Stability properties of underdominance in finite subdivided populations

IN ISOLATED populations underdominance leads to bistable evolutionary dynamics: below a certain mutant allele frequency the wildtype succeeds. Above this point, the potentially underdominant mutant allele fixes. In subdivided populations with gene flow there can be stable states with coexistence of wildtypes and mutants: polymorphism can be maintained because of a migration-selection equilibrium, i.e., selection against rare recent immigrant alleles that tend to be heterozygous. We focus on the stochastic evolutionary dynamics of systems where demographic fluctuations in the coupled populations are the main source of internal noise. We discuss the influence of fitness, migration rate, and the relative sizes of two interacting populations on the mean extinction times of a group of potentially underdominant mutant alleles. We classify realistic initial conditions according to their impact on the stochastic extinction process. Even in small populations, where demographic fluctuations are large, stability properties predicted from deterministic dynamics show remarkable robustness. Fixation of the mutant allele becomes unlikely but the time to its extinction can be long.     

Evolutionary stability and quasi-stationary strategy in stochastic evolutionary game dynamics

Stochastic evolutionary game dynamics for finite populations has recently been widely explored in the study of evolutionary game theory. It is known from the work of Traulsen et al. [2005. Phys. Rev. Lett. 95, 238701] that the stochastic evolutionary dynamics approaches the deterministic replicator dynamics in the limit of large population size. However, sometimes the limiting behavior predicted by the stochastic evolutionary dynamics is not quite in agreement with the steady-state behavior of the replicator dynamics. This paradox inspired us to give reasonable explanations of the traditional concept of evolutionarily stable strategy (ESS) in the context of finite populations. A quasi-stationary analysis of the stochastic evolutionary game dynamics is put forward in this study and we present a new concept of quasi-stationary strategy (QSS) for large but finite populations. It is shown that the consistency between the QSS and the ESS implies that the long-term behavior of the replicator dynamics can be predicted by the quasi-stationary behavior of the stochastic dynamics. We relate the paradox to the time scales and find that the contradiction occurs only when the fixation time scale is much longer than the quasi-stationary time scale. Our work may shed light on understanding the relationship between the deterministic and stochastic methods of modeling evolutionary game dynamics.     

Single-locus gametophytic incompatibility: the symmetric equilibrium is globally asymptotically stable

The deterministic dynamics of the classical single-locus multiple-allele model of gametophytic incompatibility is analyzed with the intention to prove the conjecture that the symmetric state (uniform distribution of genotypes) is the only polymorphic equilibrium and that this equilibrium is globally asymptotically stable in the interior of the frequency simplex. It is shown that the minimum allelic frequency increases strictly over the generations as long as a uniform allelic distribution is not realized. Hence, the minimum allelic frequency is a Ljapunov function for the invariant set of genotypic frequencies characterized by a uniform allelic distribution. Within this set, the uniform genotypic distribution is approached in an exponential fashion, which proves the assertion. An evolutionary optimization rule associated with the global convergence to the symmetric state is implied by the fact that at this state the overall amount of pollen elimination resulting from incompatible crosses is minimized.     

Evolutionary game dynamics with non-uniform interaction rates

The classical setting of evolutionary game theory, the replicator equation, assumes uniform interaction rates. The rate at which individuals meet and interact is independent of their strategies. Here we extend this framework by allowing the interaction rates to depend on the strategies. This extension leads to non-linear fitness functions. We show that a strict Nash equilibrium remains uninvadable for non-uniform interaction rates, but the conditions for evolutionary stability need to be modified. We analyze all games between two strategies. If the two strategies coexist or exclude each other, then the evolutionary dynamics do not change qualitatively, only the location of the equilibrium point changes. If, however, one strategy dominates the other in the classical setting, then the introduction of non-uniform interaction rates can lead to a pair of interior equilibria. For the Prisoner's Dilemma, non-uniform interaction rates allow the coexistence between cooperators and defectors. For the snowdrift game, non-uniform interaction rates change the equilibrium frequency of cooperators.     

Fixation probabilities in evolutionary game dynamics with a two-strategy game in finite diploid populations

Fixation processes in evolutionary game dynamics in finite diploid populations are investigated. Traditionally, frequency dependent evolutionary dynamics is modeled as deterministic replicator dynamics. This implies that the infinite size of the population is assumed implicitly. In nature, however, population sizes are finite. Recently, stochastic processes in finite populations have been introduced in order to study finite size effects in evolutionary game dynamics. One of the most significant studies on evolutionary dynamics in finite populations was carried out by Nowak et al. which describes “one-third law” [Nowak, et al., 2004. Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 646-650]. It states that under weak selection, if the fitness of strategy α is greater than that of strategy β when α has a frequency , strategy α fixates in a β-population with selective advantage. In their study, it is assumed that the inheritance of strategies is asexual, i.e. the population is haploid. In this study, we apply their framework to a diploid population that plays a two-strategy game with two ESSs (a bistable game). The fixation probability of a mutant allele in this diploid population is derived. A “three-tenth law” for a completely recessive mutant allele and a “two-fifth law” for a completely dominant mutant allele are found; other cases are also discussed.     

