The accumulation of beneficial mutations and convergence to a Poisson process

Abstract

We consider a model of a population with fixed size N, which is subjected to an unlimited supply of beneficial mutations at a constant rate μN. Individuals with k beneficial mutations have the fitness (1+sN)k. Each individual dies at rate 1 and is replaced by a random individual chosen with probability proportional to its fitness. We show that when μN≪1/(NlogN) and N−η≪sN≪1 for some η<1, the fixation times of beneficial mutations, after a time scaling, converge to the times of a Poisson process, even though for some choices of sN and μN satisfying these conditions, there will sometimes be multiple beneficial mutations with distinct origins in the population, competing against each other.