How should we define fitness in structured metapopulation models? Including an application to the calculation of evolutionarily stable dispersal strategies

We define a fitness concept applicable to structured metapopulations consisting of infinitely many equally coupled patches. In addition, we introduce a more easily calculated quantity Rm that relates to fitness in the same manner as R0 relates to fitness in ordinary population dynamics: the Rm of a mutant is only defined when the resident population dynamics converges to a point equilibrium and Rm is larger (smaller) than 1 if and only if mutant fitness is positive (negative). Rm corresponds to the average number of newborn dispersers resulting from the (on average less than one) local colony founded by a newborn disperser. Efficient algorithms for calculating its numerical value are provided. As an example of the usefulness of these concepts we calculate the evolutionarily stable conditional dispersal strategy for individuals that can account for the local population density in their dispersal decisions. Below a threshold density x, at which staying and leaving are equality profitable, everybody should stay and above x everybody should leave, where profitability is measured as the mean number of dispersers produced through lines of descent consisting of non-dispersers.     

Evolution of condition-dependent dispersal: A genetic-algorithm search for the ESS reaction norm

Many insects produce two types (winged and wingless) of offspring that greatly differ in dispersal ability. The ratio of the two types often depends on the quality of the local habitat and the crowding experienced by the mother. Here we studied the condition-dependent dispersal that is evolutionarily stable. The model is also applicable to annual plants that produce two types of seeds differing in dispersal rates. The model assumptions are: the population is composed of a number of sites each occupied by a single adult. The total number of offspring produced by a mother depends on the environmental quality of the site that varies over the years and between sites. The ESS fraction of dispersing type as a function of the quality of the habitat (or ESS reaction norm) states that dispersers should not be produced if habitat qualitym is smaller than a critical valuek. Ifm is larger thank, the number of dispersers should increase withm and that of nondispersers should be kept constant. Second, we developed an alternative way of searching for the ESS: the reaction norm is represented as a three-layered neural network, and the parameters (weights and biases) are chosen by genetic algorithm (GA). This method can be extended easily to the cases of multiple environmental factors. There was an optimal (relatively wide) range of mutation rates for weights and biases, outside of which the convergence of the network to the valid ESS was likely to fail. Recombination, or crossing-over, was not effective in improving the success rate. The learned network often shows several characteristic ways of deviation from the ESS. We also examined the case in which the quality of different sites was correlated. In this case the ESS fraction of dispersers increases both with the quality of the site and with the average quality of the whole population in that year.     

Evolutionarily stable strategies for a finite population and a variable contest size

This paper presents a generalization of Maynard Smith's concept of an evolutionarily stable strategy (ESS) to cover the cases of a finite population and a variable contest size. Both equilibrium and stability conditions are analysed. The standard Maynard Smith ESS with an infinite population and a contest size of two (pairwise contests) is shown to be a special case of this generalized ESS. An important implication of the generalized ESS is that in finite populations the behaviour of an ESS player is "spiteful", in the sense that an ESS player acts not only to increase his payoff but also to decrease the payoffs of his competitors. The degree of this "spiteful" behaviour is shown to increase with a decrease in the population size, and so is most likely to be observed in small populations. The paper concludes with an extended example: a symmetric two-pure-strategies two-player game for a finite population. It is shown that a mixed strategy ESS is globally stable against invasion by any one type of mutant strategist. The condition for the start of simultaneous invasion by two types of mutant is also given.     

Equilibrium structure and stability in a frequency-dependent,two-population diploid model

We investigate the equilibrium structure for an evolutionary genetic model in discrete time involving two monoecious populations subject to intraspecific and interspecific random pairwise interactions. A characterization for local stability of an equilibrium is found, related to the proximity of this equilibrium with evolutionarily stable strategies (ESS). This extends to a multi-population framework a principle initially proposed for single populations, which states that the mean population strategy at a locally stable equilibrium is as close as possible to an ESS.     

Size‐dependent resource allocation and sex allocation in herbaceous perennial plants

Individuals within a population often differ considerably in size or resource status as a result of environmental variation. In these circumstances natural selection would favour organisms not with a single, genetically determined allocation, but with a genetically determined allocation rule specifying allocation in relation to size or environment. Based on a graphical analysis of a simple evolutionarily stable strategy (ESS) model for herbaceous perennial plants, we aim to determine how cosexual plants within a population should simultaneously adjust their reproductive allocation and sex allocation to their size. We find that if female fitness gain is a linear function of resource investment, then a fixed amount of resources should be allocated to male function, and to post‐breeding survival as well, for individuals above a certain size threshold. The ESS resource allocation to male function, female function, and post‐breeding survival positively correlate if both male and female fitness gains are a saturating function of resource investment. Plants smaller than the size threshold are expected to be either nonreproductive or functionally male only.     

