Genetical ESS-models. I. Concepts and basic model

Evolutionarily Stable Strategies (ESS) in phenotypic models are used to explain the evolution of animal interactive behaviour. As the behavioural features under consideration are assumed to be genetically determined, the question arises how underlying a genetical system might affect the results of phenotypic ESS-models. This question can be fully treated in terms of ESS-theory. A method of designing Genetical ESS-Models is proposed, which transfers the question of evolutionary stability to a "lower" level, the genetical basis. Genetical ESS-models - although nonlinear even in the simplest cases - can be analysed in a way that is familiar to ESS-theorists and yield immediate results on gene pool ESSs, which then may or may not maintain ESSs on the phenotypic level. Moreover, general results can be obtained to characterize evolutionarily stable gene pool states and their interrelation with commonsense, phenotypic ESSs. This part of the article presents the basic concepts and an outline of the method of genetical ESS-models. It gives, as a demonstration, a complete analysis for phenotypic two-strategy models (linear or nonlinear) based on a diploid, diallelic single-locus system under random mating. The results in this case suggest that a phenotypic ESS should indeed be expected to evolve but, maybe, only after passing through a succession of temporarily stable states.     

When will a sexual population evolve to an ESS?

A diploid, Mendelian population is considered in which m alleles at a single autosomal locus uniquely determine the phenotype of each individual. In the population, a game-theoretical conflict is supposed. If the genetic system is able to uniquely realize the phenotypic evolutionarily stable strategy (ESS) state then the sexual population will evolve to this ESS.     

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 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.     

Coincidence of ESAD and ESS in dominant-recessive hereditary systems

The paper deals with the following question: when do the phenotypic evolutionarily stable state (ESS) and the evolutionarily stable allele distribution (ESAD) coincide? It is supposed that for a sexual population, in dominant-recessive inheritance system, n allele at one autosomal locus determine n possible pure individual phenotypes and each pure phenotype is obtained as the phenotype of a homozygote. Under these conditions, earlier results of the authors imply that, if a phenotype distribution is an ESS then the allele distribution generating it is an ESAD. In this paper, apart from a certain degenerate pay-off matrices, the inverse statement is also proved: if a distribution is an ESAD then the corresponding phenotypic distribution is an ESS.     

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.     

Evolutionary limits to the frequency of aggression between related or unrelated conspecifics in diploid species with simple mendelian inheritance

Game theory has been used by some authors to analyse evolutionary limits to the expression of aggression in theoretical haploid parthenogenetic species. Others have examined frequency dependent selection, of which aggression may be a case, by applying population genetic models to diploid species. A model is presented which attempts to combine these two approaches. Game theory is used to determine evolutionarily stable strategies and corresponding stable polymorphisms for a two-strategy game played by members of a diploid sexual species, when choice of strategy is determined by two alleles at a single locus. Results are given for dominant, co-dominant and recessive determination of choice of the more aggressive of two strategies, for two levels of relationship: unrelated players and sibs. It is found that for a range of models of single locus inheritance the evolutionarily stable strategy (ESS) determined for haploid species remains the stable population strategy for diploid sexual species, when players are unrelated. In sibling contestants aggression is reduced. The mixed strategy haploid ESS underestimates, but the pure strategy haploid ESS provides a good indication of the degree to which relatedness lessens aggression in diploid species. For both haploid and diploid species there may be a considerable advantage to confining conflicts to kin.     

On evolutionarily stable sets

As an extension of the concept of an evolutionarily stable strategy (ESS) evolutionarily stable sets are introduced, i.e. sets of equilibrium strategies (EQS) which have much of the properties of an ESS. They are primarily used with evolutionary game models that allow a continuum of EQSs, none of which can be an ESS, but also include common ESSs as a special case. For a large class even of nonlinear models it can be shown that the standard dynamics converge towards some equilibrium point in an ES set if started within a neighbourhood of the set. Important applications of ES sets include e.g. mixed-strategist models and evolutionary game models in sexual populations.     

The cost of dishonesty

The handicap principle states that stable biological signals must be honest and costly to produce. The cost of the signal should reflect the true quality of the signaller. Here, it is argued that honest signalling may be maintained although the used signals are not handicaps. A game theoretic model in the form of a game of signalling is presented: all the existing evolutionarily stable strategies (ESSs) are found. Honest and cheap signalling of male quality is shown to be evolutionarily stable if females divorce the mate if it turns out that he has cheated about his quality. However, for this ESS to apply, the cost of lost time must not be too great. The stability of the honest signalling is based on deceivers being prevented from spreading in the population because they suffer from a cost of divorce. Under some fairly strict conditions, a mixed polymorphism of dishonesty and honesty represents another possible ESS.     

