The ideal free distribution as an evolutionarily stable strategy

We examine the evolutionary stability of strategies for dispersal in heterogeneous patchy environments or for switching between discrete states (e.g. defended and undefended) in the context of models for population dynamics or species interactions in either continuous or discrete time. There have been a number of theoretical studies that support the view that in spatially heterogeneous but temporally constant environments there will be selection against unconditional, i.e. random, dispersal, but there may be selection for certain types of dispersal that are conditional in the sense that dispersal rates depend on environmental factors. A particular type of dispersal strategy that has been shown to be evolutionarily stable in some settings is balanced dispersal, in which the equilibrium densities of organisms on each patch are the same whether there is dispersal or not. Balanced dispersal leads to a population distribution that is ideal free in the sense that at equilibrium all individuals have the same fitness and there is no net movement of individuals between patches or states. We find that under rather general assumptions about the underlying population dynamics or species interactions, only such ideal free strategies can be evolutionarily stable. Under somewhat more restrictive assumptions (but still in considerable generality), we show that ideal free strategies are indeed evolutionarily stable. Our main mathematical approach is invasibility analysis using methods from the theory of ordinary differential equations and nonnegative matrices. Our analysis unifies and extends previous results on the evolutionary stability of dispersal or state-switching strategies.     

The ideal free distribution as an evolutionarily stable strategy

We examine the evolutionary stability of strategies for dispersal in heterogeneous patchy environments or for switching between discrete states (e.g. defended and undefended) in the context of models for population dynamics or species interactions in either continuous or discrete time. There have been a number of theoretical studies that support the view that in spatially heterogeneous but temporally constant environments there will be selection against unconditional, i.e. random, dispersal, but there may be selection for certain types of dispersal that are conditional in the sense that dispersal rates depend on environmental factors. A particular type of dispersal strategy that has been shown to be evolutionarily stable in some settings is balanced dispersal, in which the equilibrium densities of organisms on each patch are the same whether there is dispersal or not. Balanced dispersal leads to a population distribution that is ideal free in the sense that at equilibrium all individuals have the same fitness and there is no net movement of individuals between patches or states. We find that under rather general assumptions about the underlying population dynamics or species interactions, only such ideal free strategies can be evolutionarily stable. Under somewhat more restrictive assumptions (but still in considerable generality), we show that ideal free strategies are indeed evolutionarily stable. Our main mathematical approach is invasibility analysis using methods from the theory of ordinary differential equations and nonnegative matrices. Our analysis unifies and extends previous results on the evolutionary stability of dispersal or state-switching strategies.     

Evolutionary stability of plant communities and the maintenance of multiple dispersal types

S.A. Levin, D. Cohen, and A. Hastings (1984, Theor. Popul. Biol. 19, 169–200) and D. Cohen and S.A. Levin (1991, Theor. Popul. Biol. 39, 63–99) by analytic solution of the problem of invasion of a single dispersal type by any other, have provided a theory for evolutionarily stable strategies for seed dispersal in a random environment. Here the results of Cohen and Levin are extended to describe evolutionarily stable combinations of dispersal types. Such combinations of two types are coalitions that cannot be invaded by any other, although in isolation either of the types in the combination is invasible by others. These combinations appear when there is a negative correlation between the seed production of sites in successive years, or when environments are spatially heterogeneous, or presumably under other circumstances. In this work, we examine only the case of negative correlations. For this situation the configuration of evolutionarily stable strategies (ESS) and evolutionarily stable combinations (ESC) depends upon the ratio of (precompetitive) survival rates of dispersersing and nondispersing seeds, which is denoted by α. For low values of α, the purely nondispersing type is an ESS. At a somewhat higher value of α, the purely dispersing type can invade the nondispersing type, and the two types form an ESC, i.e., a combination that cannot be invaded by any other type. For still larger values of α, the purely nondispersing type is excluded by the ESC. Finally, for the largest values of α, pure dispersal is the ESS. In cases where a single dispersal type cannot exclude all others, the stationary distribution of types has a large spread. It can be adequately approximated by equations for conditional means of the proportions of various types at a site of a given quality, but these means must be conditioned upon the prior history at each site. For some purposes we have found that the history of as many as 8–10 generations is required for a good approximation. This phenomenon appears to preclude simple analytic approximations for the ESC.     

