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.

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.

Evolutionary game dynamics with non-uniform interaction rates in finite population.

In this study, we extend evolutionary game dynamics with non-uniform interaction rates to the situation with finite population. Our main goal is to show how the fixation probability is influenced by the non-uniform interaction rates under weak selection. Based on the diffusion approximation of the Moran process and assumption of weak selection, the stochastic dynamic properties of a two-phenotype game with non-uniform interaction rates in a finite population are investigated. By the analysis of some cases, we show that the non-uniform interaction rates may result in the potential evolutionary complexity of game dynamics in finite population.

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.

Hamilton's rule and kin competition in a finite kin population.

Kin selection means that individuals can increase their own inclusive fitness through displaying more altruistically toward their relatives. So, Hamilton's rule says kin selection will work if the coefficient of relatedness exceeds the cost-to-benefit ratio of the altruistic act. However, some studies have shown that the kin competition due to the altruism among relatives can reduce, and even totally negate, the kin-selected benefits of altruism toward relatives. In order to understand how the evolution of cooperation is influenced by both kin selection and kin competition under a general theoretical framework, we here consider the evolutionary dynamics of cooperation in a finite kin population, where kin competition is incorporated into a simple Prisoner's Dilemma game between relatives. Differently from the previous studies, we emphasize that the difference between the effects of mutually and unilaterally altruistic acts on kin competition may play an important role for the evolution of cooperation. The main results not only show the conditions that Hamilton's rule still works under the kin competition but also reveal the evolutionary biological mechanism driving the evolution of cooperation in a finite kin population.

Transitions in the cell-fate induction induced by colored noise associated with the inductive stimulus.

The cell-fate induction based on the saddle-node bifurcation is undoubtedly a very important concept in developmental biology, which provides a possible mechanism to explain the intrinsic irreversibility in the developmental process. In this paper, the effect of a colored noise, which is associated with the inductive stimulus, on the saddle-node landscape of cell-fate induction is investigated, especially, the effect of the change of correlation time of colored noise on cell-fate induction. The main results show clearly that the change of correlation time of colored noise could induce the transitions of the system. This implies that the colored noise associated with inductive stimulus may have a profound effect on the saddle-node bifurcation landscape of cell-fate induction. This will also help us to understand more deeply the role of cell-fate induction in developmental biology.

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.

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. This result provides a new perspective for a better understanding of how environmental fluctuations may contribute to the evolution of behavioral diversity.

Limiting similarity of competitive species and demographic stochasticity.

The limiting similarity of competitive species and its relationship with the competitive exclusion principle is still one of the most important concepts in ecology. In the 1970s, May [R. M. May, Stability and Complexity in Model Ecosystems (Princeton University, Princeton, NJ, 1973)] developed a concise theoretical framework to investigate the limiting similarity of competitive species. His theoretical results show that no limiting similarity threshold of competitive species can be identified in the deterministic model system whereby species more similar than this threshold never coexist. Theoretically, for competitive species coexisting in an unvarying environment, deterministic interspecific interactions and demographic stochasticity can be considered two sides of a coin. To investigate how the "tension" between these two forces affects the coexistence of competing species, a simple two-species competitive system based only on May's model system is transformed into an equivalent replicator equation. The effect of demographic stochasticity on the system stability is measured by the expected drift of the Lyapunov function. Our main results show that the limiting similarity of competitive species should not be considered to be an absolute measure. Specifically, very similar competitive species should be able to coexist in an environment with a high productivity level but big differences between competitive species should be necessary in an ecosystem with a low productivity level.

A simple rule of direct reciprocity leads to the stable coexistence of cooperation and defection in the Prisoner's Dilemma game.

The long-term coexistence of cooperation and defection is a common phenomenon in nature and human society. However, none of the theoretical models based on the Prisoner's Dilemma (PD) game can provide a concise theoretical model to explain what leads to the stable coexistence of cooperation and defection in the long-term even though some rules for promoting cooperation have been summarized (Nowak, 2006, Science 314, 1560-1563). Here, based on the concept of direct reciprocity, we develop an elementary model to show why stable coexistence of cooperation and defection in the PD game is possible. The basic idea behind our theoretical model is that all players in a PD game prefer a cooperator as an opponent, and our results show that considering strategies allowing opting out against defection provide a general and concise way of understanding the fundamental importance of direct reciprocity in driving the evolution of cooperation.

