论文标题
在健身驱动的生物进化模型中的幂律尾巴
Power-Law Tails in a Fitness-Driven Model for Biological Evolution
论文作者
论文摘要
我们研究一个离散的随机过程，该过程也可以解释为病毒进化的模型。我们过程的一个显着特征是由于类似于优先附件模型的动力学而引起的幂律尾巴。在我们研究的模型中，将一个人群分配到站点中，每个站点都在$ [0,1] $中被唯一分配的实际数字标记为“健身”。人口规模是一个离散的瞬时出生和死亡过程，概率$ p $出生和$ 1-p $死亡。健身是根据以下规则在出生时分配的：人口的新成员要么以概率$ r $ $ r $“变异”，创建了一个新的网站，在$ [0,1] $上均匀分布，或“继承”概率$ 1-r $，并与现有的网站一起以与站点大小成正比的概率。在每个死亡事件中，适合最低的现场成员被杀死。当且仅当$ pr> 1-p $时，最终的站点数量趋于无限。在这个假设下，我们表明，随着时间的趋势无穷大，位点大小和适应性的联合经验度量（给定范围内的大小和适应度的人口比例）会收敛于A.S. $ [（1-p）/（pr），1] $的修改后分布和统一分布的乘积。我们的方法基于\ cite {相似但不同}中开发的方法。 Roy和Tanemura在[RT]中独立获得了模型和结果。
We study a discrete-time stochastic process that can also be interpreted as a model for a viral evolution. A distinguishing feature of our process is power-law tails due to dynamics that resembles preferential attachment models. In the model we study, a population is partitioned into sites, with each site labeled by a uniquely-assigned real number in the interval $[0,1]$ known as fitness. The population size is a discrete-time transient birth-and-death process with probability $p$ of birth and $1-p$ of death. The fitness is assigned at birth according to the following rule: the new member of the population either "mutates" with probability $r$, creating a new site uniformly distributed on $[0,1]$ or "inherits" with probability $1-r$, joining an existing site with probability proportional to the site's size. At each death event, a member from the site with the lowest fitness is killed. The number of sites eventually tends to infinity if and only if $pr>1-p$. Under this assumption, we show that as time tends to infinity, the joint empirical measure of site size and fitness (proportion of population in sites of size and fitness in given ranges) converges a.s. to the product of a modified Yule distribution and the uniform distribution on $[(1-p)/(pr),1]$. Our approach is based on the method developed in \cite{similar-but-different}. The model and the results were independently obtained by Roy and Tanemura in [RT].
