Long time evolutionary dynamics of phenotypically structured populations in time periodic environments

TL;DR

This paper analyzes the long-term behavior of phenotypically structured populations in fluctuating environments, showing convergence to periodic solutions and population concentration as mutations diminish, with implications for biological experiments.

Contribution

It introduces a novel analysis of long-term dynamics and mutation effects in time-periodic environments using Hamilton-Jacobi theory.

Findings

Solution converges to a unique periodic state over time

Population concentrates on a single phenotype as mutations vanish

Periodic population size varies with environmental fluctuations

Abstract

We study the long time behavior of a parabolic Lotka-Volterra type equation considering a time-periodic growth rate with non-local competition. Such equation describes the dynamics of a phenotypically struc-tured population under the effect of mutations and selection in a fluctuating environment. We first prove that, in long time, the solution converges to the unique periodic solution of the problem. Next, we describe this periodic solution asymptotically as the effect of the mutations vanish. Using a theory based on Hamilton-Jacobi equations with constraint, we prove that, as the effect of the mutations vanishes, the solution concentrates on a single Dirac mass, while the size of the population varies periodically in time. When the effect of the mutations are small but nonzero, we provide some formal approximations of the moments of the population's distribution. We then show, via some examples, how such results could be compared to biological experiments.