Evolutionary principles for general frequency-dependent two-phenotype models in sexual populations

The evolutionary dynamics in general two-sex two-phenotype frequency-dependent selection models are studied with respect to underlying multi-allele one-locus genetic systems. Two classes of equilibria come into play: genotypic equilibria, with equilibrium allelic frequencies independent of the phenotype, and phenotypic equilibria, which are characterized by equal mean phenotypic fitnesses. The exact conditions for genotypic equilibria to exist and be stable and for phenotypic equilibria to exist and be evolutionarily attractive are examined. Using adequate definitions of mean fitnesses in general contexts of frequency-dependent selection in dioecious populations, we show that two phenotypes, when they can coexist in the population, tend to balance their fitnesses as far as is allowed by the genetic system as more alleles responsible for phenotype determination are introduced into the population.     

Stochastic selection in both haplophase and diplophase

Summary A population genetic model incorporating the evolutionary forces of zygotic selection, gametic selection and non-Mendelian segregation has been analyzed for the case in which all selection coefficients and the segregation parameter are assumed to be random variables that are uncorrelated from generation to generation. The diffusion approximation of the model is developed, and the subsequent analysis shows that one of four limiting outcomes of the stochastic process may obtain — an allele may be fixed or lost almost surely and irrespective of the initial gene frequency, the gene frequency may converge to a unique stationary distribution, or an allele may be fixed or lost with probabilities depending on the initial gene frequency. These outcomes correspond rather closely with the possible outcomes of the deterministic model — fixation or loss of an allele, convergence to a stable equilibrium, or the existence of an unstable equilibrium.Work supported by N. I. H. grants GM21732 and GM21623. The author is supported by Research Career Award GM2301.     

Sperm competition and the dynamics of X chromosome drive: stability and extinction

Several empirical studies of sperm competition in populations polymorphic for a driving X chromosome have revealed that Sex-ratio males (those carrying a driving X) are at a disadvantage relative to Standard males. Because the frequency of the driving X chromosome determines the population-level sex ratio and thus alters male and female mating rates, the evolutionary consequences of sperm competition for sex chromosome meiotic drive are subtle. As the SR allele increases in frequency, the ratio of females to males also increases, causing an increase in the male mating rate and a decrease in the female mating rate. While the former change may exacerbate the disadvantage of Sex-ratio males during sperm competition, the latter change decreases the incidence of sperm competition within the population. We analyze a model of the effects of sperm competition on a driving X chromosome and show that these opposing trends in male and female mating rates can result in two coexisting locally stable equilibria, one corresponding to a balanced polymorphism of the SR and ST alleles and the second to fixation of the ST allele. Stochastic fluctuations of either the population sex ratio or the SR frequency can then drive the population away from the balanced polymorphism and into the basin of attraction for the second equilibrium, resulting in fixation of the SR allele and extinction of the population.     

Domination or equilibrium

In the dynamics of an evolutionary game an equilibrium in which all strategy types occur is called an interior equilibrium. Interior equilibria fail to exist only when there is domination among the strategy types.     

Stochastic gene expression in a lentiviral positive-feedback loop: HIV-1 Tat fluctuations drive phenotypic diversity

HIV-1 Tat transactivation is vital for completion of the viral life cycle and has been implicated in determining proviral latency. We present an extensive experimental/computational study of an HIV-1 model vector (LTR-GFP-IRES-Tat) and show that stochastic fluctuations in Tat influence the viral latency decision. Low GFP/Tat expression was found to generate bifurcating phenotypes with clonal populations derived from single proviral integrations simultaneously exhibiting very high and near zero GFP expression. Although phenotypic bifurcation (PheB) was correlated with distinct genomic integration patterns, neither these patterns nor other extrinsic cellular factors (cell cycle/size, aneuploidy, chromatin silencing, etc.) explained PheB. Stochastic computational modeling successfully accounted for PheB and correctly predicted the dynamics of a Tat mutant that were subsequently confirmed by experiment. Thus, Tat stochastics appear sufficient to generate PheB (and potentially proviral latency), illustrating the importance of stochastic fluctuations in gene expression in a mammalian system.     

