Stefan A. H. Geritz

Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree

We present a general framework for modelling adaptive trait dynamics in which we integrate various concepts and techniques from modern ESS-theory. The concept of evolutionarily singular strategies is introduced as a generalization of the ESS-concept. We give a full classification of the singular strategies in terms of ESS-stability, convergence stability, the ability of the singular strategy to invade other populations if initially rare itself, and the possibility of protected dimorphisms occurring within the singular strategys neighbourhood. Of particular interest is a type of singular strategy that is an evolutionary attractor from a great distance, but once in its neighbourhood a population becomes dimorphic and undergoes disruptive selection leading to evolutionary branching. Modelling the adaptive growth and branching of the evolutionary tree can thus be considered as a major application of the framework. A haploid version of Levenes ‘soft selection’ model is developed as a specific example to demonstrate evolutionary dynamics and branching in monomorphic and polymorphic populations.

Adaptive Dynamics in Allele Space: Evolution of Genetic Polymorphism by Small Mutations in a Heterogeneous Environment

We demonstrate how a genetic polymorphism of distinctly different alleles can develop during long‐term frequency‐dependent evolution in an initially monomorphic diploid population, if mutations have only small phenotypic effect. As a specific example, we use a version of Levenes (1953) soft selection model, where stabilizing selection acts on a continuous trait within each of two habitats. If the optimal phenotypes within the habitats are sufficiently different, then two distinctly different alleles evolve gradually from a single ancestral allele. In a wide range of parameter values, the two locally optimal phenotypes will be realized by one of the homozygotes and the heterozygote, rather than by the two homozygotes. Unlike in the haploid analogue of the model, there can be multiple polymorphic evolutionary attractors with different probabilities of convergence. Our results differ from the population genetic models of short‐term evolution in two aspects: (1) a polymorphism that is population genetically stable may be invaded by a new mutant allele and, as a consequence, the population may fall back to monomorphism, (2) long‐term evolution by allele substitutions may lead from a population where polymorphism is not possible into one where polymorphism is possible.

Adaptive Dynamics in a 2-patch Environment: a Simple Model for Allopatric and Parapatric Speciation

Adaptation to an environment consisting of two patches (each with different optimal strategy) is investigated. The patches have independent density regulation (soft selection). If the patches are similar enough and migration between them is strong, then evolution ends up with a generalist ESS. If either the difference between the patches increases or migration weakens, then the generalist strategy represent a branching singularity: The initially monomorphic population first evolves towards the generalist strategy, there it undergoes branching, and finally two specialist strategies form an evolutionarily stable coalition. Further increasing the between-patch difference or decreasing migration causes the generalist to lose its convergence stability as well, and an initially monomorphic population evolves towards one of the specialists optimally adapted to one of the two patches. Bifurcation pattern of the singularities is presented as a function of patch difference and migration rate. Connection to speciation theory is discussed. The transition from the generalist ESS to the coexisting pair of specialist strategies is regarded as a clonal prototype of parapatric (if the between-patch difference increases) or allopatric (if the migration decreases) speciation. We conclude that the geographic and the competitive speciation modes are not distinct classes.

On the Coexistence of Perennial Plants by the Competition‐Colonization Trade‐Off

A competitively inferior species can coexist with a superior competitor in a metapopulation provided that the inferior species is a better colonizer of empty habitats (Levins and Culver 1971; see also Nee and May 1992; Dytham 1994; Moilanen and Hanski 1995). The same trade-off between competition and colonization can also maintain coexistence within a single population of plants that live in small patches supporting a single adult (henceforth called “sites”) and disperse their seeds randomly. With local populations interpreted as individual sites, the metapopulation model of Levins and Culver (1971) has been applied to model within-population coexistence of perennial plant species (e.g., Tilman 1994). However, Yu and Wilson (2001) pointed out that there may be an important difference between metapopulation dynamics and the site dynamics of plants: in the metapopulation model, it is assumed that dispersers of the competitively superior species displace established populations of the inferior species (displacement competition). In plants, seedlings may or may not be able to displace established adults; in the latter case, seedlings compete with one another only for the sites vacated by adult death (replacement competition). Yu and Wilson (2001) argued that under replacement competition, perennial species cannot coexist by the competitioncolonization trade-off alone. In this note, we show that the competition-colonization trade-off does maintain coexistence provided that seed numbers within individual sites are subject to demo-

A mechanistic derivation of the DeAngelis-Beddington functional response.

We give a derivation of the DeAngelis-Beddington functional response in terms of mechanisms at the individual level, and for the first time involving prey refuges instead of the usual interference between predators.

