise overview of the evolutionary dynamics under non-equilibrium
dynamics is presented by the way of evolutionary bifurcation diagram in
Figure 8. Since the current adaptive dynamics toolbox suffices only for
algebraic analyses of the cases with equilibrium or periodic population
dynamics, the evolutionary bifurcation diagram has to be complemented
by illustrating the endpoints of evolutionary simulations that are based on
procedure presented in the Appendix of (Nurmi and Parvinen, 2013).
In Figure 8, if the trade-off is sufficiently strong (θ . −2.4), the
population always evolves to a monomorphic specialist population with
chaotic population dynamics. If −2.4 . θ . −1.7, the initial strategy of
the population determines, whether the population evolves to a monomor-
phic specialist population, or, via evolutionary branching, to a dimorphic
population comprising two devoted specialist strategies. If −1.7 . θ .
−0.87, evolutionary branching occurs independent of the initial strategy,
and evolution leads to a dimorphic combination of the devoted specialist
strategies. When−0.86 . θ . −0.585, one observes evolution to trimor-
phic coexistence. In the parameter domain −0.585 . θ . 0, evolution
leads to dimorphic singular strategy pairs and evolutionary fluctuations
caused by attractor switches of the chaotic ecological dynamics.
When 0 . θ . 0.35 , evolution of specialization ends in a monomor-
phic unbiased generalist population. If 0.35 . θ, evolution still directs
towards generalism, but increasing benefit obtained from generalism to-
gether with high resource carrying capacities finally results in overly ex-
tensive resource usage exhausting both resource simultaneously, which
results in evolutionary suicide, and thus, evolutionary simulations end to
extinction at the boundary of the black area indicating unviable strategies
in the evolutionary bifurcation diagram.
63 6.3 NON-EQUILIBRIUM DYNAMICS
St
ra
te
gi
es
- -
0.5
−3 0
1
11 22
⋄⋄⋄
⋄⋄
⋄⋄
⋄⋄
⋄
⋄
⋄
∗
∗
∗
∗∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗∗
∗∗
∗
∗
∗∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗∗
∗
∗
∗
†
††††
††
†
†††
†
††
††
†
†
θ
Figure 8: Evolutionary bifurcation diagrams in the case of possibly
non-equilibrium population dynamics in the logistic model (equation
18). Singular strategies and endpoints of evolutionary simulations as a
function of the trade-off parameter θ when specialists have chaotic popu-
lation dynamics. Biased usage of two resources may stabilize population
dynamics, but high benefit of generalism enables chaotic dynamics and
even evolutionary suicide.
Thin black curve indicates evolutionary repellors and thick grey curve
branching points. The arrows indicate the expected direction of evolution
in a monomorphic population. In the black-colored parameter domain,
the population is not viable. In the grey-colored parameter domain, the
monomorphic population dynamics are (nearly) chaotic.
The evolutionary simulations are initiated in a monomorphic population
with random strategy. If an evolutionary simulation ends in a monomor-
phic population, the end-strategy is denoted by ⋄. If it ends in a dimorphic
or polymorphic population, the strategies comprising the endpoint are de-
noted by ∗-signs. If evolutionary suicide occurs, the last viable strategy is
denoted by †-sign.
Other parameter values: K1 = K2 = 3.8, α1 = α2 = 1, λ1 = λ2 = 1.
6 EVOLUTION OF RESOURCE SPECIALIZATION 64
Altogether, Figure 8 illustrates the qualitative overview of the evolu-
tionary dynamics in the discrete-time logistic population model with non-
equilibrium dynamics. The cases with local dynamics determined by the
Ricker (1954) or Hassell (1975) models are qualitatively almost similar.
In all of these models, there exists a parameter domain where, in the di-
morphic population, a secondary evolutionary branching occurs, and the
population starts to evolve towards trimorphic coexistence. According
to the observations of Nurmi and Parvinen (2013), in the logistic model
the evolution in this case always ends to trimorphic coexistence. Evo-
lution to trimorphic coexistence is possible in Ricker and Hassell mod-
els as well, but in addition, there exists also a parameter domain, where
the appearance of the generalist strategy affects the ecological conditions:
As the generalist strategy becomes more unbiased and more common,
it synchronizes the resource dynamics such that the generalist itself be-
comes unviable and goes extinct. Thus, the population becomes dimor-
phic again making way to a new evolutionary branching and evolutionary
cycles (Nurmi and Parvinen, 2013).
