School of Modern Posts, <a href="https://ror.org/04jn0td46">Xi'an University of Posts and Telecommunications</a>, Xi'an 710061, China.
Key Laboratory of Animal Ecology and Conservation Biology, Institute of Zoology, <a href="https://ror.org/034t30j35">Chinese Academy of Sciences</a>, Beijing 100101, China.
Phys Rev E. 2024 Aug;110(2-1):024211. doi: 10.1103/PhysRevE.110.024211.
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.
为了展示环境噪声对模仿动态的影响，本文研究了具有随机收益的离散时间模仿动态的随机稳定性和随机进化稳定性。基于两基因型模型中固定状态和常数内部平衡点的随机局部稳定性，我们将随机进化稳定性的概念扩展到随机模仿动态，即将这样一种策略定义为：如果种群中的所有成员都采用该策略，那么在自然选择的影响下，任何突变策略成功入侵种群的概率将任意低。我们的主要结果清楚地表明，系统的随机进化稳定性仅取决于随机收益矩阵的均值矩阵的性质，而与随机收益矩阵的随机性无关。此外，作为两个例子，我们还表明，在随机模仿动态的框架下，噪声强度会影响随机囚徒困境博弈中合作行为的进化和系统的非线性动态行为。
