Evolution of cooperation in social dilemmas with assortment in finite populations.

We investigate conditions for the evolution of cooperation in social dilemmas in finite populations with assortment of players by group founders and general payoff functions for cooperation and defection within groups. Using a diffusion approximation in the limit of a large population size that does not depend on the precise updating rule, we show that the first-order effect of selection on the fixation probability of cooperation when represented once can be expressed as the difference between time-averaged payoffs with respect to effective time that cooperators and defectors spend in direct competition in the different group states. Comparing this fixation probability to its value under neutrality and to the corresponding fixation probability for defection, we deduce conditions for the evolution of cooperation.

Stochastic viability in an island model with partial dispersal: Approximation by a diffusion process in the limit of a large number of islands.

In this paper, we investigate a finite population undergoing evolution through an island model with partial dispersal and without mutation, where generations are discrete and non-overlapping. The population is structured into D demes, each containing N individuals of two possible types, A and B, whose viability coefficients, s and s, respectively, vary randomly from one generation to the next. We assume that the means, variances and covariance of the viability coefficients are inversely proportional to the number of demes D, while higher-order moments are negligible in comparison to 1/D.

On the sign of the average effect of an allele.

The concept of the average effect of an allele pervades much of evolutionary population genetics. In this context the average effect of an allele is often considered as the main component of the "fitness" of that allele. It is widely believed that, if this fitness component for an allele is positive, then the frequency of this allele will increase, at least for one generation in discrete-time models.

Average abundancy of cooperation in multi-player games with random payoffs.

We consider interactions between players in groups of size [Formula: see text] with payoffs that not only depend on the strategies used in the group but also fluctuate at random over time. An individual can adopt either cooperation or defection as strategy and the population is updated from one time step to the next by a birth-death event according to a Moran model. Assuming recurrent symmetric mutation and payoffs to cooperators and defectors according to the composition of the group whose expected values, variances, and covariances are of the same small order, we derive a first-order approximation for the average abundance of cooperation in the selection-mutation equilibrium.

Stochastic replicator dynamics and evolutionary stability.

To develop the concept of evolutionary stability in a stochastic environment, we investigate the continuous-time dynamics of a two-phenotype linear evolutionary game with generally correlated random payoffs in pairwise interactions. By using the Gram-Schmidt orthogonalization procedure and Itô's formula, we deduce a stochastic differential equation for the phenotype frequencies that extends the replicator equation, called the stochastic replicator equation. We give conditions for stochastic stability of a fixation state or a constant interior equilibrium point with respect to the stochastic dynamics of the two phenotypes.

Stochastic evolutionary stability in matrix games with random payoffs.

Evolutionary game theory and the concept of an evolutionarily stable strategy have been not only extensively developed and successfully applied to explain the evolution of animal behavior, but also widely used in economics and social sciences. Recently, in order to reveal the stochastic dynamical properties of evolutionary games in randomly fluctuating environments, the concept of stochastic evolutionary stability based on conditions for stochastic local stability for a fixation state was developed in the context of a symmetric matrix game with two phenotypes and random payoffs in pairwise interactions [Zheng et al., Phys.

Evolution of cooperation with respect to fixation probabilities in multi-player games with random payoffs.

We study the effect of variability in payoffs on the evolution of cooperation (C) against defection (D) in multi-player games in a finite well-mixed population. We show that an increase in the covariance between any two payoffs to D, or a decrease in the covariance between any two payoffs to C, increases the probability of ultimate fixation of C when represented once, and decreases the corresponding fixation probability for D. This is also the case with an increase in the covariance between any payoff to C and any payoff to D if and only if the sum of the numbers of C-players in the group associated with these payoffs is large enough compared to the group size.

Inclusive fitness and Hamilton's rule in a stochastic environment.

The evolution of cooperation in Prisoner's Dilemmas with additive random cost and benefit for cooperation cannot be accounted for by Hamilton's rule based on mean effects transferred from recipients to donors weighted by coefficients of relatedness, which defines inclusive fitness in a constant environment. Extensions that involve higher moments of stochastic effects are possible, however, and these are connected to a concept of random inclusive fitness that is frequency-dependent. This is shown in the setting of pairwise interactions in a haploid population with the same coefficient of relatedness between interacting players.