Body condition dependent dispersal in a heterogeneous environment

We find the evolutionarily stable dispersal behaviour of a population that inhabits a heterogeneous environment where patches differ in safety (the probability that a juvenile individual survives until reproduction) and productivity (the total competitive weight of offspring produced by the local individual), assuming that these characteristics do not change over time. The body condition of clonally produced offspring varies within and between families. Offspring compete for patches in a weighted lottery, and dispersal is driven by kin competition. Survival during dispersal may depend on body condition, and competitive ability increases with increasing body condition. The evolutionarily stable strategy predicts that families abandon patches which are too unsafe or do not produce enough successful dispersers. From families that invest in retaining their natal patches, individuals stay in the patch that are less suitable for dispersal whereas the better dispersers disperse. However, this clear within-family pattern is often not reflected in the population-wide body condition distribution of dispersers or non-dispersers. This may be an explanation why empirical data do not show any general relationship between body condition and dispersal. When all individuals are equally good dispersers, then there exist equivalence classes defined by the competitive weight that remains in a patch. An equivalence class consists of infinitely many dispersal strategies that are selectively neutral. This provides an explanation why very diverse patterns found in body condition dependent dispersal data can all be equally evolutionarily stable.     

Resource Allocation and the Evolution of Self-Fertilization in Plants

This article develops a simple evolutionarily stable strategy (ESS) model of resource allocation in partially selfing plants, which incorporates reproductive and sex allocation into a single framework. The analysis shows that, if female fitness gain increases linearly with resource investment, total reproductive allocation is not affected by sex allocation, defined as the fraction of reproductive resources allocated to male function. All else being equal, the ESS total reproductive allocation increases with increasing selfing rate if the fitness of selfed progeny is more than half that of outcrossed progeny, while the ESS sex allocation is always a decreasing function of the selfing rate. Self-fertilization is much more common in annual than in perennial plants, and this association has been commonly interpreted in terms of an effect of life history on mating system. The model in this article shows that self-fertilization can itself cause the evolution of the annual habit. Incorporating the effects of pollen discounting may not have any influence on total reproductive allocation if female fitness gain is a linear function of resource investment, although the evolutionarily stable sex allocation is altered. Evolution of the selfing rate is found to be independent of reproductive and sex allocation under the mass-action assumption that self- and outcross pollen are deposited simultaneously on receptive stigmas and compete for access to ovules.     

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.     

Metapopulation persistence in dynamic landscapes: the role of dispersal distance

Species associated with transient habitats need efficient dispersal strategies to ensure their regional survival. Using a spatially explicit metapopulation model, we studied the effect of the dispersal range on the persistence of a metapopulation as a function of the local population and landscape dynamics (including habitat patch destruction and subsequent regeneration). Our results show that the impact of the dispersal range depends on both the local population and patch growth. This is due to interactions between dispersal and the dynamics of patches and populations via the number of potential dispersers. In general, long-range dispersal had a positive effect on persistence in a dynamic landscape compared to short-range dispersal. Long-range dispersal increases the number of couplings between the patches and thus the colonisation of regenerated patches. However, long-range dispersal lost its advantage for long-term persistence when the number of potential dispersers was low due to small population growth rates and/or small patch growth rates. Its advantage also disappeared with complex local population dynamics and in a landscape with clumped patch distribution.     

The evolutionary dynamics of direct phenotypic overdominance: emergence possible, loss probable

An evolutionary dynamical system with explicit diploid genetics is used to investigate the likelihood of observing phenotypically overdominant heterozygotes versus heterozygous phenotypes that are intermediate between the homozygotes. In this model, body size evolves in a population with discrete demographic episodes and with competition limiting reproduction. A genotype-phenotype map for body size is used that can generate the two qualitative types of dominance interactions (overdominance versus intermediate dominance). It is written as a single-locus model with one focal locus and parameters summarizing the effects of alleles at other loci. Two types of evolutionarily stable strategy (ESS; continuously stable strategy, CSS) occur. The ESS is generated either (1) by the population ecology; or (2) by a local maximum of the genotype-phenotype map. Overdominant heterozygotes are expected to arise if the population evolves toward the second type of ESS, where nearly maximum body sizes are found. When other loci with partially dominant inheritance also evolve, the location of the maximum in the genotype-phenotype map repeatedly changes. It is unlikely that an evolving population will track these changes; ESSs of the second type now are at best quasi-stationary states of the evolutionary dynamics. Considering the restrictions on its probability, a pattern of phenotypic overdominance is expected to be rare.     

