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 ${\mu }_N$. Individuals with $k$ beneficial mutations have the fitness $(1+s_N{)}^k$. Each individual dies at rate 1 and is replaced by a random individual chosen with probability proportional to its fitness. We show that when ${\mu }_N\ll 1\mathrm{/}(N\log N)$ and $N^{-\eta }\ll s_N\ll 1$ for some $\eta <1$, the fixation times of beneficial mutations, after a time scaling, converge to the times of a Poisson process, even though for some choices of $s_N$ and ${\mu }_N$ satisfying these conditions, there will sometimes be multiple beneficial mutations with distinct origins in the population, competing against each other.