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.     

Models on butterfly protandry: Virgin females are at risk to die

Current models on protandry in butterflies assume that females are mated instantaneously upon eclosion. However, for most butterfly species this assumption is not realistic. In this paper a model is formulated in which the mating rate depends on both male and female density. Given the female presence curve, protandry is an evolutionarily stable strategy (ESS) for males. The evolutionarily stable amount of protandry decreases with increasing death rate and decreasing encounter rate. Given the male presence curve, protandry also is an ESS for females. However, male and female ESS are not identical; moreover, in the present model a simultaneous ESS does not exist. Protandry critically depends on the assumption that females mate only once, whereas males are capable of multiple mating. If females too are capable of multiple mating, absence of protandry is the ESS for males as well as females. The model predicts that protandry depends on population density: protandry should be more pronounced in populations with high density than in populations with low density. Protandry also depends on sex ratio. It becomes more pronounced when the proportion of males among emerging adults increases.     

Modelling the adaptive dynamics of traits involved in inter- and intraspecific interactions: An assessment of three methods

In recent years, three related methods have been used to model the phenotypic dynamics of traits under the influence of natural selection. The first is based on an approximation to quantitative genetic recursion equations for sexual populations. The second is based on evolution in asexual lineages with mutation-generated variation. The third method finds an evolutionarily stable set of phenotypes for species characterized by a given set of fitness functions, assuming that the mode of reproduction places no constraints on the number of distinct types that can be maintained in the population. The three methods share the property that the rate of change of a trait within a homogeneous population is approximately proportional to the individual fitness gradient. The methods differ in assumptions about the potential magnitude of phenotypic differences in mutant forms, and in their assumptions about the probability that invasion or speciation occurs when a species has a stable, yet invadable phenotype. Determining the range of applicability of the different methods is important for assessing the validity of optimization methods in predicting the evolutionary outcome of ecological interactions. Methods based on quantitative genetic models predict that fitness minimizing traits will often be evolutionarily stable over significant time periods, while other approaches suggest this is likely to be rare. A more detailed study of cases of disruptive selection might reveal whether fitness-minimizing traits occur frequently in natural communities.     

Breeding systems in flowering plants and the control of variability

Plants have three basic means of reproduction, by outcrossing, by selfing, and asexually. In most plant populations, at least two and often all three of these options are everpresent, so that individuals adopt mixed mating strategies at evolutionarily stable strategy (ESS) threshholds. Because mating systems are genetically controlled and affect genotype structure, they are liable to feedback. Productive habitats with a large standing crop are more likely to favour outcrossing, while unproductive habitats may favour asexuality or selfing, so that mating systems may change through seral development, even within the same species. Outcrossing tends to break up linkage disequilibria, but may also favour the creation of adaptive linkage groups. Mechanisms whereby male sexual selection, small population size and selfing can influence the genetic structure of populations are examined.     

Blood groups and natural selection by genetical environment

A survey is given of a number of investigations indicating the importance in natural selection of the genetical environment of populations and individuals.In the introduction it is observed that blood group frequency patterns are very stable, even in very small populations, and appear independent of environmental factors. They appear to be race-specific, maintained by a process of natural selection which is dependent of the racial genetical composition. Indications in favour of this hypothesis are obtained from several studies carried out in populations of mixed origin:The introduction on a small scale into the populations of New Guinea of foreign elements with some S genes may result in a population with relatively high S frequencies; the frequencies of certain of the blood group genes in the mixed negroid populations of Curaçao are not in agreement with the racial compositions of the mixtures as they have been calculated from the frequencies of other blood group genes, and the same appears to be the case in the populations of the Himalayas. The marked variation of MNSsHe frequencies in Africa may perhaps be explained by a powerful selective pressure exercized by the genetical backgrounds of the various populations, as is demonstrated by the absence of some expected genotypes among male Bush Negroes in Surinam.The effects of natural selection by genetical environment can also be demonstrated by family studies:In families with elliptocytosis Rhesus segregation shows some deviation from Mendelian laws, and the ratio of elliptocytosis-positive and-negative children appears to depend on the Rh genotype of the elliptocytosis-positive parent. From blood group studies in selected pedigrees and dizygotic twins it appears that twin pairs are more often doubly concordant for both MN and Rh than is to be expected.Some implications of the observed effects of natural selection in the study of human genetics and population dynamics are briefly discussed.     

