Evolutionary dynamics in frequency-dependent two-phenotype models

General frequency-dependent selection models based on two phenotypic classes are analyzed with underlying one-locus multiallele phenotypic determination systems in diploid populations. It is proved that the mean phenotypic fitnesses tend to equality over discrete generations and genetic mutations if a phenotypic polymorphism is to be maintained. The exact conditions are examined. The present results are valid for a wide class of models whenever random groupings or assortative patterns based on phenotype and affecting fitness, linearly or not, are independent of sex, mating preferences, or kinship. They can also be applied to two-sex haploid models.     

Evolutionary stability in strategic models of single-locus frequency-dependent viability selection

The dynamic stability of an evolutionarily stable strategy (ESS) is analyzed for a diploid species under individual viability selection. An individual's viability depends on the genotypic frequencies at a single autosomal locus through a payoff matrix determined by phenotypic behaviours (i.e. strategies). It is shown that an ESS of this payoff matrix is dynamically stable if there are at most three alleles — an intuitive result that strengthens the importance of static game-theoretic methods in genetic models.Author for correspondence     

Stochastic fluctuations in frequency-dependent selection: a one-locus, two-allele and two-phenotype model

Stochastic fluctuations in a simple frequency-dependent selection model with one-locus, two-alleles and two-phenotypes are investigated. The steady-state statistics of allele frequencies for an interior stable phenotypic equilibrium are shown to be similar to the stochastic fluctuations in standard evolutionary game dynamics [Tao, Y., Cressman, R., 2007. Stochastic fluctuations through intrinsic noise in evolutionary game dynamics. Bull. Math. Biol. 69, 1377-1399]. On the other hand, for an interior stable phenotypic or genotypic equilibrium, our main results show that the deterministic model cannot be used to predict the expectation of phenotypic frequency. The variance of phenotypic frequency for an interior stable genotypic equilibrium is more sensitive to the expected population size than for an interior stable phenotypic equilibrium. Furthermore, the stochastic fluctuations of allele frequency and phenotypic frequency can be considered approximately independent of each other for these genotypic equilibria, but not for phenotypic.     

Evolutionary stability concepts for N-species frequency-dependent interactions.

The classical static concept of an evolutionarily stable strategy (ESS) for a single species gives rise to two new notions when there are more than two species (called an N-species ESS and RL-stability). The paper relates these to the dynamic stability of monomorphic and polymorphic evolutionary systems. It is shown that RL-stability implies the global asymptotic stability of either system with or without mutations. However, the N-species ESS only implies stability of the monomorphic system.     

Host-parasite evolution: general principles and avian models

edited by D.H. Clayton and J. Moore, Oxford University Press, 1997. pound60.00 (hbk)/ pound25.00 (pbk) (xi+473 pages) ISBN 0 19 854893 1/0 19 854892/3.     

Measurement of frequency-dependent sexual activity in Drosophila Melanogaster

In experiments on sexual competition in Drosophila melanogaster, the course of mating succes with time is represented by a sigmoid curve. By logarithmic transformation such curves are changed into straight lines that can be compared by covariance analysis. This method allows discrimination of the behaviour of the two types in competition, and allows us to follow it in the course of time. From a sexual competition experiment between the wild type Canton S and the mutant white-ebony we conclude that sexual activity of males and females of both types is generally frequency dependent, with evidence of rare-female advantage as well as rare-male advantage.     

Evolutionary stability for large populations

We present a revision of Maynard Smith's evolutionary stability criteria for populations which are very large (though technically finite) and of unknown size. We call this the large population ESS, as distinct from Maynard Smith's infinite population ESS and Schaffer's finite population ESS. Building on Schaffer's finite population model, we define the large population ESS as a strategy which cannot be invaded by any finite number of mutants, as long as the population size is sufficiently large. The large population ESS is not equivalent to the infinite population ESS: we give examples of games in which a large population ESS exists but an infinite population ESS does not, and vice versa. Our main contribution is a simple set of two criteria for a large population ESS, which are similar (but not identical) to those originally proposed by Maynard Smith for infinite populations.     

Evolutionary ecology,sexual conflict,and behavioral differentiation among baboon populations

A central assumption of baboon socio‐ecological models is that all populations have the same capacity to react to different environments. The burden of our argument is that this assumption needs to be reconsidered. Data suggest not only that hamadryas, but chacma as well, differ in interesting ways from the stock baboon model that has been derived, in the main, from earlier work on anubis and cynocephalus. Although environmental factors are behind these differences, much of their influence is a consequence of their effect on restricted ancestral populations, where selection for appropriate responses to the social challenges set by local conditions now constrains the nature of individual responses to contemporary environments. Available genetic evidence suggess a southern African origin for Papio at a time when climatic conditions were certainly no better than they are now and when temperatures, if nothing else, were probably lower. In light of this, a reconstruction of how climate has structured the sexual conflict between males and female charcma, which itself hinges on infanticide, can help explain not only the East African pattern, but also how the apparently anomalous hamadryas pattern has been derived.     

