适应度驱动的生物进化模型中的幂律尾
Power-Law Tails in a Fitness-Driven Model for Biological Evolution
摘要：
我们研究了一个离散时间随机过程，该过程也可以被解释为病毒进化的模型。我们的过程的一个显着特点是幂律尾由于动力学，类似于优先连接模型。在我们研究的模型中，一个种群被划分为多个位点，每个位点被一个在区间$[0,1]$中被随机分配的实数标记，称为适应度。种群规模是一个离散时间的瞬时生灭过程，其出生概率为$p$，死亡概率为$1-p$。适应度在出生时根据以下规则分配：种群的新成员以概率$r$“突变”，创建一个均匀分布在$[0,1]$上的新位点，或以概率$1-r$“继承”，加入一个与位点大小成比例的现有位点。在每个死亡事件中，来自该站点的具有最低适应度的成员被杀死。当且仅当$pr>1-p$ .在这个假设下，我们表明，随着时间趋于无穷大，网站的大小和适应度（在给定范围内的大小和适应度的网站中的人口比例）的联合经验措施收敛a.s.在$[(1-p)/(pr),1]$上，将其转化为修正的Yule分布与均匀分布的乘积。我们的方法是基于\cite{similar-but-different}中开发的方法。模型和结果由Roy和Tanemura在[RT]中独立获得。
英文摘要：
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].
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