6.4 Spatially heterogeneous models for the evolution of
specialization and the joint evolution of dispersal
propensity and specialization
Spatial heterogeneity usually makes mathematical models more difficult
to analyze. A temptingly simple approach to include spatial aspects into
the models is to resort to individual-based models where the current loca-
tion defines the resource availabilities of an individual (Kawata, 1996;
Doebeli and Dieckmann, 2000, 2004). However, even though the in-
creasing computational power of modern computers allows ever larger
and more detailed models to be analyzed, it is always easier to extract
patterns from these models when they can be backed up by theoretical
models.
Another simple way to add spatial heterogeneity is to study systems
where two (or several) patches are connected by dispersal (van Tienderen,
1991; Wilson and Yoshimura, 1994; Abrams, 1999; Kisdi and Geritz,
1999; Abrams, 2000a; Day, 2000). This approach is, naturally, not able
to include the possibility of local catastrophes, even though Kisdi (2002)
65 6.4 METAPOPULATION MODELS
included the possibility of ”good” and ”bad” years independently in each
patch.
Majority of theoretical studies considering evolution of specialization
in spatially heterogeneous environments has taken place within the con-
text of evolution of habitat specialization (Levins, 1962, 1963; van Tien-
deren, 1991; Brown and Pavlovic, 1992; Kisdi, 2002), where it is gener-
ally assumed that there is a trade-off between individual’s performances in
two different environments or habitat patch types. The results published
by Gyllenberg & Metz (Gyllenberg and Metz, 2001; Metz and Gyllen-
berg, 2001) provided tools that allowed Parvinen and Egas (2004) to con-
sider habitat specialization even within the framework of structured meta-
population models with local catastrophes and infinitely many patches
(but only two different patch types).
The habitat specialization approach has two obvious weaknesses. First
of all, it usually limits to the cases with only two different patch types and,
secondly, it does not determine the origins of the differences between the
patch types, and thus, prevents the mechanistic derivation of the local
population dynamics. In this thesis, the derivation of the metapopulation
theory for the evolution of specialization, initiated by Parvinen and Egas
(2004), is continued.
Nurmi et al. (2008) rationalize the differences between the patch types
on the basis of different resource availabilities and focus on the evolu-
tion of resource utilization. The modeling approach used by Nurmi et al.
(2008) resembles the habitat specialization or habitat usage models, but
enables the inclusion of several patch types, which, intuitively, might fa-
cilitate the coexistence of specialists and generalists.
However, price paid for the conformity with habitat usage models in
the model of Nurmi et al. (2008) is that the local resource dynamics within
the habitat patches are omitted, and the resource availabilities simply de-
termine, for each patch, a”quality” that delineates the local population
dynamics, but unfortunately does not enable mechanistic derivation of
the local population dynamics starting from the individual level. Thus,
even though the model by Nurmi et al. (2008) was a step forward from
the habitat usage models, there was still, in terms presented by Rueffler
et al. (2006), an obvious need for the evolutionary analysis of the models
that take into account the local resource–consumer dynamics. This defi-
ciency was fixed by Nurmi and Parvinen (2008), who analyzed the model
6 EVOLUTION OF RESOURCE SPECIALIZATION 66
introduced in section 4.
Both Nurmi et al. (2008) and Nurmi and Parvinen (2008) considered
evolutionary effects of various ecological parameters and sought for pos-
sibilities for trimorphic coexistence. When studying the evolutionary ef-
fects of ecological parameters, it was found that the relation between dis-
persal propensity and the evolution of specialization is rather complicated
and can even be counterintuitive. This prompted Nurmi and Parvinen
(2011) to study the joint evolution of specialization and dispersal propen-
sity. This study revealed several mechanisms that enable evolution to
trimorphic coexistence, and moreover, Nurmi and Parvinen (2011) en-
abled the analysis of the evolutionary effects of various ecological pa-
rameters in a setting where the dispersal propensity is, instead of a preas-
sumed value, assumed to have evolved to the corresponding evolutionarily
singular (attracting and uninvadable) value. Nurmi and Parvinen (2008)
showed also that evolution of specialization may, in metapopulations, end
to a dimorhic singular strategy pair. This may, as presented by Nurmi and
Parvinen (2011), be an important step on the path to trimorphic coexis-
tence.
When analyzing the joint evolution of dispersal propensity and spe-
cialization, one needs to understand also the evolution of dispersal propen-
sity. Therefore, at this point, a small interlude introducing the main as-
pects of dispersal evolution is necessary.
Evolution of dispersal propensity
Its rather typical that the evolutionary dynamics of dispersal have only a
single evolutionarily singular dispersal propensity, which is always evo-
lutionarily attracting (see, e.g., Johnson and Gaines (1990); Levin et al.
