The Accumulation of Beneficial Mutations and Convergence to a Poisson Process

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 N. 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 N 1/ (N N) and N^- sN 1 for some < 1, large numbers of beneficial mutations are present in the population at the same time, competing against each other, yet the fixation times of beneficial mutations, after a time scaling, converge to the times of a Poisson process.