Environmentally induced dispersal under heterogeneous logistic growth

We consider a single-species model which is composed of several habitats connected by linear migration rates and having logistic growth. A spatially varying, temporally constant environment is introduced by the non-homogeneity of its carrying capacity. Under this condition any type of purely diffusive behavior, characterized in our model by symmetric migration rates, produces an unbalanced population distribution, i.e. some locations receive more individuals than can be supported by the environmental carrying capacity, while others receive less. Using an evolutionarily stable strategy (ESS) approach we show that an asymmetric migration mechanism, induced by the heterogeneous carrying capacity of the environment, will be selected. This strategy balances the inflow and outflow of individuals in each habitat (balanced dispersal), as well as 'balancing' the spatial distribution relative to variation in carrying capacity (the Ideal Free Distribution from habitat selection theory). We show that several quantities are maximized or minimized by the evolutionarily stable dispersal strategy.     

Evolution of resource specialisation in competitive metacommunities

Spatial environmental heterogeneity coupled with dispersal can promote ecological persistence of diverse metacommunities. Does this premise hold when metacommunities evolve? Using a two‐resource competition model, we studied the evolution of resource‐uptake specialisation as a function of resource type (substitutable to essential) and shape of the trade‐off between resource uptake affinities (generalist‐ to specialist‐favouring). In spatially homogeneous environments, evolutionarily stable coexistence of consumers is only possible for sufficiently substitutable resources and specialist‐favouring trade‐offs. Remarkably, these same conditions yield comparatively low diversity in heterogeneous environments, because they promote sympatric evolution of two opposite resource specialists that, together, monopolise the two resources everywhere. Consumer diversity is instead maximised for intermediate trade‐offs and clearly substitutable or clearly essential resources, where evolved metacommunities are characterised by contrasting selection regimes. Taken together, our results present new insights into resource‐competition‐mediated evolutionarily stable diversity in homogeneous and heterogeneous environments, which should be applicable to a wide range of systems.     

A novel fitness proxy in structured locally finite metapopulations with diploid genetics, with an application to dispersal evolution

Many studies of evolutionarily stable strategies (ESS) for technical reasons make the simplification that reproduction is clonal. A post-hoc justification is that in the simplest eco-evolutionary models more realistic genetic assumptions, such as haploid sexual or diploid sexual cases, yield results compatible with the clonal ones. For metapopulations the technical reasons were even more poignant thanks to the lack of accessible fitness proxies for the diploid case. However, metapopulations are also precisely the sort of ecological backdrop for which one expect discrepancies between the evolutionary outcomes derived from clonal reproduction and diploid genetics, because substantially many mutant homozygotes appear locally even though the mutant is rare globally. In this paper we devise a fitness proxy applicable to the haploid sexual and diploid sexual case, in the style of Metz and Gyllenberg [Metz, J.A.J., Gyllenberg, M., 2001. How should we define fitness in structured metapopulation models? Including an application to the calculation of ES dispersal strategies. Proc. R. Soc. Lond. B 268, 499-508], that can cope with local population fluctuations due to environmental and demographic stochasticity. With the use of this fitness proxy we find that in dispersal evolution the studied clonal model is equivalent with the haploid sexual model, and that there are indeed many differences between clonal and diploid ESS dispersal rates. In a homogenous landscape the discrepancy is but minor (less than 2%), but the situation is different in a heterogeneous landscape: Not only is the quantitative discrepancy between the two types of ESSs appreciable (around 10%-20%), but more importantly, at the same parameter values, evolutionarily stability properties may differ. It is possible, that the singular strategy is evolutionarily stable in the clonal case but not in the diploid case, and vice versa.     

Evolutionary stability of ideal free dispersal strategies in patchy environments

A central question in the study of the evolution of dispersal is what kind of dispersal strategies are evolutionarily stable. Hastings (Theor Pop Biol 24:244-251, 1983) showed that among unconditional dispersal strategies in a spatially heterogeneous but temporally constant environment, the dispersal strategy with no movement is convergent stable. McPeek and Holt's (Am Nat 140:1010-1027, 1992) work suggested that among conditional dispersal strategies in a spatially heterogeneous but temporally constant environment, an ideal free dispersal strategy, which results in the ideal free distribution for a single species at equilibrium, is evolutionarily stable. We use continuous-time and discrete-space models to determine when the dispersal strategy with no movement is evolutionarily stable and when an ideal free dispersal strategy is evolutionarily stable, both in a spatially heterogeneous but temporally constant environment.     