(2003) and Ronce (2007)). This is the case also in the models studied
in this thesis. The numerical value of this propensity is primarily deter-
mined by the catastrophe probability c and the probability pi of surviving
dispersal. The higher is the probability pi, the higher is the singular dis-
persal propensity. When there remains any risk of dispersal (pi < 1),
the catastrophe probability affects the singular dispersal propensity in a
non-monotonous way: in the absence of catastrophes (c = 0), all local
populations stay at the fixed point of the local dynamics, and thus, the
strategy not to disperse is an evolutionarily attracting singular strategy, as
67 6.4 METAPOPULATION MODELS
proved by Parvinen (2006) for the class of structured discrete-time meta-
population models studied in this thesis. As the catastrophe probability
increases, the singular dispersal propensity increases in the beginning,
too. This is due to the fact that catastrophes result in empty patches,
which make dispersal profitable. As the catastrophe probability increases
further, most individuals find themselves in sparsely populated patches
with plenty of resources. This decreases the advantages of dispersal and
causes the singular dispersal propensity to diminish. The value of the sin-
gular dispersal propensity reaches zero again at the threshold where the
metapopulation loses its viability. This phenomenon has been observed
also by, e.g., Ronce et al. (2000); Gyllenberg et al. (2002); Parvinen et al.
(2003) and Parvinen (2006). In this thesis, the focus is mainly on the pa-
rameter domain in which the singular dispersal propensity appears as an
increasing function of the catastrophe probability.
Various mechanisms resulting in evolutionary branching or polymor-
phisms of dispersal have been observed in different metapopulation mod-
els. These mechanisms include temporal variation in form of cyclic (Doe-
beli and Ruxton, 1997; Parvinen, 1999) or chaotic (Holt and McPeek,
1996) local population dynamics, or temporally and spatially varying car-
rying capacities (McPeek and Holt, 1992; Mathias et al., 2001). However,
catastrophes alone, have been observed not to create enough temporal
variation to promote branching. For example, Gyllenberg et al. (2002)
did not find evolutionary branching in a structured metapopulation model
defined in continuous time with one patch type. Parvinen (2002) stud-
ied the corresponding model with several patch types, and observed that
catastrophes together with spatial heterogeneity in the sense of different
patch types can result in evolutionary branching of dispersal. The nec-
essary level of spatial heterogeneity can be obtained with differences in
growth conditions alone, as well as with differences in catastrophe rates
alone. A similar observation in a metapopulation model with small local
populations, and therefore, locally stochastic population dynamics, was
made by Parvinen et al. (2003) (one patch type) and Parvinen and Metz
(2008) (several patch types).
Parvinen (2006) studied a discrete-time metapopulation model and
found another additional mechanism, which can together with catastro-
phes result in evolutionary branching. Even though all local populations
would eventually reach an equilibrium population size, if they are not
6 EVOLUTION OF RESOURCE SPECIALIZATION 68
hit by a catastrophe, this convergence to the equilibrium can be non-
monotonous due to overcompensation in the local discrete-time dynam-
ics, such as in the Ricker model. Parvinen (2006) observed that such tem-
poral heterogeneity together with catastrophes can result in evolutionary
branching of dispersal.
In this thesis, the joint evolution of dispersal propensity and special-
ization is explored only in the case where local population dynamics of the
metapopulation are of the Beverton-Holt type, where convergence to the
population-dynamical equilibrium is monotonous. Therefore the mecha-
nism for evolutionary branching of dispersal observed by Parvinen (2006)
is not present here. Thus, the effects of non-equilibrium dynamics to this
joint evolution remains an interesting question for the future research.
In the case of Beverton–Holt-type local dynamics, the evolutionarily
singular dispersal propensity is in most cases uninvadable by mutants fea-
turing a different dispersal propensity. In accordance with the reasoning
above, Nurmi and Parvinen (2011) observed evolutionary branching of
dispersal, if individuals encounter a sufficient amount of spatial hetero-
geneity in the sense of different patch types (Parvinen, 2002).
The living conditions of generalists in a specific patch are determined
by the overall availability of the two resources, whereas the living con-
ditions of specialist are determined solely by the availability of a single
resource. With spatially and temporally varying resource availabilities,
the former naturally presents less spatial variance that the latter. There-
fore, the evolutionary branching of dispersal propensity may be impos-
sible in a generalist population even though it is possible in a specialist
population under otherwise similar ecological conditions. Especially, an
unbiased generalist regards the two resources as identical and therefore it
observes no difference between two patches with swapped resource car-
rying capacities (K11 = K22 and K12 = K21 ). Thus, Nurmi and Parvinen
(2011) conjectured that evolutionary branching of dispersal is not possi-
ble in a metapopulation comprising unbiased generalist individuals in an
environment comprising two patch types with swapped carrying capac-
ities. For a specialist, evolutionary branching of dispersal propensity in
such an environment is possible.
