Rigorous results for a population model with selection I: evolution of the fitness distribution

TL;DR

This paper rigorously analyzes a fixed-size population model with beneficial mutations and selection, deriving precise results on mutation accumulation rates and fitness distribution, confirming prior theoretical predictions.

Contribution

It provides the first rigorous mathematical results on the evolution of fitness distribution in a population with selection and beneficial mutations.

Findings

Quantifies the rate of mutation accumulation.

Describes the fitness distribution over time.

Confirms predictions of Desai and Fisher (2007).

Abstract

We consider a model of a population of fixed size $N$ undergoing selection. Each individual acquires beneficial mutations at rate $\mu_N$, and each beneficial mutation increases the individual's fitness by $s_N$. Each individual dies at rate one, and when a death occurs, an individual is chosen with probability proportional to the individual's fitness to give birth. Under certain conditions on the parameters $\mu_N$ and $s_N$, we obtain rigorous results for the rate at which mutations accumulate in the population and the distribution of the fitnesses of individuals in the population at a given time. Our results confirm predictions of Desai and Fisher (2007).