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.     

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.     

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     

ESSs with two types of players II (intra- and interspecific competition between haploid species)

The investigation of strategy evolution resulting from animal behavior is continued using a new approach to multispecies evolutionary games. A different interpretation of the known result that contestants in asymetrical games choose pure strategies is proven. This leads to a classification of all evolutionarily stable strategies (ESS) when there is no selection from interspecific competitions. In case the dimension of the strategy space is restricted as in a diallelic haploid model, the phase portrait of the evolutionary dynamics is illustrated. The results are then compared with the previous approach and to models of sex dependent selection in randomly mating diploid species.     

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.     

Genetical ESS-models. II. Multi-strategy models and multiple alleles

The problem of evolutionarily stable strategies (ESS) in sexual populations can be investigated by means of genetical ESS-models which link common sense, phenotypic ESS-models to an underlying genetical system. Thorough results are obtained for multi-strategy models in diploid, panmictic populations on the basis of multi-allelic, one-locus systems. A sexual population will be maintained at a phenotypic ESS if this can possibly be produced by the genotypes currently existing. If there is enough allelic variation, the corresponding gene pool may either be an ESS itself, or belong to an attracting, continuous set of states, which all determine the same evolutionarily stable population. The latter case allows new alleles to enter and spread in the gene pool without disturbing the phenotypic ESS. If a phenotypic ESS cannot be established, ESSs of the genetical model may be found which give rise to stable populations alternatively. Since these depend on the phenotypes determined by the currently existing genotypes, they may be destabilized by the occurrence of new mutations. In this sense, they are less durable than populations maintained at a phenotypic ESS and can be expected to evolve, in the long run, towards a phenotypic ESS.     

Deleterious Mutations and the Origin of the Meiotic Ploidy Cycle

A population genetical model is investigated in which the organism either alternates between diploid and haploid states or lives entirely in the haploid state. The behavior of the organism is determined by the genotype at a modifier locus. At an independent locus deleterious mutations occur at a low but constant frequency. It is found that the haploid behavior is always an evolutionarily attainable stable trait, while the ploidy-cyclic behavior is an evolutionarily attainable stable trait only when a certain condition holds. This condition depends on the strength of selection, the degree of "sheltering" given by the heterozygote state, and the degree of linkage between the modifier locus and the locus under selection. The last result leads to the speculation that the eukaryotes are derived from an organism which first developed more than one chromosome before it evolved the ploidy cycle.     

Host–parasite interactions and the evolution of nonrandom mating

Some species mate nonrandomly with respect to alleles underlying immunity. One hypothesis proposes that this is advantageous because nonrandom mating can lead to offspring with superior parasite resistance. We investigate this hypothesis, generalizing previous models in four ways: First, rather than only examining invasibility of modifiers of nonrandom mating, we identify evolutionarily stable strategies. Second, we study coevolution of both haploid and diploid hosts and parasites. Third, we allow for maternal parasite transmission. Fourth, we allow for many alleles at the interaction locus. We find that evolutionarily stable rates of assortative or disassortative mating are usually near zero or one. However, for one case, in which assumptions most closely match the major histocompatibility complex (MHC) system, intermediate rates of disassortative mating can evolve. Across all cases, with haploid hosts, evolution proceeds toward complete disassortative mating, whereas with diploid hosts either assortative or disassortative mating can evolve. Evolution of nonrandom mating is much less affected by the ploidy of parasites. For the MHC case, maternal transmission of parasites, because it creates an advantage to producing offspring that differ from their parents, leads to higher evolutionarily stable rates of disassortative mating. Lastly, with more alleles at the interaction locus, disassortative mating evolves to higher levels.     

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.     

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.     

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.     

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.     

Evolutionary Stability in the Asymmetric Volunteer's Dilemma

It is often assumed that in public goods games, contributors are either strong or weak players and each individual has an equal probability of exhibiting cooperation. It is difficult to explain why the public good is produced by strong individuals in some cooperation systems, and by weak individuals in others. Viewing the asymmetric volunteer''s dilemma game as an evolutionary game, we find that whether the strong or the weak players produce the public good depends on the initial condition (i.e., phenotype or initial strategy of individuals). These different evolutionarily stable strategies (ESS) associated with different initial conditions, can be interpreted as the production modes of public goods of different cooperation systems. A further analysis revealed that the strong player adopts a pure strategy but mixed strategies for the weak players to produce the public good, and that the probability of volunteering by weak players decreases with increasing group size or decreasing cost-benefit ratio. Our model shows that the defection probability of a “strong” player is greater than the “weak” players in the model of Diekmann (1993). This contradicts Selten''s (1980) model that public goods can only be produced by a strong player, is not an evolutionarily stable strategy, and will therefore disappear over evolutionary time. Our public good model with ESS has thus extended previous interpretations that the public good can only be produced by strong players in an asymmetric game.     