Evolutionarily stable sets in the single-locus frequency-dependent model of natural selection

Recent developments in the static theory of evolutionarily stable sets (ESSets) are applied to the single-locus frequency-dependent model of natural selection. Particular emphasis is paid to the ESSet properties of the preimage of an ESS (or ESSet) under the genotype-phenotype map. When an ESS is realized in genetic equilibrium with redundancy in a diploid sexual population, the basic problem in biological terms is whether the corresponding set of allele frequencies is an evolutionarily stable set. The interesting question of the dynamic stability of this preimage is also discussed and a geometric condition developed which implies its evolutionary and dynamic stability.The authors appreciate detailed suggestions for improvement made by the reviewers of the original version of this paper. Financial assistance from the Natural Sciences and Engineering Research Council of Canada and from the Hungarian National Scientific Research Fund (OTKA Projects T029320 and T037271) is also gratefully acknowledged.     

An explicit approach to evolutionarily stable dispersal strategies: no cost of dispersal

The evolution of dispersal is examined by looking at evolutionarily stable strategies (ESS) for dispersal parameters in discrete time multisite models without any cost of dispersal. ESS are investigated analytically, based on explicit results on sensitivity analysis of matrix models. The basic model considers an arbitrary number of sites and a single age class. An ESS for dispersal parameters is obtained when the spatial reproductive values, calculated at the density-dependent population equilibrium, are equal across sites. From this basic formulation, one derives equivalently that all local populations should be at equilibrium in the absence of migration, and that dispersal between sites should be balanced, i.e., the numbers of individuals arriving to and leaving a site are equal. These results are then generalized to a model with several age classes. Equal age-specific reproductive values do not however imply balanced dispersal in this case. Our results generalize to any number of sites and age classes those available ?M. Doebeli, Dispersal and dynamics, Theoret. Popul. 47 (1995) 82 for two sites and one age class.     

Evolutionarily stable strategies and short-term selection in Mendelian populations re-visited

This note concerns a one locus, two allele, random mating diploid population, subject to frequency-dependent viability selection. It is already known that in such a population, any evolutionarily stable strategies (ESS), if only accessible by the genotype-to-phenotype mapping, is the phenotypic image of a stable genetic equilibrium (Eshel, I. 1982. Evolutionarily stable strategies and viability selection in Mendelian populations. Theor. Popul. Biol. 22(2), 204-217; Cressman et al. 1996. Evolutionary stability in strategic models of single-locus frequency-dependent viability selection. J. Math. Biol. 34, 707-733). The opposite is not true. We find necessary and sufficient parametric conditions for global convergence to the ESS, but we also demonstrate conditions under which, although a unique, genetically accessible ESS exists, there is another, "non-phenotypic" genetically stable equilibrium.     

The unavoidable costs and unexpected benefits of parasitism: population and metapopulation models of parasite-mediated competition

When faced with limited resources, organisms have to determine how to allocate their resources to maximize fitness. In the presence of parasites, hosts may be selected for their ability to balance between the two competing needs of reproduction and immunity. These decisions can have consequences not only for host fitness, but also for the ability of parasites to persist within the population, and for the competitive dynamics between different host species. We develop two mathematical models to investigate how resource allocation strategies evolve at both population and metapopulation levels. The evolutionarily stable strategy (ESS) at the population level is a balanced investment between reproduction and immunity that maintains parasites, even though the host has the capacity to eliminate parasites. The host exhibiting the ESS can always invade other host populations through parasite-mediated competition, effectively using the parasites as biological weapons. At the metapopulation level, the dominant strategy is sometimes different from the population-level ESS, and depends on the ratio of local extinction rate to host colonization rate. This study may help to explain why parasites are as common as they are, and can serve as a modeling framework for investigating parasite-mediated ecological invasions. Furthermore, this work highlights the possibility that the ‘introduction of enemies’ process may facilitate species invasion.     

Effects of phenological constraints on sex allocation in cosexual monocarpic plants

The effects of two phenological constraints in resource investment to reproduction – resource limitation at the flowering stage and unpredictability of resources gained after flowering – on the resource allocation between male and female functions in monocarpic plants are considered using the ESS (evolutionarily stable strategy) approach. The model predicts that the sex allocation including the seed maturation stage has a female bias, when the quantity of reproductive resources available at flowering is small compared with that which is obtained after flowering, or when the cost of seed maturation relative to ovule production is low. The fluctuation of the quantity of resources available for seed maturation favors overproduction of ovules. As a result, more resources are allocated to female function and less to male function at flowering. The ESS allocation depends on the variability of resources and the cost of seed maturation relative to ovule production. The probability that total resource allocation has a female bias becomes higher than 0.5, and it depends on the cost of seed maturation relative to ovule production rather than resource variability. On the other hand, the probability that resource allocation has a female bias decreases with resource variability if we assume that the floral sex ratio is fixed. Future studies of plant sex allocation would profit by taking account of the phenological process of reproduction such as ovule production or seed maturation.     