A spatially explicit model of sex ratio evolution in response to sex-biased dispersal

Sex-biased dispersal occurs in all seed plants and many animal species. Theoretical models have shown that sex-biased dispersal can lead to evolutionarily stable biased sex ratios. Here, we use a spatially explicit chessboard model to simulate the evolution of sex ratio in response to sex-biased dispersal range and sex-biased dispersal rate. Two life cycles are represented in the model: one in which both sexes disperse before mating (DDM), the other in which males disperse before mating and mated females or zygotes disperse after mating (DMD). Model parameters include factors like dispersal rate, dispersal range, number of individuals per patch, and habitat heterogeneity.When dispersal range is sex biased, we find that, in a homogeneous environment, the sex ratio is generally biased towards the sex that disperses more widely (sex ratio range: 0.47–0.52). In a heterogeneous environment, the sex ratio is generally biased towards the more dispersive sex in good habitats, and towards the less dispersive sex in poor habitats (sex ratio range: 0–1). This is opposite to the effect of sex-biased dispersal rate, which favours the production of the more dispersive sex in poor habitats and the less dispersive sex in good habitats (sex ratio range: 0–1). To allow for a comparison with theoretical predictions, data concerning sex-biased dispersal and habitat-dependent sex ratios should thus incorporate information about the spatial scale of both dispersal and environmental heterogeneity.     

A simple evolutionary model of dormancy and dispersal in heterogeneous patches with special reference to phytophagous lady beetles: I. Stable environments

A simple evolutionary model of dormancy and dispersal is presented with special reference to phytophagous lady beetles. In order to investigate spatially heterogeneous environments, we assume the simplest patch structure, that is, there are only two patches, main and sub. Environments are also assumed to be temporally constant. The main patch is superior to the sub patch, but density effect at the main patch is higher than at the sub patch. Optimal dormancy and dispersal are obtained at the same time by the method of evolutionarily stable strategy (ESS). In the univoltine life cycle, dormancy strategy vanishes because dormant individuals do not reproduce at all but suffer from a certain mortality rate during winter hibernation. In the bivoltine life cycle, the dormancy and dispersal rates constitute a trade-off: the rates change together with a negative correlation when the mortality rate during dispersal or during winter hibernation changes. When suitability of the main patch gradually deteriorates, the optimal strategy changes as follows: neither dormancy nor dispersal is adopted at the most suitable condition, the dispersal rate is increased without dormancy in the intermediate condition, and then the dormancy rate is increased with a constant dispersal rate. We discuss the field observation data of lady beetles in the light of results of our model.     

Dispersal: risk spreading versus local adaptation

I investigate how risk spreading in stochastic environments and adaptation to permanent properties of local habitats interplay in the simultaneous evolution of dispersal and habitat specialization. In a simple two-patch model, I find many types of locally evolutionarily stable attractors of dispersal and of a trait involved in habitat specialization, including a single habitat specialist and a coalition of two specialists with low dispersal, a generalist with high dispersal, and several types of dispersal polymorphisms. In general, only one attractor is a global evolutionarily stable strategy (ESS). In addition to the ESS analysis, I also present some examples of the dynamics of evolution that exhibit adaptive diversification by evolutionary branching.     

Evolution in heterogeneous environments: Effects of migration on habitat specialization

Summary Richard Levins introduced fitness sets as a tool for investigating evolution within heterogeneous environments. Evolutionary game theory permits a synthesis and generalization of this approach by considering the evolutionary response of organisms to any scale of habitat heterogeneity. As scales of heterogeneity increase from fine to coarse, the evolutionary stable strategy (ESS) switches from a single generalist species to several species that become increasingly specialized on distinct habitats. Depending upon the organisms' ecology, the switch from one to two species may occur at high migration rates (relatively fine-grained environment), or may only occur at very low migration rates (coarse-grained environment). At the ESS, the evolutionary context of a species is the entire landscape, while its ecological context may be a single habitat.Evolution towards the ESS can be represented with adaptive landscapes. In the absence of frequency-dependence, shifting from a single strategy ESS to a two strategy ESS poses the problem of evolving across valleys in the adaptive surface to occupy new peaks (hence, Sewell Wright's shifting balance theory). Frequency-dependent processes facilitate evolution across valleys. If a system with a two strategy ESS is constrained to possess a single strategy, the population may actually evolve a strategy that minimizes fitness. Because the population now rests at the bottom of a valley, evolution by natural selection can drive populations to occupy both peaks.     

Adaptive diversification of germination strategies.

Evolution of the germination rate (the proportion of newly produced and dormant seeds that germinates every year) of annual plants is investigated, when the environment is temporally stochastic and spatially heterogeneous. The environment consists of two habitats with synchronous stochastic variation in the annual yield and permanent difference in constant seed survival rates. Density dependence operates within the habitats, which are connected via restricted seed dispersal. We find that instead of a single common evolutionarily stable strategy the coexistence of several germination strategies is possible and that in an initially monomorphic population evolutionary branching may occur. During evolutionary branching the population undergoes disruptive selection and splits into two branches of different lineages that converge to the evolutionarily stable coalition of different germination strategies. It is shown that spatial heterogeneity and restricted dispersal are essential for evolutionary branching. Disruptive selection on the germination rate presents yet another possibility for parapatric speciation.     