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.     

A theory for the evolutionary game

We present an evolutionary game theory. This theory differs in several respects from current theories related to Maynard Smith's pioneering work on evolutionary stable strategies (ESS). Most current work deals with two person matrix games. For these games the strategy set is finite. We consider evolutionary games which are defined over a continuous strategy set and which permit any number of players. Matrix games are included as a bilinear continuous game. However, under our definition, such games will not posses an ESS on the interior of the strategy set. We extend previous work on continuous games by developing an ESS definition which permits the ESS to be composed of a coalition of several strategies. This definition requires that the coalition must not only be stable with respect to perturbations in strategy frequencies which comprise the coalition, but the coalition must also satisfy the requirement that no mutant strategies can invade. Ecological processes are included in the model by explicitly considering population size and density dependent selection.     

Properties of a mixed ESS candidate in continuous strategy sets

An evolutionarily stable strategy (ESS) is a strategy that if almost all members of the population adopt, then this population cannot be invaded by any mutant strategy. An ESS is not necessarily a possible end point of the evolutionary process. Moreover, there are cases where the population evolves towards a strategy that is not an ESS. This paper studies the properties of a unique mixed ESS candidate in a continuous time animal conflict. A member of a group sized three finds itself at risk and needs the assistance of another group member to be saved. In this conflict, a player's strategy is to choose the probability distribution of the interval between the beginning of the game and the moment it assists the player which is at risk. We first assume that a player is only allowed to choose an exponential distribution, and show that in this case the ESS candidate is an attracting ESS; the population will always evolve towards this strategy, and once it is adopted by most members of the population it cannot be invaded by mutant strategies. Then, we extend the strategy sets and allow a player to choose any continuous distribution. We show that although this ESS candidate may no longer be an ESS, under fairly general conditions the population will tend towards it. This is done by characterizing types of strategies that if established in the population, can be invaded by this ESS candidate, and by presenting possible paths of transition from other types of common strategies to this ESS candidate.     

PLOIDY AND EVOLUTION BY SEXUAL SELECTION: A COMPARISON OF HAPLOID AND DIPLOID FEMALE CHOICE MODELS NEAR FIXATION EQUILIBRIA

We compare the stability properties of haploid and diploid models of Fisherian sexual selection (with male contribution limited to sperm) by examining both models at equilibria for which a male trait is fixed or absent. Haploid and diploid two locus diallelic models share the property that the stability of such fixation equilibria is determined by the relationship between the harmonic mean of relative preference values for the common male trait, weighted by the frequency of the preferences, and the relative viability associated with the common male trait. When diploid females with heterozygotic-based preferences express preference strengths intermediate between homozygote-based preferences, then boundary equilibria of haploid and diploid models share many stability properties. However, even with intermediate heterozygote preferences, haploid and diploid models do differ: (1) for a particular frequency of the preference allele, both fixation boundaries can be stable for the diploid model, and (2) with over- or underdominance at the preference locus (a possibility precluded in the haploid model), a fixation boundary in the diploid model may show two switches in its stability state for increasing frequencies of one of the preference alleles. These differences are due not just to the impossibility of dominance in haploid models, but also to the larger number of diploid genotypes.     

Evolutionarily stable strategy and invader strategy in matrix games

In this paper, we consider the concepts of evolutionarily stable strategy (ESS), neighborhood invader strategy (NIS) and global invader strategy (GIS) in single species with frequency-dependent interactions. We find some general relationships among the three concepts in matrix games. The main conclusion is that ESS and NIS are equivalent to each other and are both equivalent to local superiority; a strategy with global superiority must be a GIS; a GIS may not be equivalent to its global superiority in games with more than two players; and in any two-player matrix game a GIS is just equivalent to its global superiority. In two-player games, globally asymptotic stability in the replicator dynamics has also been shown. Equivalent conditions for the three concepts stated by payoff comparisons are given and are applied to examples involved.     

Stability of evolutionarily stable strategies in discrete replicator dynamics with time delay

We construct two models of discrete-time replicator dynamics with time delay. In the social-type model, players imitate opponents taking into account average payoffs of games played some units of time ago. In the biological-type model, new players are born from parents who played in the past. We consider two-player games with two strategies and a unique mixed evolutionarily stable strategy. We show that in the first type of dynamics, it is asymptotically stable for small time delays and becomes unstable for big ones when the population oscillates around its stationary state. In the second type of dynamics, however, evolutionarily stable strategy is asymptotically stable for any size of a time delay.     

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.