Conditional dispersal under kin competition: extension of the Hamilton-May model to brood size-dependent dispersal

I consider a site-based model with contest competition among siblings, and assume that dispersal is conditional on the number of offspring in the natal site. Evolutionarily stable populations contain threshold dispersal strategies, which retain a certain number of offspring in the natal site and disperse the rest (if the actual number of offspring is less than the threshold, then all offspring are retained). Due to the discrete nature of the strategy set (the threshold must be integer), the ESS may not be unique or may not exist. In the latter case, two neighboring threshold strategies coexist in the evolutionarily stable population. Dispersal first decreases and then increases as a function of dispersal mortality, such that all but one offspring should be dispersed both when dispersal mortality is very small or very high. Population-level dispersal fractions are often similar to the unconditional ESS, but differ strongly when fecundity is small and dispersal mortality is high.     

Evolutionarily Stable Dispersal Rates Do Not Always Increase with Local Extinction Rates

Earlier models on the evolution of dispersal have suggested that evolutionarily stable dispersal rates should increase with the frequency of local extinctions. Most metapopulation models assume site saturation (i.e., no local population dynamics), yet the majority of species distributed as metapopulations rarely attain carrying capacity in all occupied patches. In this article, we relax this assumption and examine the evolutionarily stable dispersal rate under nonsaturated but still competitive demographic conditions. Contrary to previous predictions, we show that increasing local extinction rates may allow decreasing dispersal rates to evolve.     

Evolution of dispersal in metapopulations with local density dependence and demographic stochasticity

In this paper, we predict the outcome of dispersal evolution in metapopulations based on the following assumptions: (i) population dynamics within patches are density-regulated by realistic growth functions; (ii) demographic stochasticity resulting from finite population sizes within patches is accounted for; and (iii) the transition of individuals between patches is explicitly modelled by a disperser pool. We show, first, that evolutionarily stable dispersal rates do not necessarily increase with rates for the local extinction of populations due to external disturbances in habitable patches. Second, we describe how demographic stochasticity affects the evolution of dispersal rates: evolutionarily stable dispersal rates remain high even when disturbance-related rates of local extinction are low, and a variety of qualitatively different responses of adapted dispersal rates to varied levels of disturbance become possible. This paper shows, for the first time, that evolution of dispersal rates may give rise to monotonically increasing or decreasing responses, as well as to intermediate maxima or minima.     

What can genetics tell us about population connectivity?

Genetic data are often used to assess ‘population connectivity’ because it is difficult to measure dispersal directly at large spatial scales. Genetic connectivity, however, depends primarily on the absolute number of dispersers among populations, whereas demographic connectivity depends on the relative contributions to population growth rates of dispersal vs. local recruitment (i.e. survival and reproduction of residents). Although many questions are best answered with data on genetic connectivity, genetic data alone provide little information on demographic connectivity. The importance of demographic connectivity is clear when the elimination of immigration results in a shift from stable or positive population growth to negative population growth. Otherwise, the amount of dispersal required for demographic connectivity depends on the context (e.g. conservation or harvest management), and even high dispersal rates may not indicate demographic interdependence. Therefore, it is risky to infer the importance of demographic connectivity without information on local demographic rates and how those rates vary over time. Genetic methods can provide insight on demographic connectivity when combined with these local demographic rates, data on movement behaviour, or estimates of reproductive success of immigrants and residents. We also consider the strengths and limitations of genetic measures of connectivity and discuss three concepts of genetic connectivity that depend upon the evolutionary criteria of interest: inbreeding connectivity, drift connectivity, and adaptive connectivity. To conclude, we describe alternative approaches for assessing population connectivity, highlighting the value of combining genetic data with capture‐mark‐recapture methods or other direct measures of movement to elucidate the complex role of dispersal in natural populations.     

Evolution of unconditional dispersal in periodic environments

Organisms modulate their fitness in heterogeneous environments by dispersing. Prior work shows that there is selection against 'unconditional' dispersal in spatially heterogeneous environments. 'Unconditional' means individuals disperse at a rate independent of their location. We prove that if within-patch fitness varies spatially and between two values temporally, then there is selection for unconditional dispersal: any evolutionarily stable strategy (ESS) or evolutionarily stable coalition (ESC) includes a dispersive phenotype. Moreover, at this ESS or ESC, there is at least one sink patch (i.e. geometric mean of fitness less than one) and no sources patches (i.e. geometric mean of fitness greater than one). These results coupled with simulations suggest that spatial-temporal heterogeneity is due to abiotic forcing result in either an ESS with a dispersive phenotype or an ESC with sedentary and dispersive phenotypes. In contrast, the spatial-temporal heterogeneity due to biotic interactions can select for higher dispersal rates that ultimately spatially synchronize population dynamics.     

