Environmental Noise Could Promote Stochastic Local Stability of Behavioral Diversity Evolution

Evolutionary dynamics of cooperation coupled with ecological feedback compensation

A simple theoretical model (or a demonstrative example) was developed to illustrate how the evolution of cooperation can be affected by the density-dependent survival competition, in which we assume that the fertility of an individual depends only on the pairwise interaction between him and other individuals based on Prisoner's Dilemma game, while its viability is only related to the density-dependent survival competitiveness. Our results show that not only cooperation could be evolutionarily stable if the advantage of cooperators in viability can compensate for the cost they pay for their fertility, but also the long-term stable coexistence of cooperation and defection is possible if none of cooperation and defection is evolutionarily stable. Moreover, for the stochastic evolutionary dynamics in a finite population, our analysis shows that the increase (or decrease) of the survival competitiveness of cooperators (or defectors) should be conductive to the evolutionary emergence of cooperation.

Impact of environmental stochastic fluctuations on the evolutionary stability of imitation dynamics

To show the impact of environmental noise on imitation dynamics, the stochastic stability and stochastic evolutionary stability of a discrete-time imitation dynamics with random payoffs are studied in this paper. Based on the stochastic local stability of fixation states and constant interior equilibria in a two-phenotype model, we extend the concept of stochastic evolutionary stability to the stochastic imitation dynamics, which is defined as a strategy such that, if all the members of the population adopt it, then the probability for any mutant strategy to invade the population successfully under the influence of natural selection is arbitrarily low. Our main results show clearly that the stochastic evolutionary stability of the system depends only on the properties of the mean matrix of the random payoff matrix and is independent of the randomness of the random payoff matrix. Moreover, as two examples, we show also that under the framework of stochastic imitation dynamics, the noise intensity affects the evolution of cooperative behavior in a stochastic prisoner's dilemma game and the system's nonlinear dynamic behavior.

Weak selection and stochastic evolutionary stability in a stochastic replicator dynamics

It is a very challenging problem whether natural selection is able to effectively resist the continuous disturbance of environmental noise such that the direction or outcome of evolution determined by the deterministic selection pressure will not be changed. By analyzing the impact of weak selection on the evolutionary stability of a stochastic replicator dynamics with n possible pure strategies, we found that the weak selection is able to enhance the evolutionary stability, that is, under weak selection, the stochastic evolutionary stability of the system is determined by the mean payoff matrix. This finding strongly implies that the weak selection should be regarded as an important mechanism to ensure evolutionary stability in stochastic environments.

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.

Spatial evolution of cooperation with variable payoffs

In the evolution of cooperation, the individuals' payoffs are commonly random in real situations, e.g., the social networks and the economic regions, leading to unpredictable factors. Therefore, there are chances for each individual to obtain the exceeding payoff and risks to get the low payoff. In this paper, we consider that each individual's payoff follows a specific probability distribution with a fixed expectation, where the normal distribution and the exponential distribution are employed in our model. In the simulations, we perform the models on the weak prisoner's dilemmas (WPDs) and the snowdrift games (SDGs), and four types of networks, including the hexagon lattice, the square lattice, the small-world network, and the triangular lattice are considered. For the individuals' normally distributed payoff, we find that the higher standard deviation usually inhibits the cooperation for the WPDs but promotes the cooperation for the SDGs. Besides, with a higher standard deviation, the cooperation clusters are usually split for the WPDs but constructed for the SDGs. For the individuals' exponentially distributed payoff, we find that the small-world network provides the best condition for the emergence of cooperators in WPDs and SDGs. However, when playing SDGs, the small-world network allows the smallest space for the pure cooperative state while the hexagon lattice allows the largest.

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. We show that, if a fixation state is stochastically stable, then the pure strategy corresponding to this fixation state must be stochastically evolutionarily stable with respect to mixed strategies. However, this is not the case for a mixed strategy that corresponds to a stochastically stable constant interior equilibrium point with respect to the two phenotypes.

Disease-Induced Cooperation Mitigates Populations Against Decline: The Cascade Effect of Cooperation Evolution

Due to density-dependent selection, the ecological factors impacting population dynamics can play an important role in promoting cooperation, and accordingly, benefit a population from the eco-evolutionary feedback. This implies that cooperation between individuals could help resist the attack of infectious diseases. Yet, little is known about how cooperation evolves in response to infections. We here examined theoretically the impact of disease infections with various transmission types on cooperation evolution and its feedback to population dynamics. Results show that infected populations can evolve to be more cooperative, and the level of cooperation increases with the transmission rate, which can protect the population against decline due to infection and prevent population extinction driven by defection. A high transmission rate can stabilize population fluctuation, while a relatively low transmission rate could destabilize population dynamics. We argue that the mechanism underlying such stress-induced cooperation is analogous to the cascade effect of trophic interactions in food webs: reduction in selfishness from environmental stress indirectly relaxes the exploitation of cooperators by defectors. These findings emphasize the role of eco-evolutionary feedback in evolving cooperation and the ecological significance of cooperation evolution for populations withstanding disease infection.

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. Rev. E 96, 032414 (2017)2470-004510.1103/PhysRevE.96.032414]. In this paper, we extend this study to more general situations, namely, multiphenotype symmetric as well as asymmetric matrix games with random payoffs. Conditions for stochastic local stability and stochastic evolutionary stability are established. Conditions for a fixation state to be stochastically unstable and almost everywhere stochastically unstable are distinguished in a multiphenotype setting according to the initial population state. Our results provide some alternative perspective and a more general theoretical framework for a better understanding of the evolution of animal behavior in a stochastic environment.

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.

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. In an infinite population, fixation of cooperation is stochastically stable if a mean geometric inclusive fitness of defection when rare is negative, while fixation of defection is stochastically unstable if a mean geometric inclusive fitness of cooperation when rare is positive, and these conditions are generally not equivalent. In a finite population, the probability for cooperation to ultimately fix when represented once exceeds the probability under neutrality or the corresponding probability for defection if the mean inclusive fitness of cooperation when its frequency is 1/3 or 1/2, respectively, exceeds 1. All these results rely on the simplifying assumption of a linear fitness function. It is argued that meaningful applications of random inclusive fitness in complex settings (multi-player game, diploidy, population structure) would generally require conditions of weak selection and additive gene action.

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.

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. Moreover, although stochastic local stability or instability of an equilibrium may not depend on environmental noise if selection is weak enough, the growth rate near an equilibrium not only depends on environmental noise, but can even be enhanced by environmental noise if selection is weak. This is the case, for instance, when the variances of the random payoffs as well as the covariances are equal. These results suggest that natural selection could be able to filter (or resist) the effect of environmental noise on the evolution of animal behavior if selection is weak.

Cooperation dynamics in networked geometric Brownian motion

Recent works suggest that pooling and sharing may constitute a fundamental mechanism for the evolution of cooperation in well-mixed fluctuating environments. The rationale is that, by reducing the amplitude of fluctuations, pooling and sharing increases the steady-state growth rate at which individuals self-reproduce. However, in reality interactions are seldom realized in a well-mixed structure, and the underlying topology is in general described by a complex network. Motivated by this observation, we investigate the role of the network structure on the cooperative dynamics in fluctuating environments, by developing a model for networked pooling and sharing of resources undergoing a geometric Brownian motion. The study reveals that, while in general cooperation increases the individual steady state growth rates (i.e., is evolutionary advantageous), the interplay with the network structure may yield large discrepancies in the observed individual resource endowments. We comment possible biological and social implications and discuss relations to econophysics.