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, sA and sB, 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. We use a discrete-time Markov chain with two timescales to model the evolutionary process, and we demonstrate that as the number of demes D approaches infinity, the accelerated Markov chain converges to a diffusion process for any deme size N≥2. This diffusion process allows us to evaluate the fixation probability of type A following its introduction as a single mutant in a population that was fixed for type B. We explore the impact of increasing the variability in the viability coefficients on this fixation probability. At least when N is large enough, it is shown that increasing this variability for type B or decreasing it for type A leads to an increase in the fixation probability of a single A. The effect of the population-scaled variances, σA2 and σB2, can even cancel the effects of the population-scaled means, µA and µB. We also show that the fixation probability of a single A increases as the deme-scaled migration rate increases. Moreover, this probability is higher for type A than for type B if the population-scaled geometric mean viability coefficient is higher for type A than for type B, which means that µA-σA2/2>µB-σB2/2.

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. In classical social dilemmas with random cost and benefit for cooperation, the evolution of C is more likely to occur if the variances of the cost and benefit, as well as the group size, are small, while the covariance between cost and benefit is large.

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. In general, we show that increasing the variance of any payoff for defection or decreasing the variance of any payoff for cooperation increases the average abundance of cooperation. As for the effect of the covariance between any payoff for cooperation and any payoff for defection, we show that it depends on the number of cooperators in the group associated with these payoffs. We study in particular the public goods game, the stag hunt game, and the snowdrift game, all social dilemmas based on random benefit b and random cost c for cooperation, which lead to correlated payoffs to cooperators and defectors within groups. We show that a decrease in the scaled variance of b or c, or an increase in their scaled covariance, makes it easier for weak selection to favor the abundance of cooperation in the stag hunt game and the snowdrift game. The same conclusion holds for the public goods game except that the variance of b has no effect on the average abundance of C. Moreover, while the mutation rate has little effect on which strategy is more abundant at equilibrium, the group size may change it at least in the stag hunt game with a larger group size making it more difficult for cooperation to be more abundant than defection under weak selection.

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. Introducing identity measures under neutrality that depend on the mutation rate and calculating these in the limit of a large population size by using the coalescent process, we study the first-order effect of the means, variances and covariances of the payoffs on average abundance in the stationary state under mutation and selection. This shows how the average abundance of a strategy is driven not only by its mean payoffs but also by the variances and covariances of its payoffs. In Prisoner's Dilemmas with additive cost and benefit for cooperation, where constant payoffs always favor the abundance of defection, stochastic fluctuations in the payoffs can change the strategy that is more abundant on average in the stationary state. The average abundance of cooperation is increased if the variance of any payoff to cooperation against cooperation or defection, or their covariance, is decreased, or if the variance of any payoff to defection against cooperation or defection, or their covariance, is increased. This is also the case for a Prisoner's Dilemma with independent payoffs that is repeated a random number of times. As for the mutation rate, it comes into play in the coefficients of the variances and covariances that determine average abundance. Increasing the mutation rate can enhance or lessen the condition for a strategy to be more abundant on average than another.

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. The population can be well mixed in the sense that everyone is equally likely to interact with everyone else, but the behavioral strategies of the individuals can depend on their distance in the phenotype space. A cooperator can choose to help other individuals exhibiting the same phenotype and defects otherwise. Cooperation is said to be favored by selection if it is more abundant than defection in the stationary state. This means that the average frequency of cooperators in the stationary state strictly exceeds 1/2. Antal et al. (2009c) found conditions that ensure that cooperation is more abundant than defection in a one-dimensional (i.e. n = 1) and an infinite-dimensional (i.e. n = ∞) phenotype space in the case of the Prisoner's Dilemma under weak selection. However, reality lies between these two limit cases. In this paper, we derive the corresponding condition in the case of a phenotype space of any finite dimension. This is done by applying a perturbation method to study a mutation-selection equilibrium under weak selection. This condition is obtained in the limit of a large population size by using the ancestral process. The best scenario for cooperation to be more likely to evolve is found to be a high population-scaled phenotype mutation rate, a low population-scaled strategy mutation rate and a high phenotype space dimension. The biological intuition is that a high population-scaled phenotype mutation rate reduces the quantity of interactions between cooperators and defectors, while a high population-scaled strategy mutation rate introduces newly mutated defectors that invade groups of cooperators. Finally it is easier for cooperation to evolve in a phenotype space of higher dimension because it becomes more difficult for a defector to migrate to a group of cooperators. The difference is significant from n = 1 to n = 2 and from n = 2 to n = 3, but becomes small as soon as n ≥ 3.