Evolution of dispersal is, in a sense, an inviting field of research, since
externally determined trade-offs are not necessary but the evolution of
dispersal always takes place in the balance between the costs and benefits
69 6.4 METAPOPULATION MODELS
of dispersal (risks and costs of dispersal versus the benefits gained from,
e.g., the colonization of new areas (Hamilton and May, 1977)). Hence,
there has been a wide range of research focusing in the evolution of dis-
persal. However, when both dispersal and ecological specialization may
evolve, the mathematical models become notably more complex. Thus,
there have been only a few studies exploring this area (Kisdi, 2002; Han-
ski and Heino, 2003; Heinz et al., 2009; Scheiner et al., 2012).
(Scheiner et al., 2012) analyze the joint evolution of phenotypic plas-
ticity and dispersal using individual-based simulations. They do not ob-
serve evolutionary branching of dispersal, which is the key ingredient of
all the non-trivial results of Nurmi and Parvinen (2013). Instead in their
results, high dispersal is always accompanied with phenotypic plasticity
whereas low dispersal leads to genetic differentiation, especially in the
presence of cost of plasticity. Hanski and Heino (2003) have carried out a
simulation-based case study on the evolution of dispersal and host-plant
preference (specialization) among Glanville fritillary butterflies (Melitaea
cinxia). Their model is parametrized on the basis of observing the actual
metapopulation in the A˚land Islands in south-western Finland. This field-
biologically inclined approach differs notably from the approach of this
thesis, where the aim is to explore the biologically realistic parameter
domain in order to find different possible evolutionary scenarios. Heinz
et al. (2009) have studied the joint evolution of dispersal distance and lo-
cal adaptation in an environment with a continuously varying character by
means of individual-based simulation models both with clonal and sexual
reproduction. Their viewpoint is different from the viewpoint of this the-
sis, but noteworthily in their model, predictions based on asexual model
are, qualitatively speaking, principally consistent with the predictions de-
rived from the sexual model. Kisdi (2002) explores a two-patch model in
which the evolving traits are dispersal propensity and the adaption to the
local conditions in different patches. Compared to the metapopulation
models with local catastrophes, she assumes rather mild temporal vari-
ations: ”good” and ”bad” years that occur randomly and independently
in each patch. These temporal variations are not influential enough to
allow selection for high dispersal. Thus, a high degree of dispersal or
generalism usually appeared only as a response to the competition with
low-dispersal specialists. Kisdi (2002) also observes evolution to the tri-
morphic coexistence of the specialists and generalists, but only on an ex-
6 EVOLUTION OF RESOURCE SPECIALIZATION 70
tremely narrow parameter domain.
Evolution to a singular dimorphic strategy pair
In a well-mixed population with globally convex (concave) trade-off func-
tions (given by, e.g., equation 20), the usage of two resources evolves to
generalism with concave trade-off curves (θ > 0), and to devoted special-
ism with convex trade-off curves (θ < 0). Under frequency-dependent
selection, it is also possible that evolutionary branching occurs and the
population evolves to the combination of the two devoted specialists.
However, evolution to any other singular dimorphic strategy combination
requires rather complex trade-off structures (Zu et al., 2011a, 2011b).
Figure 9A illustrates two evolutionary scenarios brought in by the
metapopulation structure (when only the specialization strategy evolves).
On one hand, a metapopulation structure may enable evolutionary branch-
ing also when it is not possible in well-mixed populations (θ > 0, i.e.,
weak trade-off). Note that, sometimes, spatial structure may also inhibit
diversification (Day, 2000, 2001). On the other hand, Figure 9A illus-
trates that, within a metapopulation structure, the evolution of specializa-
tion may, even for simple trade-off curves, end in a singular dimorphic
strategy combination, where the involved strategies are not devoted spe-
cialists.
Evolution to trimorphic coexistence
In structured metapopulation models of the type characterized by the
equations 21 and 22, the competitive exclusion principle (Mesze´na et al.,
2006), i.e., the dimension of the environmental interaction variable never
limits the number of coexisting strategies. In fact, the possibility for eco-
logical coexistence of a generalist strategy with two specialist strategies is
rather firmly built into the metapopulation models where patch types are
determined by the carrying capacities (or availabilities) of two resources.
