In evolutionary dynamics, a key measure of a mutant trait’s success is the probability that it takes over the population given some initial mutant-appearance distribution. This “fixation probability” is difficult to compute in general, as it depends on the mutation’s effect on the organism as well as the population’s spatial structure, mating patterns, and other factors. In this study, we consider weak selection, which means that the mutation’s effect on the organism is small. We obtain a weak-selection perturbation expansion of a mutant’s fixation probability, from an arbitrary initial configuration of mutant and resident types. Our results apply to a broad class of stochastic evolutionary models, in which the size and spatial structure are arbitrary (but fixed). The problem of whether selection favors a given trait is thereby reduced from exponential to polynomial complexity in the population size, when selection is weak. We conclude by applying these methods to obtain new results for evolutionary dynamics on graphs.

在进化动力学中，突变特征成功的一个关键衡量标准是它在给定一些初始突变外观分布的情况下接管种群的概率。这种“固定概率”一般很难计算，因为它取决于突变对生物体的影响以及种群的空间结构、交配模式和其他因素。在本研究中，我们考虑弱选择，这意味着突变对生物体的影响很小。我们从突变体和常驻类型的任意初始配置中获得了突变体固定概率的弱选择扰动扩展。我们的结果适用于一大类随机进化模型，其中大小和空间结构是任意的（但固定的）。The problem of whether selection favors a given trait is thereby reduced from exponential to polynomial complexity in the population size, when selection is weak. 我们通过应用这些方法来获得图上进化动力学的新结果。