Traulsen Arne, Claussen Jens Christian, Hauert Christoph
Program for Evolutionary Dynamics, Harvard University, One Brattle Square, Cambridge, Massachusetts 02138, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2006 Jul;74(1 Pt 1):011901. doi: 10.1103/PhysRevE.74.011901. Epub 2006 Jul 5.
Coevolving and competing species or game-theoretic strategies exhibit rich and complex dynamics for which a general theoretical framework based on finite populations is still lacking. Recently, an explicit mean-field description in the form of a Fokker-Planck equation was derived for frequency-dependent selection with two strategies in finite populations based on microscopic processes [A. Traulsen, J. C. Claussen, and C. Hauert, Phys. Rev. Lett. 95, 238701 (2005)]. Here we generalize this approach in a twofold way: First, we extend the framework to an arbitrary number of strategies and second, we allow for mutations in the evolutionary process. The deterministic limit of infinite population size of the frequency-dependent Moran process yields the adjusted replicator-mutator equation, which describes the combined effect of selection and mutation. For finite populations, we provide an extension taking random drift into account. In the limit of neutral selection, i.e., whenever the process is determined by random drift and mutations, the stationary strategy distribution is derived. This distribution forms the background for the coevolutionary process. In particular, a critical mutation rate uc is obtained separating two scenarios: above uc the population predominantly consists of a mixture of strategies whereas below uc the population tends to be in homogeneous states. For one of the fundamental problems in evolutionary biology, the evolution of cooperation under Darwinian selection, we demonstrate that the analytical framework provides excellent approximations to individual based simulations even for rather small population sizes. This approach complements simulation results and provides a deeper, systematic understanding of coevolutionary dynamics.
共同进化和相互竞争的物种或博弈论策略展现出丰富而复杂的动态变化，然而目前仍缺乏一个基于有限种群的通用理论框架。最近，基于微观过程，针对有限种群中两种策略的频率依赖选择，推导出了一种以福克 - 普朗克方程形式呈现的显式平均场描述[A. 特劳尔森、J. C. 克劳森和C. 豪尔特，《物理评论快报》95, 238701 (2005)]。在此，我们以两种方式对该方法进行推广：其一，将框架扩展至任意数量的策略；其二，在进化过程中考虑突变。频率依赖的莫兰过程在无限种群规模下的确定性极限产生了调整后的复制者 - 突变者方程，该方程描述了选择和突变的综合效应。对于有限种群，我们给出了一种考虑随机漂移的扩展形式。在中性选择的极限情况下，即只要过程由随机漂移和突变决定，就可以推导出平稳策略分布。这种分布构成了共同进化过程的背景。特别地，我们得到了一个临界突变率(u_c)，它区分了两种情况：高于(u_c)时，种群主要由多种策略的混合组成；低于(u_c)时，种群倾向于处于同质状态。对于进化生物学中的一个基本问题，即达尔文选择下合作的进化，我们证明即使对于相当小的种群规模，该分析框架也能为基于个体的模拟提供出色的近似。这种方法补充了模拟结果，并为共同进化动态提供了更深入、系统的理解。
