In evolutionary game theory, it is customary to be partial to the dynamical models possessing fixed points so that they may be understood as the attainment of evolutionary stability, and hence, Nash equilibrium. Any show of periodic or chaotic solution is many a time perceived as a shortcoming of the corresponding game dynamic because (Nash) equilibrium play is supposed to be robust and persistent behaviour, and any other behaviour in nature is deemed transient. Consequently, there is a lack of attempt to connect the non-fixed point solutions with the game theoretic concepts. Here we provide a way to render game theoretic meaning to periodic solutions. To this end, we consider a replicator map that models Darwinian selection mechanism in unstructured infinite-sized population whose individuals reproduce asexually forming non-overlapping generations. This is one of the simplest evolutionary game dynamic that exhibits periodic solutions giving way to chaotic solutions (as parameters related to reproductive fitness change) and also obeys the folk theorems connecting fixed point solutions with Nash equilibrium. Interestingly, we find that a modified Darwinian fitness -- termed heterogeneity payoff -- in the corresponding population game must be put forward as (conventional) fitness times the probability that two arbitrarily chosen individuals of the population adopt two different strategies. The evolutionary dynamics proceeds as if the individuals optimize the heterogeneity payoff to reach an evolutionarily stable orbit, should it exist. We rigorously prove that a locally asymptotically stable period orbit must be heterogeneity stable orbit -- a generalization of evolutionarily stable state.
Mathematical Models of Evolution and Replicator Systems Dynamics
Moscow Center for Fundamental and Applied Mathematics, Lomonosov Moscow State University（莫斯科国立大学基础与应用数学莫斯科中心）；A. I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation（阿·伊·阿利哈尼扬国家科学实验室（耶烈万物理研究所）基金会）
Eco-evolutionary games for harvesting self-renewing common resource: Effect of growing harvester population
公地悲剧（英语：Tragedy of the Commons，TOC）是一个普遍存在的社会困境，在一群生物与共享资源之间的互动中被见证：群体中的个体倾向于自私地过度开发公共资源，因为这可以说是理性的选择，或者在非人类的情况下，这可能是进化上不可侵犯的行为。如何避免TOC是一个关系到节约资源的重大问题。不难设想这样的情况：资源可以自我更新，人口的规模可能取决于资源的状况，因为人口的不同部分采用不同的开采率。如果资源的自我更新率位于最大和最小开采率之间，那么在什么条件下可以避免TOC就不是先验的。在本文中，我们解决这个问题的分析和数值模拟使用的进化博弈理论复制子方程模型的达尔文主义的自然选择的宗旨设置。通过复制子方程，我们在研究复制子群体如何利用共享资源的同时，也不能忽略后者对前者的动态反馈。我们还提出了一个透明的自下而上的游戏资源反馈模型的推导，以方便未来的研究对本文提出的研究结果的随机效应。
Max Planck Institute of Molecular Cell Biology and Genetics（马克斯·普朗克分子细胞生物学与遗传学研究所）；Center for Systems Biology Dresden（德累斯顿系统生物学中心）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学数学研究所）
