Dirac concentrations in Lotka-Volterra parabolic PDEs

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.
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Indiana University Mathematics Journal, Indiana University Mathematics Journal, 2008, 57 (7), pp.3275-3301
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  • HAL Id : hal-00168404, version 1
  • ARXIV : 0708.3720

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Benoît Perthame, Guy Barles. Dirac concentrations in Lotka-Volterra parabolic PDEs. Indiana University Mathematics Journal, Indiana University Mathematics Journal, 2008, 57 (7), pp.3275-3301. 〈hal-00168404〉

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