Maximization principles for frequency-dependent selection I: the one-locus two-allele case

In this article we study the one-locus two-allele version of the pairwise-interaction model of frequency-dependent selection in discrete and continuous time. Our main aim is to provide necessary and sufficient conditions for the validity of maximization principles. We provide a systematic approach that covers all possible facets of the dynamical behavior of the model, and we illustrate our results by concrete examples. We show that the mean fitness of the population is nondecreasing if the interaction coefficients are symmetric and positive. Moreover, monotonic convergence to the set of equilibria always occurs, which is not true if we also consider negative interaction coefficients. For asymmetric interaction, we provide necessary conditions when the mean fitness is nondecreasing and sufficient conditions when it is not. Furthermore, in discrete time, we show that limit cycles cannot occur, unless some interaction coefficients are negative.     

Maximization principles for frequency-dependent selection II: the one-locus multiallele case

In this article we study a single-locus multiallele version of the pairwise-interaction model (PIM) in discrete and continuous time and a density-dependent version of this model (D-PIM) in continuous time. The PIM assumes that the fitnesses of genotypes are proportional to the average amount of competition resulting from pairwise interactions. Hence, fitness is frequency dependent. Our main aim is to provide necessary and sufficient conditions for the validity of maximization principles analogous to Fisher’s Fundamental Theorem for constant selection. We provide a systematic analysis and illustrate our results by concrete examples. We show that in discrete time the mean fitness is nondecreasing along every trajectory provided the interaction coefficients are nonnegative and symmetric. For asymmetric interactions this is in general not true. However, for what we call pseudo-symmetric interactions a function similar to, but in general not identical to, the mean fitness: the adjusted-mean fitness, is nondecreasing along trajectories. For asymmetric interactions, we also provide sufficient conditions for the mean fitness, and more generally for the adjusted-mean fitness, to be nondecreasing and sufficient conditions when it is not. In continuous time, we provide similar but stronger results. If the interaction coefficients are pseudo-symmetric, the adjusted-mean fitness is nondecreasing in the D-PIM.     

Evolutionary games for spatially dispersed populations

An evolutionary game model is developed that incorporates both spatial dispersion and density effects in the evolutionary dynamic. It is shown that a stable equilibrium (e.g. an evolutionarily stable strategy) of the non-dispersed frequency dynamic becomes a stable equilibrium of the larger system if population density stabilizes at these fixed frequencies. It is also shown, by example, that other equilibria, whose frequencies change from one location to another, may appear when dispersal rates are relatively small.Research supported by Natural Sciences and Engineering Research Council of Canada Operating Grant A6187Research supported by Natural Sciences and Engineering Research Council of Canada Operating Grant A7822     

Evolutionary dynamics in structured populations

Evolutionary dynamics shape the living world around us. At the centre of every evolutionary process is a population of reproducing individuals. The structure of that population affects evolutionary dynamics. The individuals can be molecules, cells, viruses, multicellular organisms or humans. Whenever the fitness of individuals depends on the relative abundance of phenotypes in the population, we are in the realm of evolutionary game theory. Evolutionary game theory is a general approach that can describe the competition of species in an ecosystem, the interaction between hosts and parasites, between viruses and cells, and also the spread of ideas and behaviours in the human population. In this perspective, we review the recent advances in evolutionary game dynamics with a particular emphasis on stochastic approaches in finite sized and structured populations. We give simple, fundamental laws that determine how natural selection chooses between competing strategies. We study the well-mixed population, evolutionary graph theory, games in phenotype space and evolutionary set theory. We apply these results to the evolution of cooperation. The mechanism that leads to the evolution of cooperation in these settings could be called ‘spatial selection’: cooperators prevail against defectors by clustering in physical or other spaces.     

Polymorphism of melanic ladybirds maintained by frequency-dependent sexual selection

Sexual selection, whether by female preference or male competition, is almost inevitably frequency-dependent. Female preference gives rise to a 'rare male effect', by which the rarer male phenotypes gain a relatively greater selective advantage. In addition to this effect, the proportion of females expressing a preference may also be frequency-dependent.
Frequency-dependent expression of mating preference can arise in at least two ways: (1) when females encounter a succession of courting males while searching for a male they prefer; (2) when females chose a male from within a lek. Models of mating behaviour reveal a clear distinction between the frequency dependence in the expression of female preference and the frequency dependence in the consequent selection of the males. When expression of preference is highly dependent on frequency, the selection of males is constant or only slightly frequency-dependent: constant expression of preference produces high frequency dependence of selection. Analysis of general models shows that genetic polymorphisms can be maintained under a wide range of conditions.
The ladybird, Adalia bipunctata , is polymorphic for several melanic and non-melanic phenotypes. Females have a genetically determined preference for melanic males. Non-melanic phenotypes mate assortatively. By estimating the parameters of a detailed model of natural selection, sexual selection and assortative mating, it has been shown that the Adalia bipunctata polymorphism will be maintained at frequencies observed in the wild.     

