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1 LJLL - Laboratoire Jacques-Louis Lions 2 LMPT - Laboratoire de Mathématiques et Physique Théorique

Abstract : We consider parabolic partial differential equations of Lotka-Volterra type, with a non-local nonlinear term. This models, at the population level, the darwinian evolution of a population; the Laplace term represents mutations and the nonlinear birth-death term represents competition leading to selection. Once rescaled with a small diffusion, we prove that the solutions converge to a moving Dirac mass. The velocity and weights cannot be obtained by a simple expression, e.g., an ordinary differential equation. We show that they are given by a constrained Hamilton-Jacobi equation. This extends several earlier results to the parabolic case and to general nonlinearities. Technical new ingredients are a $BV$ estimate in time on the non-local nonlinearity, a characterization of the concentration point in a monomorphic situation and, surprisingly, some counter-examples showing that jumps on the Dirac locations are indeed possible.

Keywords : population dynamics Integral parabolic equations adaptive dynamics asymptotic behavior Dirac concentrations population dynamics.

Autor: Benoît Perthame - Guy Barles -

Fuente: https://hal.archives-ouvertes.fr/


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