On several conjectures from evolution of dispersal

We address several conjectures raised in Cantrell et al. [Evolution of dispersal and ideal free distribution, Math. Biosci. Eng. 7 (2010), pp. 17-36 [ 9 ]] concerning the dynamics of a diffusion-advection-competition model for two competing species. A conditional dispersal strategy, which results in the ideal free distribution of a single population at equilibrium, was found in Cantrell et al. [ 9 ]. It was shown in [ 9 ] that this special dispersal strategy is a local evolutionarily stable strategy (ESS) when the random diffusion rates of the two species are equal, and here we show that it is a global ESS for arbitrary random diffusion rates. The conditions in [ 9 ] for the coexistence of two species are substantially improved. Finally, we show that this special dispersal strategy is not globally convergent stable for certain resource functions, in contrast with the result from [ 9 ], which roughly says that this dispersal strategy is globally convergent stable for any monotone resource function.     

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.     

Evolution of reproductive effort in a metapopulation with local extinctions and ecological succession

Abstract Using a metapopulation model, we study how local extinctions, limited population life span, and local demographic disequilibrium affect the evolution of the reproductive effort in a species with overlapping generations but no senescence. We show that in a metapopulation with saturation of all sites and an infinite deme maximal life span (no succession), local extinctions simply constitute an additional source of extrinsic mortality. When either the hypothesis of an infinite deme maximal life span or the saturation hypothesis is relaxed, nontrivial predictions arise. in particular, we find interactions between the evolutionarily stable reproductive effort strategy and the demographic dynamics in the metapopulation. We predict that larger reproductive effort may be selected for in habitats of poorer productivity, contrary to what would be predicted in a single population. Also, we predict that higher dispersal rates should favor selection for lower reproductive efforts. However, metapopulation parameters that favor high dispersal rates also favor larger reproductive efforts. Conflicting selection pressures in the metapopulation also allow maintaining evolutionarily stable polymorphism between a low and high reproductive effort for particular trade-offs between survival and fecundity.     

Emigration to new habitats by voles: the cost of dispersal paradox

We tested the hypothesis that dispersal and philopatry are components of a mixed evolutionarily stable strategy (ESS). The hypothesis predicts that fitness of dispersers should be equal to that of philopatric individuals. Alternatively, fitness of dispersers could be lower (the resident fitness hypothesis) or greater (the cost of dispersal hypothesis) than that of philopatric individuals. We compared fitness of individuals that moved to new habitats (emigrants) and those that remained within habitat boundaries (residents) in populations of the prairie vole, Microtus ochrogaster, and the meadow vole, M. pennsylvanicus. We established vole populations in four enclosures (). Within each enclosure, voles were free to move between four types of habitats that varied in the availability of supplemental food and the amount of vegetative cover. We analysed two fitness components: the survival rates of all individuals, and pregnancy rates of females. Our study showed that emigrants generally had greater fitness than residents and that the difference in fitness was habitat dependent (i.e. was greater when individuals were emigrating from low-quality habitats than from high-quality habitats). High-food, high-cover habitats were the only habitat types for which fitness of emigrants was lower than that of residents. Similar patterns occurred in both prairie voles and meadow voles. Our results support the cost of dispersal hypothesis.     

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.     

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.     

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.     

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.     

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.     

Testing the mechanisms by which source-sink dynamics alter competitive outcomes in a model system

Dispersal among sites can affect within-site competitive outcomes via source-sink dynamics. Source-sink dynamics are thought to affect competitive outcomes primarily via spatial subsidies: by redistributing individuals from sources to sinks, source-sink dynamics can alter competitive outcomes in both sources and sinks. However, dispersal also can affect competitive outcomes via demography modification, which occurs when dispersal alters the parameters governing species' per capita demographic rates. For instance, dispersal of exploitative competitors might cause extinction of some of the resources for which competition occurs, thereby altering the competition coefficients. I used protist microcosms as a model system to test whether spatial subsidies alone could explain the effects of source-sink dynamics on competitive outcomes. I examined the long-term outcome of exploitative competition among three bacterivorous ciliate protists in microcosms of high enrichment (sources) and low enrichment (sinks) in both the presence and the absence of dispersal. Dispersal altered competitive outcomes. Fitting mathematical models to the population dynamics revealed that spatial subsidies were insufficient to account for the effects of dispersal. Fitting alternative models strongly suggested that demography modification was an important determinant of competitive outcomes. These results provide the first evidence that dispersal does not simply redistribute competitors but can alter their per capita demographic rates.     