Evolutionary game dynamics in finite populations

We introduce a model of stochastic evolutionary game dynamics in finite populations which is similar to the familiar replicator dynamics for infinite populations. Our focus is on the conditions for selection favoring the invasion and/or fixation of new phenotypes. For infinite populations, there are three generic selection scenarios describing evolutionary game dynamics among two strategies. For finite populations, there are eight selection scenarios. For a fixed payoff matrix a number of these scenarios can occur for different population sizes. We discuss several examples with unexpected behavior.     

Interplay of Darwinian and frequency-dependent selection in the host-associated microbial populations

In order to analyze the microevolutionary processes in host-associated microorganisms, we simulated the dynamics of rhizobia populations composed of a parental strain and its mutants possessing the altered fitness within "plant-soil" system. The population dynamics was presented as a series of cycles (each one involves "soil-->rhizosphere-->nodules-->soil" succession) described using recurrent equations. For representing the selection and mutation pressures, we used a universal approach based on calculating the shifts in the genetic ratios of competing bacterial genotypes within the particular habitats and across several habitats. Analysis of the model demonstrated that a balanced polymorphism may be established in rhizobia population: mutants with an improved fitness do not supplant completely the parental strain while mutants with a decreased fitness may be maintained stably. This polymorphism is caused by a rescue of low-fitted genotypes via negative frequency-dependent selection (FDS) that is implemented during inoculation of nodules and balances the Darwinian selection that occurs during multiplication or extinction of bacteria at different habitats. The most diverse populations are formed if the rhizobia are equally successful in soil and nodules, while a marked preference for any of these habitats results in the decrease of diversity. Our simulation suggests that FDS can maintain the mutualistic rhizobia-legume interactions under the stress conditions deleterious for surviving the bacterial strains capable for intensive N2 fixation. Genetic consequences of releasing the modified rhizobia strains may be addressed using the presented model.     

Evolutionary inevitability of sexual antagonism

Sexual antagonism, whereby mutations are favourable in one sex and disfavourable in the other, is common in natural populations, yet the root causes of sexual antagonism are rarely considered in evolutionary theories of adaptation. Here, we explore the evolutionary consequences of sex-differential selection and genotype-by-sex interactions for adaptation in species with separate sexes. We show that sexual antagonism emerges naturally from sex differences in the direction of selection on phenotypes expressed by both sexes or from sex-by-genotype interactions affecting the expression of such phenotypes. Moreover, modest sex differences in selection or genotype-by-sex effects profoundly influence the long-term evolutionary trajectories of populations with separate sexes, as these conditions trigger the evolution of strong sexual antagonism as a by-product of adaptively driven evolutionary change. The theory demonstrates that sexual antagonism is an inescapable by-product of adaptation in species with separate sexes, whether or not selection favours evolutionary divergence between males and females.     

Evolutionary game dynamics in diploid populations

Selection that influences behaviour can be studied using game theory if individual behavioural success depends on the frequencies of various behavioural types in the population. The evolutionarily stable strategy of J. Maynard Smith and G. R. Price (1973. Nature (London) 246, 15–18) is an equilibrium concept like the solution of a game. The dynamic model of Taylor and Jonker, studied in detail by Zeeman, goes beyond game theory using fitness to cause evolution, perhaps towards an equilibrium. A diploid version of their haploid model is considered and it is found that diploid evolution can be quite different. For example “catastrophic” bifurcations can occur between stable internal polymorphisms when the game matrix entries are changed slowly. A slight drop in food supply may cause extinction. Totally unfit altruistic genotypes can be maintained if they help the rest of the population. The relation of haploid game models to constant selection in diploids is also discussed.     

Stochastic models for toxicant-stressed populations

We obtain conditions for the existence of an invariant distribution on (0, ∞) for stochastic growth models of Ito type. We interpret the results in the case where the intrinsic growth rate is adjusted to account for the impact of a toxicant on the population. Comparisons with related results for ODE models by Hallamet al. are given, and consequences of taking the Stratonovich interpretation for the stochastic models are mentioned.     

Primate models for australopithecine sexual dimorphism

Several different models of sexual dimorphism in the South African australopithecines are compared with sexual dimorphism in the living primates. Australopithecine dimorphism is placed in an evolutionary context, and contrasting trends in the hominid and pongid lineages are shown. Evidence suggesting that the australopithecines were an extremely polytypic taxon is presented, and a high level of both inter- and intra-population variation is indicated.