Trimorphic coexistence may occur, for example, in landscapes that
consist of equal amounts of three different patch types such that in one
patch type resource 1 is abundant and resource 2 scarce, in one patch type
resource 2 is abundant and resource 1 scarce and in one patch type both of
the resources are equally abundant. If furthermore, the generalist has even
71 6.4 METAPOPULATION MODELS
A) B)
0.2
0.4
0.6
0.8
1
2000 4000 6000
0.2
0.4
0.6
0.8
1
2000 4000 6000
Figure 9: Result of evolutionary simulations where only specialization
evolves in a metapopulation model with local dynamics of the Beverton–
Holt type (equation 16).
Panel A: Evolution ends to a singular dimorphic strategy combination in
an environment comprising two patch types with moderate catastrophe
probability. Parameter values: θ = 0.1, c = 0.05, λ = 3, e = 0.3, pi =
0.8, K11 = K
2
2 = 3, K
2
1 = K
1
2 = 1 and p1 = p2 = 0.5.
Panel B: Evolution ends to the trimorphic coexistence of two partially
specialized strategies and the unbiased generalist strategy in an environ-
ment comprising three patch types when catastrophes are extremely rare.
Parameter values: θ = 0.1, c = 0.0001, λ = 3, e = 0.3, pi = 0.9, K11 =
K22 = 3, K
3
1 = K
3
2 = 2, p1 = p2 = 0.25 and p3 = 0.5.
a small advantage (θ > 0), then the generalist is a superior competitor in
patches with equal amount of the two resources, whereas the specialists
are superior competitors in patches rich in resource they are specialized
to. Now, all the three morphs have patches that they can take over in the
long run (if the patch avoids local catastrophes sufficiently long).
Assume now, that catastrophes are extremely rare. Then it is possible
to assume an extremely low dispersal propensity without losing the viabil-
ity of the metapopulation. Then the local dynamics within the patches are
virtually independent of each other, with dispersal only allowing slow re-
colonization of the patches emptied by the local catastrophes. Moreover,
due to extremely small catastrophe probability, the patches are virtually
always fully occupied, which means, that the type that is best adapted to
a certain patch, will outcompete all the other types from this patch, and
the immigrants adapted to other patch types will be rapidly ousted, which
makes the ecological dynamics in different patches virtually independent.
6 EVOLUTION OF RESOURCE SPECIALIZATION 72
This mechanism allows the evolution to trimorphic coexistence as il-
lustrated in figure 9B. A similar mechanism may also affect the rich vari-
ety of the patterns of local adaptation observed in some species (or clades)
inhabiting extremely isolated but stable habitats, such as the Gala´pagos
finches (Darwin, 1845; Grant and Grant, 2002).
Altogether, in the models studied in this thesis, the evolution of spe-
cialization hardly ever leads to trimorphic coexistence under equilibrium
population dynamics with moderate catastrophe probabilities when only
specialization can evolve. This is largely related to the dispersal pro-
cess: dispersal is assumed to be completely global and random. Distance-
limited dispersal together with spatially aggregated resource availabili-
ties is known to enable trimorphic coexistence (De´barre and Lenormand,
2011; Karonen, 2011). Moreover, the evolutionary dynamics of special-
ization, and thus, the possibilities of trimorphic coexistence, are affected
by form of habitat selection, i.e., whether a dispersing individual is able to
assess different habitats and choose its target patch according to its char-
acteristics (Rosenzweig, 1981, 1987, 1991; Richards and De Roos, 2001;
Ravigne´ et al., 2004, 2009).
In this thesis, evolution to trimorphic coexistence is more likely un-
der equilibrium population dynamics when both dispersal propensity and
specialization may evolve. A typical evolutionary scenario resulting in
such coexistence is illustrated in Figure 10. In this figure, local dynamics
are of the Beverton–Holt type and the environment is symmetric, i.e., the
two resources are, on average, equally abundant. However, since there are
three patch types, it is possible to find such parameter combinations that
the branching of the dispersal propensity is possible in a metapopulation
comprising generalists, and that the dispersal propensity significantly af-
fects the invadability of the generalist strategy.
After the initial phase of evolutionary branching of dispersal propen-
sity, the two branches diverge further apart from each other and, given
that trade-off parameter θ has an appropriate value, the generalist strat-
egy may turn from an ESS to an evolutionary branching point for the less
dispersive morph. This results in evolutionary branching of the special-
ization strategy employed by the scarcely dispersing morph, and finally
in trimorphic coexistence, in which each of the three morphs has patches
where it is a superior competitor and can outcompete the other morphs
given that the patch avoids local catastrophes sufficiently long.