Adaptive Dynamics of Speciation: Ecological Underpinnings

Speciation occurs when a population splits into ecologically differentiated and reproductively isolated lineages. In this chapter, we focus on the ecological side of nonallopatric speciation: Under what ecological conditions is speciation promoted by natural selection? What are the appropriate tools to identify speciation-prone ecological systems? For speciation to occur, a population must have the potential to become polymorphic (i.e., it must harbor heritable variation). Moreover, this variation must be under disruptive selection that favors extreme phenotypes at the cost of intermediate ones. With disruptive selection, a genetic polymorphism can be stable only if selection is frequency dependent (Pimm 1979; see Chapter 3). Some appropriate form of frequency dependence is thus an ecological prerequisite for nonallopatric speciation. Frequency-dependent selection is ubiquitous in nature. It occurs, among many other examples, in the context of resource competition (Christiansen and Loeschcke 1980; see Box 4.1), predator–prey systems (Marrow et al. 1992), multiple habitats (Levene 1953), stochastic environments (Kisdi and Meszena 1993; Chesson 1994), asymmetric competition (Maynard Smith and Brown 1986), mutualistic interactions (Law and Dieckmann 1998), and behavioral conflicts (Maynard Smith and Price 1973; Hofbauer and Sigmund 1990). The theory of adaptive dynamics is a framework devised to model the evolution of continuous traits driven by frequency-dependent selection. It can be applied to various ecological settings and is particularly suitable for incorporating ecological complexity. The adaptive dynamic analysis reveals the course of long-term evolution expected in a given ecological scenario and, in particular, shows whether, and under which conditions, a population is expected to evolve toward a state in which disruptive selection arises and promotes speciation. To achieve analytical tractability in ecologically complex models, many adaptive dynamic models (and much of this chapter) suppress genetic complexity with the assumption of clonally reproducing phenotypes (also referred to as strategies or traits). This enables the efficient identification of interesting features of the engendered selective pressures that deserve further analysis from a genetic perspective.

Evolutionary disarmament in interspecific competition

Competitive asymmetry, which is the advantage of having a larger body or stronger weaponry than a contestant, drives spectacular evolutionary arms races in intraspecific competition. Similar asymmetries are well documented in interspecific competition, yet they seldom lead to exaggerated traits. Here we demonstrate that two species with substantially different size may undergo parallel coevolution towards a smaller size under the same ecological conditions where a single species would exhibit an evolutionary arms race. We show that disarmament occurs for a wide range of parameters in an ecologically explicit model of competition for a single shared resource; disarmament also occurs in a simple Lotka–Volterra competition model. A key property of both models is the interplay between evolutionary dynamics and population density. The mechanism does not rely on very specific features of the model. Thus, evolutionary disarmament may be widespread and may help to explain the lack of interspecific arms races.

Adaptive dynamics: a framework to model evolution in the ecological theatre

The astonishing diversity of life has evolved over many millions of years of natural selection. Yet natural selection, as often taught in introductory courses and textbooks on population genetics, seems to explain why diversity should not exist: the survival of the fittest is the loss of everything else. Simple models of natural selection predict the fixation of the best allele in every locus.1 Yet the Earth (or any part of it) is not ruled by a single Darwinian monster. Frequency-dependent selection (the advantage of an allele being rare, regardless of which allele it is) solves the problem of diversity, but then solves it all too well: by choosing the frequency-dependent fitness function appropriately, the model can produce any conceivable result and, hence, lacks predictive power.2 In between the survival of the fittest and the survival of anybody, the problem of adaptive diversity slips away. The solution to this paradox of classical population genetics is in the Ecology textbooks. Fitness describing survival and reproduction must be derived from realistic ecological interactions,3 which explain how different genotypes or species coexist.

Mathematical ecology: why mechanistic models?

Everyone knows the logistic model of population growth. Populations cannot grow indefinitely; hence the per capita growth rate must decrease as the population density N increases; the simplest decreasing function is linear; and so we have d N dt = [r(1 − N/K )]N , where r is the intrinsic growth rate and K is called the carrying capacity of the environment. Conscientious textbooks will add in parentheses, “(assuming r > 0)”. Indeed, with r < 0, the equation predicts unbounded growth, a biological nonsense, for populations whose initial size exceeds K . But an experimentalist is free to choose any positive initial density, and r < 0 is certainly not nonsense: r < 0 holds when the death rate exceeds the birth rate even at low population densities, as for an unfit mutant or a species outside its region of viability. This shows that we cannot meaningfully vary r and K independently. But how they should be connected is unclear: The information is simply not in the equation. A population is an ensemble of individuals, and its behaviour is ultimately a consequence of the behaviour of the individuals. In order to predict population behaviour, we should first describe the behaviour of the individuals and then use it to derive a model of the population. When the logistic model is derived in this way, it is clear that K is not an externally fixed parameter (the “carrying capacity of the environment”).

Group defence and the predator’s functional response

We derive from first principles the functional response of the predator and the reproduction rate of the prey in the case that the prey form groups as a defence against the predator and the latter captures only single prey. We also give some examples of the resulting predator–prey population dynamics.