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.

Keywords: Reaction–diffusion equations, Asymptotic analysis, Hamilton–Jacobi equation, Adaptive dynamics, Population biology, Homogenization

@article{AIHPC_2015__32_1_41_0, author = {Mirrahimi, Sepideh and Perthame, Beno{\^\i}t and Souganidis, Panagiotis E.}, title = {Time fluctuations in a population model of adaptive dynamics}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {41--58}, publisher = {Elsevier}, volume = {32}, number = {1}, year = {2015}, doi = {10.1016/j.anihpc.2013.10.001}, mrnumber = {3303941}, zbl = {1312.35011}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2013.10.001/} }

TY - JOUR AU - Mirrahimi, Sepideh AU - Perthame, Benoît AU - Souganidis, Panagiotis E. TI - Time fluctuations in a population model of adaptive dynamics JO - Annales de l'I.H.P. Analyse non linéaire PY - 2015 SP - 41 EP - 58 VL - 32 IS - 1 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2013.10.001/ DO - 10.1016/j.anihpc.2013.10.001 LA - en ID - AIHPC_2015__32_1_41_0 ER -

%0 Journal Article %A Mirrahimi, Sepideh %A Perthame, Benoît %A Souganidis, Panagiotis E. %T Time fluctuations in a population model of adaptive dynamics %J Annales de l'I.H.P. Analyse non linéaire %D 2015 %P 41-58 %V 32 %N 1 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2013.10.001/ %R 10.1016/j.anihpc.2013.10.001 %G en %F AIHPC_2015__32_1_41_0

Mirrahimi, Sepideh; Perthame, Benoît; Souganidis, Panagiotis E. Time fluctuations in a population model of adaptive dynamics. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 1, pp. 41-58. doi : 10.1016/j.anihpc.2013.10.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2013.10.001/

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