MUTANT INVASIONS AND ADAPTIVE DYNAMICS IN VARIABLE ENVIRONMENTS

The evolution of natural organisms is ultimately driven by the invasion and possible fixation of mutant alleles. The invasion process is highly stochastic, however, and the probability of success is generally low, even for advantageous alleles. Additionally, all organisms live in a stochastic environment, which may have a large influence on what alleles are favorable, but also contributes to the uncertainty of the invasion process. We calculate the invasion probability of a beneficial, mutant allele in a monomorphic, large population subject to stochastic environmental fluctuations, taking into account density‐ and frequency‐dependent selection, stochastic population dynamics and temporal autocorrelation of the environment. We treat both discrete and continuous time population dynamics, and allow for overlapping generations in the continuous time case. The results can be generalized to diploid, sexually reproducing organisms embedded in communities of interacting species. We further use these results to derive an extended canonical equation of adaptive dynamics, predicting the rate of evolutionary change of a heritable trait on long evolutionary time scales.     

Evolutionary dynamics of finite populations in games with polymorphic fitness equilibria

The hawk-dove (HD) game, as defined by Maynard Smith [1982. Evolution and the Theory of Games. Cambridge University Press, Cambridge], allows for a polymorphic fitness equilibrium (PFE) to exist between its two pure strategies; this polymorphism is the attractor of the standard replicator dynamics [Taylor, P.D., Jonker, L., 1978. Evolutionarily stable strategies and game dynamics. Math. Biosci. 40, 145-156; Hofbauer, J., Sigmund, K., 1998. Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge] operating on an infinite population of pure-strategists. Here, we consider stochastic replicator dynamics, operating on a finite population of pure-strategists playing games similar to HD; in particular, we examine the transient behavior of the system, before it enters an absorbing state due to sampling error. Though stochastic replication prevents the population from fixing onto the PFE, selection always favors the under-represented strategy. Thus, we may naively expect that the mean population state (of the pre-absorption transient) will correspond to the PFE. The empirical results of Fogel et al. [1997. On the instability of evolutionary stable states. BioSystems 44, 135-152] show that the mean population state, in fact, deviates from the PFE with statistical significance. We provide theoretical results that explain their observations. We show that such deviation away from the PFE occurs when the selection pressures that surround the fitness-equilibrium point are asymmetric. Further, we analyze a Markov model to prove that a finite population will generate a distribution over population states that equilibrates selection-pressure asymmetry; the mean of this distribution is generally not the fitness-equilibrium state.     

Phenotypic variation,sexual reproduction and evolutionary population dynamics

I studied the effects of introducing phenotypic variation into a well-known single species model for a population with discrete, non-overlapping generations. The phenotypes differed in their dynamic behaviour. The analysis was made under the assumption that the population was in an evolutionary stable state. Differences in the timing of the competitive impacts of the phenotypes on each other had a strong simplifying effect on the dynamics. This result could also be applied to competition between species. The effect of sexual reproduction on the dynamics of the population was analysed by assuming the simplest genetic model of one locus with two alleles. Sexual reproduction made the system much more stable in the (mathematical) sense that the number of attractors was reduced and their basins of attraction enlarged. In a dominant system sex tended to increase the frequency of the recessive allele, and in an overdominant system it induced gene frequencies of 1/2. Whether the attractors in the dominant system tended to be simpler or more complex than the attractors in the asexual system depended on the phenotype of the recessive homozygote. The overdominant sexual system tended to have simpler dynamics than the corresponding asexual population. A 2-locus model was used to study whether sexuals can invade an asexual population and vice versa. One locus coded for sexual and asexual reproduction, while the other coded for the dynamics. Enhanced stability through sexual reproduction seemed to be the reason why there was a clear asymmetry favouring sex in this evolutionary context.     

Roles of maternal effects in maintaining genetic variation: Maternal storage effect

Understanding the maintenance of genetic variation remains a central challenge in evolutionary biology. Recent empirical studies suggest the importance of temporally varying selection, as allele frequencies have been found to fluctuate substantially in the wild. However, previous theory suggests that the conditions for the maintenance of genetic variation under temporally fluctuating selection are quite restrictive. Using mathematical models, we demonstrate that maternal genetic effects, whereby maternal genotypes affect offspring phenotypes, can facilitate the maintenance of polymorphism in temporally varying environments. Maternal effects result in mismatches between genotypes and phenotypes, thereby buffering the influence of selection on allele frequency. This decreases the magnitude of allele‐frequency fluctuations and creates conditions for the maintenance of variation when selection causes fluctuations. Therefore, maternal effects may result in a temporal storage effect (“maternal storage effect”). On the other hand, when selection does not cause fluctuations (e.g., linear negative frequency‐dependent selection), maternal genetic effects moderate the relative importance of selection compared to genetic drift and promote stochastic allele extinction in finite populations. Thus, maternal effects can play an important role in the maintenance of polymorphism, but the direction of the effect depends on the nature of selection.     