Effects of local interactions and local migration on stability

Mathematical models can help to resolve the longstanding question of whether more diverse communities are more stable. Here, I focus on how local dispersal and local interactions--hallmarks of spatial communities--affect stability in a spatially implicit model with demographic stochasticity. The results are based on a novel way to analyze moment equations. The main conclusion is that the type and strength of density-dependent factors, such as fecundity and competition, determine whether local dispersal and local interactions increase or decrease stability. Local dispersal has a stabilizing effect when fecundity is high, interspecific competition is either low or high, and the number of species is small. Effects of local migration on stability are amplified when space is explicit.     

Joint evolution of altruistic cooperation and dispersal in a metapopulation of small local populations

We investigate the joint evolution of public goods cooperation and dispersal in a metapopulation model with small local populations. Altruistic cooperation can evolve due to assortment and kin selection, and dispersal can evolve because of demographic stochasticity, catastrophes and kin selection. Metapopulation structures resulting in assortment have been shown to make selection for cooperation possible. But how does dispersal affect cooperation and vice versa, when both are allowed to evolve as continuous traits? We found four qualitatively different evolutionary outcomes. (1) Monomorphic evolution to full defection with positive dispersal. (2) Monomorphic evolution to an evolutionarily stable state with positive cooperation and dispersal. In this case, parameter changes selecting for increased cooperation typically also select for increased dispersal. (3) Evolutionary branching can result in the evolutionarily stable coexistence of defectors and cooperators. Although defectors could be expected to disperse more than cooperators, here we show that the opposite case is also possible: Defectors tend to disperse less than cooperators when the total amount of cooperation in the dimorphic population is low enough. (4) Selection for too low cooperation can cause the extinction of the evolving population. For moderate catastrophe rates dispersal needs to be initially very frequent for evolutionary suicide to occur. Although selection for less dispersal in principle could prevent such evolutionary suicide, in most cases this rescuing effect is not sufficient, because selection in the cooperation trait is typically much stronger. If the catastrophe rate is large enough, a part of the boundary of viability can be evolutionarily attracting with respect to both strategy components, in which case evolutionary suicide is expected from all initial conditions.     

Evolutionary consequences of asymmetric dispersal rates

We study the consequences of asymmetric dispersal rates (e.g., due to wind or current) for adaptive evolution in a system of two habitat patches. Asymmetric dispersal rates can lead to overcrowding of the "downstream" habitat, resulting in a source-sink population structure in the absence of intrinsic quality differences between habitats or can even cause an intrinsically better habitat to function as a sink. Source-sink population structure due to asymmetric dispersal rates has similar consequences for adaptive evolution as a source-sink structure due to habitat quality differences: natural selection tends to be biased toward the source habitat. We demonstrate this for two models of adaptive evolution: invasion of a rare allele that improves fitness in one habitat but reduces it in the other and antagonistic selection on a quantitative trait determined by five additive loci. If a habitat can sustain a population without immigration, the conditions for adaptation to that habitat are most favorable if there is little or no immigration from the other habitat; the influence of emigration depends on the magnitude of the allelic effects involved and other parameters. If, however, the population is initially unable to persist in a given habitat without immigration, our model predicts that the population will be most likely to adapt to that habitat if the dispersal rates in both directions are high. Our results highlight the general message that the effect of gene flow upon local adaptation should depend profoundly on the demographic context of selection.     

The ideal free distribution: a review and synthesis of the game-theoretic perspective

The Ideal Free Distribution (IFD), introduced by Fretwell and Lucas in [Fretwell, D.S., Lucas, H.L., 1970. On territorial behavior and other factors influencing habitat distribution in birds. Acta Biotheoretica 19, 16-32] to predict how a single species will distribute itself among several patches, is often cited as an example of an evolutionarily stable strategy (ESS). By defining the strategies and payoffs for habitat selection, this article puts the IFD concept in a more general game-theoretic setting of the “habitat selection game”. Within this game-theoretic framework, the article focuses on recent progress in the following directions: (1) studying evolutionarily stable dispersal rates and corresponding dispersal dynamics; (2) extending the concept when population numbers are not fixed but undergo population dynamics; (3) generalizing the IFD to multiple species.For a single species, the article briefly reviews existing results. It also develops a new perspective for Parker’s matching principle, showing that this can be viewed as the IFD of the habitat selection game that models consumer behavior in several resource patches and analyzing complications involved when the model includes resource dynamics as well. For two species, the article first demonstrates that the connection between IFD and ESS is now more delicate by pointing out pitfalls that arise when applying several existing game-theoretic approaches to these habitat selection games. However, by providing a new detailed analysis of dispersal dynamics for predator-prey or competitive interactions in two habitats, it also pinpoints one approach that shows much promise in this general setting, the so-called “two-species ESS”. The consequences of this concept are shown to be related to recent studies of population dynamics combined with individual dispersal and are explored for more species or more patches.