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.     

Evolutionary stability: States and strategies

Different aspects and modifications of the definition of an evolutionarily stable (ES) strategy that have been considered in the literature can be incorporated in a unifying concept which regards the population context. This concept of evolutionary stability will generally characterize population states in both pure- and mixed-strategist models. In particular, it includes ES strategies, represented as a phenotype unique in an ES population. For an important class of mixed-strategist models, no strict ESS can exist. This will be the case whenever the success of an individual strategy is considered to follow as an average from the successes of its behavioural components. Instead, ESS results may be obtained from what will be called a “degenerate” form of the model, which is simply an ESS model on the level of elementary actions. Then, however, the correct interpretation of an ESS is not an individual phenotype but rather a population mixture of elementary actions. If an ES state exists in a mixed-strategist model it may be determined by an equilibrium condition; if there is an ES strategy, a different approach—mainly maximum considerations—is needed for finding it. An equilibrium condition does not hold for the components of an ES strategy straightforwardly; but it can be derived in terms of an auxiliary ESS model that considers first-order effects of the components. Several examples illustrate the significance of these results. Particularly, two models on “Games between Relatives” are reconsidered in order to display both their formal interrelation and the different meaning of their results in the context of mixed-strategist models.     

Population dynamics and harvesting of semelparous species with phenotypic and genotypic variability in reproductive age

We study the evolution of polymorphic life histories in anadromous semelparous salmon and the effects of harvesting. We derive dynamic phenotypic and genetic ESS models for describing the evolutionary dynamics. We show in our deterministic analysis that polymorphisms are not possible in a panmictic random mating population. Instead, genetic or behavioral polymorphisms may be observed in populations with assortative mating systems. Positive assortative mating may be supported and generated by behavioral and phenotypic traits like male mate choice, spawning ground selection by phenotype, or within-river homing-migration-distance by size. In the case of an evolutionarily stable dimorphism, the ESS is characterized by a reproductive ideal free distribution such that at an equilibrium the individuals are indifferent from the fitness point of view between the two life histories of early and late reproduction. Different strategy models - that is, phenotypic and genetic ESS models - yield identical behavioral predictions and, consequently, genetics does not seem to play an important role in the present model. An evolutionary response to increased fishing mortality is obvious and may have resource management implications. High sea fishing mortalities drive the populations toward early spawning. Thus it is possible that unselective harvesting at sea may eliminate, depending on the biological system, behavioral polymorphisms or genetic heterozygozity and drive the population to a monomorphic one. If within-river homing migration distances depend on the size of fish, unselective harvesting at sea, or selective harvesting of spawning runs in rivers, may reduce local population sizes on spawning grounds high up rivers. Finally, harvesting in a population may cause a switch in a dominant life-history strategy in a population so that anticipated sustainable yields cannot be realized in practice.     

Optimal floating and queuing strategies: consequences for density dependence and habitat loss

Field studies of many vertebrates show that some individuals (floaters) do not defend territories even when there is space for them to do so. We show that the evolutionarily stable strategy (ESS) for the threshold territory quality at which floating takes place is that which maximizes the size of the floating population (but not the total population, breeding population, or reproductive output). The ESS is solved separately for two assumptions: whether individuals wait to occupy a single territory or multiple territories and whether queuing rules are strict or if all waiting individuals are equally likely to obtain the next territory. The four combinations of these assumptions all give the same evolutionarily stable population size of both floaters and breeders. At the ESS, only territories with expected lifetime reproductive success (LRS) exceeding 1 should be occupied, which introduces a limit to ideal habitat selection. The behavioral decision to float alters the shape of the density-dependent response, reduces the equilibrium population size, and affects the response of the population to habitat loss. Specifically, the floater: breeder ratio is directly related to average breeding habitat quality, and the floater population size will decrease more than the breeding population size if better than average quality habitat is lost.     

Asynchronous pupation of univoltine insects as evolutionarily stable phenology

Evolutionarily stable seasonal timing of larval feeding stages is studied theoretically for univoltine insects. In the evolutionarily stable (or ESS) population, each individual maximizes its own lifetime reproductive success by choosing the hatching and pupation dates, given the resource availability curve with a peak in the middle of a year, a higher daily mortality in the feeding stages, and the growth rate decreasing with the larval biomass in the population. If growth rate is proportional to the body size, the population at the ESS is composed of a mixture of phenotypes differing in hatching and pupation, but pupation interval over which some popation occur every day is much longer than hatching interval. If growth rate increases with the body size at a speed slower than linearly, large sized larvae should pupate earlier than small ones.