Long Time Evolutionary Dynamics of Phenotypically Structured Populations in Time-Periodic Environments

We study the long time behavior of a parabolic Lotka--Volterra type equation considering a time-periodic growth rate with nonlocal competition. Such an equation describes the dynamics of a phenotypically structured 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 effects 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 is 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.