Time fluctuations in a population model of adaptive dynamics

  • Sepideh Mirrahimi

    CNRS, Institut de Mathématiques de Toulouse UMR 5219, 31062 Toulouse, France, Université de Toulouse, UPS, INSA, UT1, UTM, IMT, 31062 Toulouse, France
  • Benoît Perthame

    UPMC Université Paris 06, Laboratoire Jacques-Louis Lions, CNRS UMR7598, 4 pl. Jussieu, 75005 Paris, France, INRIA EPI BANG and Institut Universitaire de France, France
  • Panagiotis E. Souganidis

    The University of Chicago, Department of Mathematics, 5734 S. University Avenue, Chicago, IL 60637, USA

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.

Cite this article

Sepideh Mirrahimi, Benoît Perthame, Panagiotis E. Souganidis, Time fluctuations in a population model of adaptive dynamics. Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 1, pp. 41–58

DOI 10.1016/J.ANIHPC.2013.10.001