Dirac concentrations in Lotka-Volterra parabolic PDEs

By Benoit Perthame, Guy Barles. 2007

150 citations recorded by the source corpus.

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.

Subfields: Genetics, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/06d83559db086f7f0457bc3937ed3226d495a26f

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Citations in the corpus

A Hamilton-Jacobi approach for a model of population structured by space and trait (2013, methodology): “Our objective is to generalize the methods developed recently on models structured only by a phenotypical trait [28, 5, 26] to spatial models, to be able to use the previous results in more general frameworks.”
Concentration in Lotka-Volterra Parabolic or Integral Equations: A General Convergence Result (2009, methodology): “We recall the following existence results for nǫ and a priori bounds for Iǫ (see also [8, 17]).”
Super-linear spreading in local and non-local cane toads equations (2015, unknown): “The Hopf-Cole transformation is an eﬀective tool in the analysis of front propagation for reaction-diﬀusion equations – see, for instance, [4, 22, 25], including parabolic integro-diﬀerential equations modeling populations structured by a phenotypical trait (see [14,21,41]).”