Evolutionary Stability for Interactions among Kin under Quantitative Inheritance

The theory of evolutionarily stable strategies (ESS) predicts the long-term evolutionary outcome of frequency-dependent selection by making a number of simplifying assumptions about the genetic basis of inheritance. I use a symmetrized multilocus model of quantitative inheritance without mutation to analyze the results of interactions between pairs of related individuals and compare the equilibria to those found by ESS analysis. It is assumed that the fitness changes due to interactions can be approximated by the exponential of a quadratic surface. The major results are the following. (1) The evolutionarily stable phenotypes found by ESS analysis are always equilibria of the model studied here. (2) When relatives interact, one of the two conditions for stability of equilibria differs between the two models; this can be accounted for by positing that the inclusive fitness function for quantitative characters is slightly different from the inclusive fitness function for characters determined by a single locus. (3) The inclusion of environmental variance will in general change the equilibrium phenotype, but the equilibria of ESS analysis are changed to the same extent by environmental variance. (4) A class of genetically polymorphic equilibria occur, which in the present model are always unstable. These results expand the range of conditions under which one can validly predict the evolution of pairwise interactions using ESS analysis.     

Evolution of dispersal in a predator-prey metacommunity

Dispersal is crucial to allowing species inhabiting patchy or spatially subdivided habitats to persist globally despite the possibility of frequent local extinctions. Theoretical studies have repeatedly demonstrated that species that exhibit a regional metapopulation structure and are subject to increasing rates of local patch extinctions should experience strong selective pressures to disperse more rapidly despite the costs such increased dispersal would entail in terms of decreased local fitness. We extend these studies to consider how extinctions arising from predator-prey interactions affect the evolution of dispersal for species inhabiting a metacommunity. Specifically, we investigate how increasing a strong extinction-prone interaction between a predator and prey within local patches affects the evolution of each species' dispersal. We found that for the predator, as expected, evolutionarily stable strategy (ESS) dispersal rates increased monotonically in response to increasing local extinctions induced by strong predator top-down effects. Unexpectedly for the prey, however, ESS dispersal rates displayed a nonmonotonic response to increasing predator-induced extinction rates-actually decreasing for a significant range of values. These counterintuitive results arise from how extinctions resulting from trophic interactions play out at different spatial scales: interactions that increase extinction rates of both species locally can, at the same time, decrease the frequency of interaction between the prey and predator at the metacommunity scale.     

THE EVOLUTIONARILY STABLE PHENOTYPE DISTRIBUTION IN A RANDOM ENVIRONMENT

In an unpredictably changing environment, phenotypic variability may evolve as a “bet-hedging” strategy. We examine here two models for evolutionarily stable phenotype distributions resulting from stabilizing selection with a randomly fluctuating optimum. Both models include overlapping generations, either survival of adults or a dormant propagule pool. In the first model (mixed-strategies model) we assume that individuals can produce offspring with a distribution of phenotypes, in which case, the evolutionarily stable population always consists of a single genotype. We show that there is a unique evolutionarily stable strategy (ESS) distribution that does not depend on the amount of generational overlap, and that the ESS distribution generically is discrete rather than continuous; that is, there are distinct classes of offspring rather than a continuous distribution of offspring phenotypes. If the probability of extreme fluctuations in the optimum is sufficiently small, then the ESS distribution is monomorphic: a single type fitted to the mean environment. At higher levels of variability, the ESS distribution is polymorphic, and we find stability conditions for dimorphic distributions. For an exponential or similarly broad-tailed distribution of the optimum phenotype, the ESS consists of an infinite number of distinct phenotypes. In the second model we assume that an individual produces offspring with a single, genetically determined phenotype (pure-strategies model). The ESS population then contains multiple genotypes when the environmental variance is sufficiently high. However the phenotype distributions are similar to those in the mixed-strategies model: discrete, with an increasing number of distinct phenotypes as the environmental variance increases.     

