A mutation-selection model for evolution of random dispersal

TL;DR

This paper analyzes a mutation-selection model for population evolution, showing that as mutation rate decreases, the population concentrates at the lowest diffusion trait, extending prior two-species results to multiple traits.

Contribution

It extends existing two-species diffusion results to models with many traits, demonstrating concentration phenomena at the lowest diffusion rate in the small mutation limit.

Findings

Population concentrates at lowest diffusion rate as mutation rate approaches zero.

The model's steady states remain regular in space but concentrate in trait variable.

Results generalize previous two-species diffusion dominance findings to multiple traits.

Abstract

We consider a mutation-selection model of a population structured by the spatial variables and a trait variable which is the diffusion rate. Competition for resource is local in spatial variables, but nonlocal in the trait variable. We focus on the asymptotic profile of positive steady state solutions. Our result shows that in the limit of small mutation rate, the solution remains regular in the spatial variables and yet concentrates in the trait variable and forms a Dirac mass supported at the lowest diffusion rate. [Hastings, Theor. Pop. Biol. 24, 244-251, 1983] and [Dockery et al., J. Math. Biol. 37, 61-83, 1998] showed that for two competing species in spatially heterogeneous but temporally constant environment, the slower diffuser always prevails, if all other things are held equal. Our result suggests that their findings may hold for arbitrarily many traits.