A Stochastic Model for the Evolution of a Quasispecies

We introduce a discrete time model for a virus-like evolving population with high mutation probability. Different genomes correspond to different points (or sites) in the interval [0, 1]. Each site has one or more individual on it (corresponding to the number of individuals with that genome). When a birth with mutation occurs a new site is selected uniformly in [0, 1] and we put one individual on it. When a birth is without mutation we select one existing site at random and increase its population (or size) by 1. When a death occurs we kill the smallest site and all its population. From previous work we know that there is a critical value in (0, 1) such that the distribution of sites in converges to a uniform distribution as time tends to infinity. We prove here that the number of individuals per site converges to a geometric distribution whose parameter can be computed exactly, and that the size of the population at a site above is independent of the location of the site. We argue that this picture is consistent with quasispecies theory.