73 6.4 METAPOPULATION MODELS
Note that although panel A in Figure 10 may seem to indicate a de-
generate case in which specialization divides in three branches, this is
not the case. Instead, after evolutionary branching of dispersal, both
morphs employ the same specialization strategy, s = 0.5. The morph with
low dispersal propensity undergoes branching of specialization into two
branches, while the specialization strategy of the high-dispersal morph
remains at s = 0.5 as illustrated in Figures 10B-F.
Each evolutionary path leading to trimorphic coexistence observed by
Nurmi and Parvinen (2011) involves an evolutionary branching of disper-
sal propensity in a nearly generalist metapopulation and such parameter
combinations that dispersal propensity affects the invadability of the (un-
biased or biased) generalist strategies. In environments comprising only
two different patch types, it is rather difficult to find such ecological set-
tings, and evolution rarely leads to the coexistence of specialists and gen-
eralists. However, this possible at least in two ways (in narrow parameter
domains).
Figure 11 illustrates the scenario, where evolution to trimorphic co-
existence is possible even in a symmetric environment (resources are on
averages equally abundant) comprising only two patch types. There, in
a monomorphic population, evolution leads to a singular dimorphic strat-
egy combination. Even though evolutionary branching of the dispersal
propensity is not possible in a metapopulation comprising unbiased gen-
eralists (since they observe only one patch type), it is possible for both of
these partially specialized strategies. As the environment is symmetric,
and thus, the evolutionary forces acting on both branches are symmetric,
also the events of evolutionary branching occur fairly simultaneously (for
most sequences of stochastic mutation events).
Thus for a while, the population becomes quadrimorphic, and the
more dispersive morphs start to evolve towards generalism while the less
dispersive morphs become more specialized. Finally, either both of the
more dispersive morphs converge to generalism or one of them dies out
and the other converges to generalism. In the resulting trimorphism the
more dispersive morph finds its niche by efficiently colonizing patches
emptied by catastrophes. On the other hand, the low dispersal special-
ists get along as, in the long run, they can take over the patches rich in
the resource they are specialized in. Recently, Nagelkerke and Menken
(2013) showed, in a Levins-type metapopulation model, that this kind of
6 EVOLUTION OF RESOURCE SPECIALIZATION 74
ecological coexistence may possible even without differences in the dis-
persal propensities if the specialists can live only on a single patch type
while generalists can inhabit any patch type, since in this case the general-
ists can colonize new patches efficiently because they have more patches
(different types) to colonize.
Figure 12 illustrates the scenario, in which the asymmetricity of the
environment enables the evolutionary branching of dispersal propensity
in a metapopulation utilizing the slightly biased singular generalist strat-
egy that is evolutionarily attracting in a monomorphic population. In fig-
ure 12, the environment consists of unequal amount of two patch types
with swapped carrying capacities. An unbiased generalist observes no
differences between such patches, and hence evolutionary branching of
the dispersal propensity is not possible in a metapopulation using the un-
biased generalist strategy. Due to the asymmetricity, the singular special-
ization strategy is, however, sufficiently distant from the unbiased strat-
egy in order to enable evolutionary branching of the dispersal propensity.
In Figure 12, one actually observes two successive events of evolution-
ary branching of dispersal. In both cases, the dispersal propensity at the
branching point is rather large. Therefore, the dispersal propensity of one
of the emerging morphs cannot increase much more and this morph re-
mains nearly generalist, while the dispersal propensity of the other emerg-
ing morph decreases substantially. During the first event of evolutionary
branching, the morph with decreasing dispersal propensity specializes in
the less abundant resource 1 (s = 1), whereas during the second evolu-
tionary branching the newly appeared morph with decreasing dispersal
propensity specializes in the more abundant resource (s = 0). Finally,
the metapopulation reaches a trimorphic state comprised of one abun-
dantly dispersing generalist and two scarcely dispersing specialists. The
exploited niches are qualitatively similar to those in the case involving
symmetric environments (Figure 11).
75 6.4 METAPOPULATION MODELS
A)
0.5
0
1
6000 12000
D
isp
er
sa
lp
ro
pe
n
sit
y
B) 0-120 C) 120-2800 D) 2800-6000
0.5
0.500
1
1
0.5
0.500
1
1
0.5
0.500
1
1
E) 6000-11000 F) 11000-50000
0.5
0.500
1
1
0.5
0.500
1
1
Specialization
Figure 10: Panel A illustrates the strategies present in the metapopulation
with local dynamics of the Beverton–Holt type as a function of evolution-
ary time. Grey curve = the specialization component s of the strategy,
black curve = the dispersal component e. Each dot in Panels B-F rep-
resents a strategy that has been present in the metapopulation during the
corresponding evolutionary time interval. The vertical axis illustrates the
dispersal propensity e and the horizontal axis illustrates specialization s.
