Time fluctuations in a population model of adaptive dynamics

TL;DR

This paper analyzes how populations with phenotypic traits evolve in fluctuating environments, revealing concentration on optimal traits and effects of environmental oscillations on population size using advanced mathematical techniques.

Contribution

It introduces novel methods from Hamilton-Jacobi theory to study population dynamics under time oscillations, extending previous results to non-homogeneous environments.

Findings

Population concentrates on traits with maximum effective growth rate.

Time oscillations can lead to an increase in long-term population size.

Solutions converge to a Dirac mass driven by a Hamilton-Jacobi equation.

Abstract

We study the dynamics of phenotypically structured populations in environments with fluctuations. In particular, using novel arguments from the theories of Hamilton-Jacobi equations with constraints and homogenization, we obtain results about the evolution of populations in environments with time oscillations, the development of concentrations in the form of Dirac masses, the location of the dominant traits and their evolution in time. Such questions have already been studied in time homogeneous environments. More precisely we consider the dynamics of a phenotypically structured population in a changing environment under mutations and competition for a single resource. The mathematical model is a non-local parabolic equation with a periodic in time reaction term. We study the asymptotic behavior of the solutions in the limit of small diffusion and fast reaction. Under concavity assumptions on the reaction term, we prove that the solution converges to a Dirac mass whose evolution in time is driven by a Hamilton-Jacobi equation with constraint and an effective growth/death rate which is derived as a homogenization limit. We also prove that, after long-time, the population concentrates on a trait where the maximum of an effective growth rate is attained. Finally we provide an example showing that the time oscillations may lead to a strict increase of the asymptotic population size.