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

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.

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Cited by in corpus (7)

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• Ideal Free Dispersal under General Spatial heterogeneity and Time Periodicity
• A comparative study between discrete and continuum models for the evolution of competing phenotype-structured cell populations in dynamical environments
• Asymptotic analysis of selection-mutation models in the presence of multiple fitness peaks
• Dynamics of dirac concentrations in the evolution of quantitative alleles with sexual reproduction
• Asymptotics of the principal eigenvalue for a linear time-periodic parabolic operator II: Small diffusion
• Harris's method for non-conservative periodic semiflows and application to some non-local PDEs