Population dynamics in two-patch environments: Some anomalous consequences of an optimal habitat distribution

The effect of dispersal on population size and stability is explored for a population that disperses passively between two discrete habitat patches. Two basic models are considered. In the first model, a single population experiences density-dependent growth in both patches. A graphical construction is presented which allows one to determine the spatial pattern of abundance at equilibrium for most reasonable growth models and rates of dispersal. It is shown under rather general conditions that this equilibrium is unique and globally stable. In the second model, the dispersing population is a food-limited predator that occurs in both a source habitat (which contains a prey population) and a sink habitat (which does not). Passive dispersal between source and sink habitats can stabilize an otherwise unstable predator-prey interaction. The conditions allowing this are explored in some detail. The theory of optimal habitat selection predicts the evolutionarily stable distribution of a population, given that individuals can freely move among habitats so as to maximize individual fitness. This theory is used to develop a heuristic argument for why passive dispersal should always be selectively disadvantageous (ignoring kin effects) in a spatially heterogeneous but temporally constant environment. For both the models considered here, passive dispersal may lead to a greater number of individuals in both habitats combined than if there were no dispersal. This implies that the evolution of an optimal habitat distribution may lead to a reduction in population size; in the case of the predator-prey model, it may have the additional effect of destabilizing the interaction. The paper concludes with a discussion of the disparate effects habitat selection might have on the geographical range occupied by a species.     

Evolution of dispersal in open advective environments

We consider a two-species competition model in a one-dimensional advective environment, where individuals are exposed to unidirectional flow. The two species follow the same population dynamics but have different random dispersal rates and are subject to a net loss of individuals from the habitat at the downstream end. In the case of non-advective environments, it is well known that lower diffusion rates are favored by selection in spatially varying but temporally constant environments, with or without net loss at the boundary. We consider several different biological scenarios that give rise to different boundary conditions, in particular hostile and “free-flow” conditions. We establish the existence of a critical advection speed for the persistence of a single species. We derive a formula for the invasion exponent and perform a linear stability analysis of the semi-trivial steady state under free-flow boundary conditions for constant and linear growth rate. For homogeneous advective environments with free-flow boundary conditions, we show that populations with higher dispersal rate will always displace populations with slower dispersal rate. In contrast, our analysis of a spatially implicit model suggest that for hostile boundary conditions, there is a unique dispersal rate that is evolutionarily stable. Nevertheless, both scenarios show that unidirectional flow can put slow dispersers at a disadvantage and higher dispersal rate can evolve.     

The dynamical attainability of ESS in evolutionary games

In this paper, the attainability of ESS of the evolutionary game among n players under the frequency-independent selection is studied by means of a mathematical model describing the dynamical development and a concept of stability (strongly determined stability). It is assumed that natural selection and small mutations cause the phenotype to change gradually in the direction of fitness increasing. It is shown that (1) the ESS solution is not always evolutionarily attainable in the evolutionary dynamics, (2) in the game where the interaction between two species is completely competitive, the Nash solution is always attainable, and (3) one of two species may attain the state of minimum fitness as a result of evolution. The attainability of ESS is also examined in two game models on the sex ratio of wasps and aphids in light of our criterion of the attainability of ESS.     

Spatial patterns in population conflicts

We consider a dynamical model for evolutionary games, and enquire how the introduction of diffusion may lead to the formation of stationary spatially inhomogeneous solutions, that is patterns. For the model equations being used it is already known that if there is an evolutionarily stable strategy (ESS), then it is stable. Equilibrium solutions which are not ESS's and which are stable with respect to spatially constant perturbations may be unstable for certain choices of the dispersal rates. We prove by a bifurcation technique that under appropriate conditions the instability leads to patterns. Computations using a curve-following technique show that the bifurcations exhibit a rich structure with loops joined by symmetry-breaking branches.     

Emergence of long‐term balanced polymorphism under cyclic selection of spatially variable magnitude

A fundamental question in evolutionary biology is what promotes genetic variation at nonneutral loci, a major precursor to adaptation in changing environments. In particular, balanced polymorphism under realistic evolutionary models of temporally varying environments in finite natural populations remains to be demonstrated. Here, we propose a novel mechanism of balancing selection under temporally varying fitnesses. Using forward‐in‐time computer simulations and mathematical analysis, we show that cyclic selection that spatially varies in magnitude, such as along an environmental gradient, can lead to elevated levels of nonneutral genetic polymorphism in finite populations. Balanced polymorphism is more likely with an increase in gene flow, magnitude and period of fitness oscillations, and spatial heterogeneity. This polymorphism‐promoting effect is robust to small systematic fitness differences between competing alleles or to random environmental perturbation. Furthermore, we demonstrate analytically that protected polymorphism arises as spatially heterogeneous cyclic fitness oscillations generate a type of storage effect that leads to negative frequency dependent selection. Our findings imply that spatially variable cyclic environments can promote elevated levels of nonneutral genetic variation in natural populations.