The arrows in Panels B-F indicate the direction of evolution. The ini-
tial strategy (e, s) = (0.1, 0.1). The simulation ended in a trimorphic
metapopulation using strategies (e, s) ≈ (0.1, 0), (0.1, 1) and (0.8, 0.5).
Parameter values: θ = 0.1, pi = 0.99, λ = 3, c = 0.05, K11 = 5, K12 =
1, K21 = 1, K
2
2 = 5, K
3
1 = K
3
2 = 1, p1 = p2 = 0.25, p3 = 0.5
6 EVOLUTION OF RESOURCE SPECIALIZATION 76
A)
2000 4000
0
0.5
1
D
isp
er
sa
lp
ro
pe
n
sit
y
B) 0-120 C) 120-500 D) 500-1100
0 0.5 1
0
0.5
1
0 0.5 1
0
0.5
1
0 0.5 1
0
0.5
1
E) 1100-2000 F) 2000-2800 G) 2800-3000
0 0.5 1
0
0.5
1
0 0.5 1
0
0.5
1
0 0.5 1
0
0.5
1
H) 3000-4500 I) 4500-40000
0 0.5 1
0
0.5
1
0 0.5 1
0
0.5
1
Specialization
Figure 11: Panel A illustrates the strategies present in the metapopulation
with local dynamics of the Beverton–Holt type as a function of evolution-
ary time. Grey curve = the specialization component s of the strategy,
black curve = the dispersal component e. Each dot in Panels B-I repre-
sents a strategy that has been present in the metapopulation during the
corresponding evolutionary time interval. The vertical axis illustrates the
dispersal propensity e and the horizontal axis illustrates specialization s.
The arrows in Panels B-I indicate the direction of evolution. The initial
strategy (e, s) = (1, 1). The simulation ended in a trimorphic population
using strategies (e, s) ≈ (0.2, 0.5), (0.1, 0.1) and (0.1, 0.9). Parameter
values: θ = 0.1, pi = 0.8, λ = 3, c = 0.05, K11 = K22 = 3, K21 = K12 =
1, p1 = p2 = 0.5
77 6.4 METAPOPULATION MODELS
A)
0.5
0
1
10000 20000
D
isp
er
sa
lp
ro
pe
n
sit
y
B) 0-350 C) 350-10000 D) 10000-12000
0.5
0.500
1
1
0.5
0.500
1
1
0.5
0.500
1
1
E) 12000-14000 F) 14000-20000 G) 20000-25000
0.5
0.500
1
1
0.5
0.500
1
1
0.5
0.500
1
1
Specialization
Figure 12: Panel A illustrates the strategies present in the metapopulation
with local dynamics of the Beverton–Holt type as a function of evolution-
ary time. Grey curve = the specialization component s of the strategy,
black curve = the dispersal component e. Each dot in Panels B-G rep-
resents a strategy that has been present in the metapopulation during the
corresponding evolutionary time interval. The vertical axis illustrates the
dispersal propensity e and the horizontal axis illustrates specialization s.
The arrows in Panels B-G indicate the direction of evolution. The ini-
tial strategy (e, s) = (0.1, 0.1). The simulation ended in a trimorphic
population using strategies (e, s) ≈ (0.25, 0), (1, 0.4) and (0.1, 1). Pa-
rameter values: c = 0.1, pi = 0.99, θ = 0.1, λ = 1.5, K11 = K22 = 10,
K21 = K
1
2 = 1, p1 = 0.2, p2 = 0.8.
6 EVOLUTION OF RESOURCE SPECIALIZATION 78
Evolutionary effects of ecological parameters
In this thesis, the evolutionary dynamics of specialization are dominated
by the trade-off parameter θ. For low values of θ, the evolutionary dy-
namics of specialization always converge to a specialist strategy. As θ
increases, the generalist strategy first turns from an evolutionary repel-
lor into a branching point. For even greater values of θ the generalist
strategy becomes an evolutionary endpoint, after which increasing θ does
not cause any further qualitative changes under equilibrium ecological
dynamics (when evolutionary suicide is not possible). Thus, there are
always at least two critical values of θ:
• At θ∗1, the generalist strategy turns from an evolutionary repellor
into a branching point.
