论文标题
在一般有限特征空间上具有功率法突变率的随机基于个体的模型
Stochastic individual-based models with power law mutation rate on a general finite trait space
论文作者
论文摘要
我们考虑了一个基于随机的个体模型，用于单倍体，无性繁殖的种群的演变。可能的特征空间由（可能有限的）有限图$ g =（v，e）$的顶点给出。人口的演变是由$ g $边缘的出生，死亡，竞争和突变驱动的。我们对系统的负能力$ k $给出的突变率$μ_k$ $μ_k$感兴趣：$μ_k= k = k^{ - 1/α}，α> 0 $。这导致同时存在几种突变特征，并竞争入侵居民人群。我们将每个子群体在$ \ log k $时间尺度上的数量级的时间演变描述为$ k $倾向于无穷大。使用[Champagnat，Méléard，Tran，2019年]开发的技术，我们表明这些是分段仿射连续功能，其斜率由算法描述，该算法描述了由于新居民或新兴居民或新兴居民的连续性而导致健身景观的变化。这项工作将[Kraut，Bovier，2019]概括为随机设置，并将[Bovier，Coquille，Smadi，2018]定理3.2到任何有限的突变图。我们通过一系列示例来说明我们的定理，这些例子描述了图形和/或突变速率产生的令人惊讶的现象。
We consider a stochastic individual-based model for the evolution of a haploid, asexually reproducing population. The space of possible traits is given by the vertices of a (possibly directed) finite graph $G=(V,E)$. The evolution of the population is driven by births, deaths, competition, and mutations along the edges of $G$. We are interested in the large population limit under a mutation rate $μ_K$ given by a negative power of the carrying capacity $K$ of the system: $μ_K=K^{-1/α},α>0$. This results in several mutant traits being present at the same time and competing for invading the resident population. We describe the time evolution of the orders of magnitude of each sub-population on the $\log K$ time scale, as $K$ tends to infinity. Using techniques developed in [Champagnat, Méléard, Tran, 2019] we show that these are piecewise affine continuous functions, whose slopes are given by an algorithm describing the changes in the fitness landscape due to the succession of new resident or emergent types. This work generalises [Kraut, Bovier, 2019] to the stochastic setting, and Theorem 3.2 of [Bovier, Coquille, Smadi, 2018] to any finite mutation graph. We illustrate our theorem by a series of examples describing surprising phenomena arising from the geometry of the graph and/or the rate of mutations.