Observability in dynamic evolutionary models

In the paper observability problems are considered in basic dynamic evolutionary models for sexual and asexual populations. Observability means that from the (partial) knowledge of certain phenotypic characteristics the whole evolutionary process can be uniquely recovered. Sufficient conditions are given to guarantee observability for both sexual and asexual populations near an evolutionarily stable state.     

Quantifying male attractiveness

Genetic models of sexual selection are concerned with a dynamic process in which female preference and male trait values coevolve. We present a rigorous method for characterizing evolutionary endpoints of this process in phenotypic terms. In our phenotypic characterization the mate-choice strategy of female population members determines how attractive females should find each male, and a population is evolutionarily stable if population members are actually behaving in this way. This provides a justification of phenotypic explanations of sexual selection and the insights into sexual selection that they provide. Furthermore, the phenotypic approach also has enormous advantages over a genetic approach when computing evolutionarily stable mate-choice strategies, especially when strategies are allowed to be complex time-dependent preference rules. For simplicity and clarity our analysis deals with haploid mate-choice genetics and a male trait that is inherited phenotypically, for example by vertical cultural transmission. The method is, however, easily extendible to other cases. An example illustrates that the sexy son phenomenon can occur when there is phenotypic inheritance of the male trait.     

Co-evolution of seed size and seed predation

Using the evolutionarily stable strategy (ESS) approach in a model for the co-evolution of seed size and seed predation, I show that seed size variation within individual plants is favoured if there is a trade-off in the predator's attack rate for different seed sizes. A single seed size is not evolutionarily stable because a predator that is optimally adapted to one particular seed size cannot prevent invasion by plants with a different seed size. The model generates the following predictions. The ESS consists of a continuous range of seed sizes. Small seeds tend to be attacked more frequently than big seeds. Plants with many resources and plants with low (frequency-independent) juvenile mortality have more variable seeds than plants with few resources and a high juvenile mortality. Seed size variation is higher in fluctuating populations regulated by seed predation alone than in stable populations (partially) regulated by seedling competition. Predator searching behaviour does not directly affect the ESS seed size range, but may have an indirect effect by affecting population stability or the significance of seedling competition as a population regulating mechanism. Moreover, seed size distributions are found to be more skewed in favour of small seeds if predation is spatially non-uniform than if predation is more even. Application of the model to systems of several co-evolving plant and predator species is discussed.     

The evolution of parasite virulence and transmission rate in a spatially structured population

If the transmission occurs through local contact of the individuals in a spatially structured population, the evolutionarily stable (ESS) traits of parasite might be quite different from what the classical theory with complete mixing predicts. In this paper, we theoretically study the ESS virulence and transmission rate of a parasite in a lattice-structured host population, in which the host can send progeny only to its neighboring vacant site, and the transmission occurs only in between the infected and the susceptible in the nearest-neighbor sites. Infected host is assumed to be infertile. The analysis based on the pair approximation and the Monte Carlo simulation reveal that the ESS transmission rate and virulence in a lattice-structured population are greatly reduced from those in completely mixing population. Unlike completely mixing populations, the spread of parasite can drive the host to extinction, because the local density of the susceptible next to the infected can remain high even when the global density of host becomes very low. This demographic viscosity and group selection between self-organized spatial clusters of host individuals then leads to an intermediate ESS transmission rate even if there is no tradeoff between transmission rate and virulence. The ESS transmission rate is below the region of parasite-driven extinction by a finite amount for moderately large reproductive rate of host; whereas, the evolution of transmission rate leads to the fade out of parasite for small reproductive rate, and the extinction of host for very large reproductive rate.     

Short- vs. long-range disperser: the evolutionarily stable allocation in a lattice-structured habitat

The population dynamics of two types of organisms in a lattice-structured habitat are studied and the evolutionarily stable allocation between short- and long-range disperser is calculated. Offsprings of short-range dispersal stay in the vicinity of their parent and cause local competition. Using pair approximation, I derive a closed system of ordinary differential equations of global and local densities (or mean crowding), and calculate the condition for one type to invade the population dominated by the other type. The evolutionarily stable strategy (ESS) of resource allocation is derived for the case in which there is a linear trade-off between short- and long-range dispersers. The maximum equilibrium abundance of the population may be achieved by a mixture of both types of dispersers, but it is in general different from the ESS resource allocation calculated from the invasibility condition. For the same parameter values, the ESS invests a larger fraction of resources to short-range disperser than the optimal allocation which maximizes the equilibrium population density. This difference can be explained by the fact that long-range disperser is more effective in the preoccupation of space than short-range disperser. The predictions are confirmed by the direct computer simulations of the lattice stochastic models. Copyright 1999 Academic Press.