Sample Christine, Allen Benjamin
Department of Mathematics, Emmanuel College, 400 Fenway, Boston, MA, 02115, USA.
Program for Evolutionary Dynamics, Harvard University, One Brattle Square, Cambridge, MA, 02138, USA.
J Math Biol. 2017 Nov;75(5):1285-1317. doi: 10.1007/s00285-017-1119-4. Epub 2017 Mar 28.
Evolutionary game theory is a mathematical approach to studying how social behaviors evolve. In many recent works, evolutionary competition between strategies is modeled as a stochastic process in a finite population. In this context, two limits are both mathematically convenient and biologically relevant: weak selection and large population size. These limits can be combined in different ways, leading to potentially different results. We consider two orderings: the [Formula: see text] limit, in which weak selection is applied before the large population limit, and the [Formula: see text] limit, in which the order is reversed. Formal mathematical definitions of the [Formula: see text] and [Formula: see text] limits are provided. Applying these definitions to the Moran process of evolutionary game theory, we obtain asymptotic expressions for fixation probability and conditions for success in these limits. We find that the asymptotic expressions for fixation probability, and the conditions for a strategy to be favored over a neutral mutation, are different in the [Formula: see text] and [Formula: see text] limits. However, the ordering of limits does not affect the conditions for one strategy to be favored over another.
进化博弈论是一种研究社会行为如何演变的数学方法。在许多近期的研究中，策略之间的进化竞争被建模为有限种群中的一个随机过程。在这种情况下，有两个极限在数学上既方便又具有生物学相关性：弱选择和大种群规模。这些极限可以以不同的方式组合，从而可能导致不同的结果。我们考虑两种排序：[公式：见正文]极限，即在大种群极限之前应用弱选择；以及[公式：见正文]极限，其中顺序相反。给出了[公式：见正文]和[公式：见正文]极限的形式化数学定义。将这些定义应用于进化博弈论的莫兰过程，我们得到了这些极限下固定概率的渐近表达式和成功条件。我们发现，在[公式：见正文]和[公式：见正文]极限中，固定概率的渐近表达式以及一个策略优于中性突变的条件是不同的。然而，极限的排序并不影响一个策略优于另一个策略的条件。