The effect of variability in payoffs on average abundance in two-player linear games under symmetric mutation.

Classical studies in evolutionary game theory assume constant payoffs. Randomly fluctuating environments in real populations make this assumption idealistic. In this paper, we study randomized two-player linear games in a finite population in a succession of birth-death events according to a Moran process and in the presence of symmetric mutation.

The effect of the opting-out strategy on conditions for selection to favor the evolution of cooperation in a finite population.

We consider a Prisoner's Dilemma (PD) that is repeated with some probability 1-ρ only between cooperators as a result of an opting-out strategy adopted by all individuals. The population is made of N pairs of individuals and is updated at every time step by a birth-death event according to a Moran model. Assuming an intensity of selection of order 1/N and taking 2N birth-death events as unit of time, a diffusion approximation exhibiting two time scales, a fast one for pair frequencies and a slow one for cooperation (C) and defection (D) frequencies, is ascertained in the limit of a large population size.

Randomized matrix games in a finite population: Effect of stochastic fluctuations in the payoffs on the evolution of cooperation.

A diffusion approximation for a randomized 2 × 2-matrix game in a large finite population is ascertained in the case of random payoffs whose expected values, variances and covariances are of order given by the inverse of the population size N. Applying the approximation to a Randomized Prisoner's Dilemma (RPD) with independent payoffs for cooperation and defection in random pairwise interactions, conditions on the variances of the payoffs for selection to favor the evolution of cooperation, favor more the evolution of cooperation than the evolution of defection, and disfavor the evolution of defection are deduced. All these are obtained from probabilities of ultimate fixation of a single mutant.

Weak selection can filter environmental noise in the evolution of animal behavior.

Weak selection is an important assumption in theoretical evolutionary biology, but its biological significance remains unclear. In this study, we investigate the effect of weak selection on stochastic evolutionary stability in a two-phenotype evolutionary game dynamics with a random payoff matrix assuming an infinite, well-mixed population undergoing discrete, nonoverlapping generations. We show that, under weak selection, both stochastic local stability and stochastic evolutionary stability in this system depend on the means of the random payoffs but not on their variances.

Diffusion approximation for an age-class-structured population under viability and fertility selection with application to fixation probability of an advantageous mutant.

In this paper, we ascertain the validity of a diffusion approximation for the frequencies of different types under recurrent mutation and frequency-dependent viability and fertility selection in a haploid population with a fixed age-class structure in the limit of a large population size. The approximation is used to study, and explain in terms of selection coefficients, reproductive values and population-structure coefficients, the differences in the effects of viability versus fertility selection on the fixation probability of an advantageous mutant.

First-order effect of frequency-dependent selection on fixation probability in an age-structured population with application to a public goods game.

In this paper, we deduce the first-order effect of frequency-dependent viability and fertility selection on the probability of fixation of a mutant in a large finite haploid population with a fixed age structure by applying a direct small perturbation method to the neutral two-timescale genealogical process. This effect is expressed in terms of fixation-fitness coefficients times ancestry coefficients that are related to the effective population size. In the case of constant selection, the fixation-fitness coefficients are functions of the coefficients of viability and fertility selection weighted by reproductive values and population-structure coefficients for the different age classes.

The left-hand side of the Fundamental Theorem of Natural Selection: A reply.

In a recent paper, Grafen (2018) discussed the left-hand side in the equation stating Fisher's (1930, 1958) "Fundamental Theorem of Natural Selection" (FTNS). Fisher's original statement of the FTNS is, in effect, "The rate of increase in fitness of any organism is equal to its genetic variance in fitness at that time" with the rate of increase in fitness understood as the one "due to all changes in gene ratios" (Fisher, 1930, p. 35).

Environmental Noise Could Promote Stochastic Local Stability of Behavioral Diversity Evolution.