The stationary distribution of a continuously varying strategy in a class-structured population under mutation-selection-drift balance

Many traits and/or strategies expressed by organisms are quantitative phenotypes. Because populations are of finite size and genomes are subject to mutations, these continuously varying phenotypes are under the joint pressure of mutation, natural selection and random genetic drift. This article derives the stationary distribution for such a phenotype under a mutation-selection-drift balance in a class-structured population allowing for demographically varying class sizes and/or changing environmental conditions. The salient feature of the stationary distribution is that it can be entirely characterized in terms of the average size of the gene pool and Hamilton's inclusive fitness effect. The exploration of the phenotypic space varies exponentially with the cumulative inclusive fitness effect over state space, which determines an adaptive landscape. The peaks of the landscapes are those phenotypes that are candidate evolutionary stable strategies and can be determined by standard phenotypic selection gradient methods (e.g. evolutionary game theory, kin selection theory, adaptive dynamics). The curvature of the stationary distribution provides a measure of the stability by convergence of candidate evolutionary stable strategies, and it is evaluated explicitly for two biological scenarios: first, a coordination game, which illustrates that, for a multipeaked adaptive landscape, stochastically stable strategies can be singled out by letting the size of the gene pool grow large; second, a sex-allocation game for diploids and haplo-diploids, which suggests that the equilibrium sex ratio follows a Beta distribution with parameters depending on the features of the genetic system.     

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     

Effects of survival thresholds upon one-dimensional dynamics of single-species populations

A simple one-dimensional model of single-species populations is studied by means of computer simulations. Although the model has a rich spectrum of dynamics including chaotic behavior, the introduction of survival thresholds makes the chaotic region so small that it can be hardly observed. Stochastic fluctuations further reduce the chaotic region because they accidentally lead populations to extinction. The model thus naturally explains the observation that the majority of natural populations do not show chaotic behavior but a monotonic return to a stable equilibrium point following a disturbance.     

What pair formation can do to the battle of the sexes: towards more realistic game dynamics

In the various dynamic models of Dawkin's Battle of the Sexes, payoff matrices serve as the basic ingredients for the specification of a game-dynamic model. Here I model the sex war mechanistically, by expressing the costs of raising the offspring and performing a prolonged courtship via a time delay for the corresponding individuals, instead of via payoff matrices. During such a time delay an individual is not able to have new matings. Only after the delay has occurred, an individual (and its offspring) appears on the mating market again. From these assumptions I derive a pair-formation submodel, and a system of delay-differential equations describing the dynamics of the game. By a time-scale argument, I obtain an approximation of this system by means of a much simpler system of ordinary differential equations. Analysis of this simplified system shows that the model can give rise to two non-trivial asymptotically stable equilibrium points: an interior equilibrium where both female strategies and both male strategies are present, and a boundary equilibrium where only one of the female strategies and both male strategies are present. This behaviour is qualitatively different from that of models of the battle of the sexes formulated in the traditional framework of game-dynamic equations. In other words, the addition of a most elementary further assumption about individual life history fundamentally changes the model predictions. These results show that in analysing evolutionary games one should pay careful attention to the specific mechanisms involved in the conflict. In general, I advocate deriving simple models for evolutionary games, starting from more complex, mechanistic building blocks. The wide-spread method of modelling games at a high phenomenological level, through payoff matrices, can be misleading.     

随机进化稳定性研究进展

在过去的三十多年, 演化博弈理论及其进化稳定对策的概念不仅被广泛地应用于解释动物行为的进化, 而且也被成功地应用于分子生物学、经济学、政治学和社会学等诸多学科。然而, 在随机波动环境中演化博弈动态的随机动力学性质始终没有被清晰地认识, 并且这是一个极具挑战性的理论问题。本文简单介绍了我们最近所提出的随机进化稳定性(stochastic evolutionary stability, SES)的概念。随机进化稳定性不仅是经典进化稳定对策(evolutionarily stably strategy, ESS)概念在随机环境下的自然扩展, 而且为揭示在随机环境中动物行为的演化动态提供一个基本的理论框架。     

Evolutionary stability in Lotka-Volterra systems

The Lotka-Volterra model of population ecology, which assumes all individuals in each species behave identically, is combined with the behavioral evolution model of evolutionary game theory. In the resultant monomorphic situation, conditions for the stability of the resident Lotka-Volterra system, when perturbed by a mutant phenotype in each species, are analysed. We develop an evolutionary ecology stability concept, called a monomorphic evolutionarily stable ecological equilibrium, which contains as a special case the original definition by Maynard Smith of an evolutionarily stable strategy for a single species. Heuristically, the concept asserts that the resident ecological system must be stable as well as the phenotypic evolution on the "stationary density surface". The conditions are also shown to be central to analyse stability issues in the polymorphic model that allows arbitrarily many phenotypes in each species, especially when the number of species is small. The mathematical techniques are from the theory of dynamical systems, including linearization, centre manifolds and Molchanov's Theorem.