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. Based on this model, we develop the concepts of stochastic evolutionary stability (SES) and stochastic convergence stability (SCS). We show that the condition for a pure strategy to be SES and SCS is more stringent than in a constant environment, while the condition for a constant mixed strategy to be SES is less stringent than the condition to be SCS, which is less stringent than the condition in a constant environment.

Opting out against defection leads to stable coexistence with cooperation.

Cooperation coexisting with defection is a common phenomenon in nature and human society. Previous studies for promoting cooperation based on kin selection, direct and indirect reciprocity, graph selection and group selection have provided conditions that cooperators outcompete defectors. However, a simple mechanism of the long-term stable coexistence of cooperation and defection is still lacking. To reveal the effect of direct reciprocity on the coexistence of cooperation and defection, we conducted a simple experiment based on the Prisoner's Dilemma (PD) game, where the basic idea behind our experiment is that all players in a PD game should prefer a cooperator as an opponent. Our experimental and theoretical results show clearly that the strategies allowing opting out against defection are able to maintain this stable coexistence.

Kinship as a frequency dependent strategy.

Humans divide themselves up into separate cultures, which is a unique and ubiquitous characteristic of our species. Kinship norms are one of the defining features of such societies. Here we show how norms of marital residence can evolve as a frequency-dependent strategy, using real-world cases from southwestern China and an evolutionary game model. The process of kinship change has occurred in the past and is also occurring now in southwestern China. Our data and models show how transitions between residence types can occur both as response to changing costs and benefits of co-residence with kin, and also due to the initial frequency of the strategies adopted by others in the population: patrilocal societies can become matrilocal, and neolocal societies can become duolocal. This illustrates how frequency-dependent selection plays a role both in the maintenance of group-level cultural diversity and in cultural extinction.

Stochastic analysis of gene expression.

In this chapter, stochasticity in gene expression is investigated using Ω-expansion technique. Two theoretical models are considered here, one concern the stochastic fluctuations in a single-gene network with negative feedback regulation, and the other the additivity of noise propagation in a protein cascade. All of these theoretical analyses may provide a basic framework for understanding stochastic gene expression.

Bistability, probability transition rate and first-passage time in an autoactivating positive-feedback loop.

A hallmark of positive-feedback regulation is bistability, which gives rise to distinct cellular states with high and low expression levels, and that stochasticity in gene expression can cause random transitions between two states, yielding bimodal population distribution (Kaern et al., 2005, Nat Rev Genet 6: 451-464). In this paper, the probability transition rate and first-passage time in an autoactivating positive-feedback loop with bistability are investigated, where the gene expression is assumed to be disturbed by both additive and multiplicative external noises, the bimodality in the stochastic gene expression is due to the bistability, and the bistability determines that the potential of the Fokker-Planck equation has two potential wells. Our main goal is to illustrate how the probability transition rate and first-passage time are affected by the maximum transcriptional rate, the intensities of additive and multiplicative noises, and the correlation of additive and multiplicative noises. Our main results show that (i) the increase of the maximum transcription rate will be useful for maintaining a high gene expression level; (ii) the probability transition rate from one potential well to the other one will increase with the increase of the intensity of additive noise; (iii) the increase of multiplicative noise strength will increase the amount of probability in the left potential well; and (iv) positive (or negative) cross-correlation between additive and multiplicative noises will increase the amount of probability in the left (or right) potential well.

Costly punishment does not always increase cooperation.

In a pairwise interaction, an individual who uses costly punishment must pay a cost in order that the opponent incurs a cost. It has been argued that individuals will behave more cooperatively if they know that their opponent has the option of using costly punishment. We examined this hypothesis by conducting two repeated two-player Prisoner's Dilemma experiments, that differed in their payoffs associated to cooperation, with university students from Beijing as participants. In these experiments, the level of cooperation either stayed the same or actually decreased when compared with the control experiments in which costly punishment was not an option. We argue that this result is likely due to differences in cultural attitudes to cooperation and punishment based on similar experiments with university students from Boston that found cooperation did increase with costly punishment.