In this Letter, we investigate stochastic stability in a two-phenotype evolutionary game model for an infinite, well-mixed population undergoing discrete, nonoverlapping generations. We assume that the fitness of a phenotype is an exponential function of its expected payoff following random pairwise interactions whose outcomes randomly fluctuate with time. We show that the stochastic local stability of a constant interior equilibrium can be promoted by the random environmental noise even if the system may display a complicated nonlinear dynamics.

Evolutionary stability concepts in a stochastic environment.

Over the past 30 years, evolutionary game theory and the concept of an evolutionarily stable strategy have been not only extensively developed and successfully applied to explain the evolution of animal behaviors, but also widely used in economics and social sciences. Nonetheless, the stochastic dynamical properties of evolutionary games in randomly fluctuating environments are still unclear. In this study, we investigate conditions for stochastic local stability of fixation states and constant interior equilibria in a two-phenotype model with random payoffs following pairwise interactions.

Frequency-dependent growth in class-structured populations: continuous dynamics in the limit of weak selection.

In this paper we consider class-structured populations in discrete time in the limit of weak selection and with the inverse of the intensity of selection as unit of time. The aim is to establish a continuous model that approximates the discrete model. More precisely, we study frequency-dependent growth in an infinite haploid population structured into a finite number of classes such that individuals in each class contribute to a given subset of classes from one time step to the next.

A coalescent dual process for a Wright-Fisher diffusion with recombination and its application to haplotype partitioning.

Duality plays an important role in population genetics. It can relate results from forwards-in-time models of allele frequency evolution with those of backwards-in-time genealogical models; a well known example is the duality between the Wright-Fisher diffusion for genetic drift and its genealogical counterpart, the coalescent. There have been a number of articles extending this relationship to include other evolutionary processes such as mutation and selection, but little has been explored for models also incorporating crossover recombination.

Fixation probability in a two-locus intersexual selection model.

We study a two-locus model of intersexual selection in a finite haploid population reproducing according to a discrete-time Moran model with a trait locus expressed in males and a preference locus expressed in females. We show that the probability of ultimate fixation of a single mutant allele for a male ornament introduced at random at the trait locus given any initial frequency state at the preference locus is increased by weak intersexual selection and recombination, weak or strong. Moreover, this probability exceeds the initial frequency of the mutant allele even in the case of a costly male ornament if intersexual selection is not too weak.

Definitions of fitness in age-structured populations: Comparison in the haploid case.

Fisher's (1930) Fundamental Theorem of Natural Selection (FTNS), and in particular the development of an explicit age-structured version of the theorem, is of everlasting interest. In a recent paper, Grafen (2015a) argues that Fisher regarded his theorem as justifying individual rather than population fitness maximization. The argument relies on a new definition of fitness in age-structured populations in terms of individual birth and death rates and age-specific reproductive values in agreement with a principle of neutrality.

On the interpretation and relevance of the Fundamental Theorem of Natural Selection.

The attempt to understand the statement, and then to find the interpretation, of Fisher's "Fundamental Theorem of Natural Selection" caused problems for generations of population geneticists. Price's (1972) paper was the first to lead to an understanding of the statement of the theorem. The theorem shows (in the discrete-time case) that the so-called "partial change" in mean fitness of a population between a parental generation and an offspring generation is the parental generation additive genetic variance in fitness divided by the parental generation mean fitness.

Evolution of cooperation in a multidimensional phenotype space.

The emergence of cooperation in populations of selfish individuals is a fascinating topic that has inspired much theoretical work. An important model to study cooperation is the phenotypic model, where individuals are characterized by phenotypic properties that are visible to others. The phenotype of an individual can be represented for instance by a vector x = (x1,…,xn), where x1,…,xn are integers.

Effect of epistasis and linkage on fixation probability in three-locus models: an ancestral recombination-selection graph approach.

We study the probability of ultimate fixation of a single new mutant arising in an individual chosen at random at a locus linked to two other loci carrying previously arisen mutations. This is done using the Ancestral Recombination-Selection Graph (ARSG) in a finite population in the limit of a large population size, which is also known as the Ancestral Influence Graph (AIG). An analytical expansion of the fixation probability with respect to population-scaled recombination rates and selection intensities is obtained.