• At θ∗2, the generalist strategy turns from a branching point into an
evolutionary endpoint (ESS)
Since the trade-off parameter θ measures the additional benefit or cost
of generalism (see equation (20)), the critical values of θ can be exploited
when studying how changes in different ecological parameters affect the
evolutionary dynamics of specialization. If a certain change in ecological
parameters causes both of the critical values to decrease, this change can
be interpreted to favor the spread of the generalist strategy. Correspond-
ingly, a change that causes an increase in both critical values favors the
spread of the specialist strategies. If θ∗1 decreases and θ∗2 increases, the
parameter domain where evolutionary branching occurs becomes larger.
Nurmi and Parvinen (2008) did this kind of investigation for a variety
of different metapopulation models assuming constant dispersal propen-
sity, whereas Nurmi and Parvinen (2011) assumed the dispersal propen-
sity always to have the corresponding evolutionarily singular value. Both
studies focused on metapopulations where within-patch dynamics have
fixed-point attractors. The results of Nurmi and Parvinen (2008) and
Nurmi and Parvinen (2011) are qualitatively similar concerning the fol-
lowing conclusions:
• Increasing environmental heterogeneity, i.e., increasing difference
between the resource carrying capacities K1 and K2 among the
patches enlarges the parameter domain where evolutionary branch-
ing may occur.
79 6.4 METAPOPULATION MODELS
• Increasing fecundity λ and increasing dispersal survival pi favor
the spread of the generalist strategy. In the case of joint evolu-
tion (Nurmi and Parvinen, 2011), this is natural, since also the sin-
gular dispersal propensity increases, which again is natural since,
on one hand, increasing dispersal survival obviously increases dis-
persal propensity, and on the other hand, increasing fecundity in-
creases crowding within the patches, which makes dispersal more
profitable. More surprising is the observation that increasing fecun-
dity favors the spread of the generalist strategy even with constant
dispersal propensity (Nurmi and Parvinen, 2008).
Nurmi and Parvinen (2008) observed, that with constant dispersal propen-
sity, the evolutionary effects of decreasing catastrophe probability depend
on the details of the within-patch dynamics. However, Nurmi and Parvi-
nen (2011) deduced that decreasing catastrophe probability always results
in decreasing dispersal propensity, which always enlarges the parameter
domain where evolutionary branching may occur.
Clonal interference and the joint evolution of dispersal propensity
and specialization
Besides, the results described above, Nurmi and Parvinen (2011) demon-
strated that the evolution of dispersal is usually slower than the evolution
of specialization, i.e., evolutionary forces influencing specialization are
stronger than those influencing dispersal. This phenomenon is rather nat-
ural, since the degree of specialization always affects reproduction. Dis-
persal affects both the reproduction of the dispersers and the reproduction
of those remaining. However, the effect on the dispersers’ fecundity de-
pends crucially on how the original patch and the target patch differ in
terms of quality and crowdedness. Thus, it requires several generations
and dispersal events to be able to observe the average effect of dispersal
on the dispersers’ fecundity. Moreover, the fecundity of the remaining
individuals is increased by dispersals only in crowded patches.
When two traits are evolving and there are significant differences in
the strength of the evolutionary forces influencing them, it is even possi-
ble that the evolution of the faster evolving trait slows down or halts the
evolution of the other. For example, in Figures 10, 11 and 12 the evolution
of specialization halts the evolution of dispersal at the initial phase. This
6 EVOLUTION OF RESOURCE SPECIALIZATION 80
may occur, since mutations affect only one trait at a time (no pleiotropy).
When a new mutant dispersal propensity comes up, it has initially a very
small population size that increases rather slowly even if the mutant is
capable to invade the population. New mutants usually come up before
this mutant population has reached a significant size. Consequently, the
new mutants usually have a dispersal propensity inherited from the initial
resident population. If any of these mutants has a specialization strategy
that is capable to invade the resident, this mutant (carrying the original
dispersal propensity) will increase rapidly in population size (due to the
stronger evolutionary forces) and outcompete the other strategies, includ-
ing the one in which the new dispersal propensity results in higher inva-
sion fitness compared to the initial resident population.
This phenomenon is based on clonal interference. It is possible, since
there is no pleiotropy or recombination (Gerrish and Lenski, 1998). In
this thesis, pleiotropy is not under consideration, since already the case
without pleiotropy involves the main evolutionary feature the search of
which motivated the analysis of the joint evolution of dispersal propen-
sity and specialization: the evolutionary attainability of the trimorphic
coexistence of a generalist strategy with two specialist strategies.
81
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