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.

Classification: 35B25, 35K57, 49L25, 92D15

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}, publisher = {Elsevier}, volume = {32}, number = {1}, year = {2015}, pages = {41-58}, doi = {10.1016/j.anihpc.2013.10.001}, zbl = {1312.35011}, mrnumber = {3303941}, language = {en}, url = {http://www.numdam.org/item/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://www.numdam.org/item/AIHPC_2015__32_1_41_0/

[1] Concentration in Lotka–Volterra parabolic or integral equations: a general convergence result, Methods Appl. Anal. 16 no. 3 (2009), 321 -340 | MR 2650800 | Zbl 1204.35027

, , ,[2] Concentrations and constrained Hamilton–Jacobi equations arising in adaptive dynamics, Contemp. Math. 439 (2007), 57 -68 | MR 2359020 | Zbl 1137.49027

, ,[3] Comparison of the Perron and Floquet eigenvalues in monotone differential systems and age structured equations, C. R. Math. Acad. Sci. Paris 345 no. 10 (2007), 549 -555 | MR 2374463 | Zbl 1141.34326

, , ,[4] On mutation-selection dynamics for continuous structured populations, Commun. Math. Sci. 6 no. 3 (2008), 729 -747 | MR 2455473 | Zbl 1176.45009

, , , ,[5] The dynamics of adaptation: an illuminating example and a Hamilton–Jacobi approach, Theor. Popul. Biol. 67 no. 4 (2005), 257 -271 | Zbl 1072.92035

, , , ,[6] Evolutionary and continuous stability, J. Theor. Biol. 103 no. 1 (1983), 99 -111 | MR 714279

,[7] On selection dynamics for competitive interactions, J. Math. Biol. 63 no. 3 (2011), 493 -517 | MR 2824978 | Zbl 1230.92038

, ,[8] Bacterial persistence: a model of survival in changing environments, Genetics 169 (2005), 1807 -1814

, , , ,[9] Natural selection and random genetic drift in phenotypic evolution, Evolution 30 (1976), 314 -334

,[10] Dirac mass dynamics in multidimensional nonlocal parabolic equations, Commun. Partial Differ. Equ. 36 no. 6 (2011), 1071 -1098 | MR 2765430 | Zbl 1229.35113

, , ,[11] The logic of animal conflict, Nature 246 (1973), 15 -18

, ,[12] Dirac concentrations in Lotka–Volterra parabolic PDEs, Indiana Univ. Math. J. 57 no. 7 (2008), 3275 -3301 | MR 2492233 | Zbl 1172.35005

, ,[13] Long time evolution of populations under selection and vanishing mutations, Acta Appl. Math. 114 (2011), 1 -14 | MR 2781076 | Zbl 1213.35112

,[14] Population models in a periodically fluctuating environment, J. Math. Biol. 9 (1980), 23 -36 | MR 648843 | Zbl 0426.92018

,[15] The evolutionarily stable phenotype distribution in a random environment, Evolution 49 no. 2 (1995), 337 -350

, ,[16] Comparing environmental and genetic variance as adaptive response to fluctuating selection, Evolution 65 (2011), 2492 -2513

, , ,