Methods and Applications of Analysis

Concentration in Lotka-Volterra Parabolic or Integral Equations: A General Convergence Result

Guy Barles, Sepideh Mirrahimi, and Benoît Perthame

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We study two equations of Lotka-Volterra type that describe the Darwinian evolution of a population density. In the first model a Laplace term represents the mutations. In the second one we model the mutations by an integral kernel. In both cases, we use a nonlinear birth-death term that corresponds to the competition between the traits leading to selection.

In the limit of rare or small mutations, we prove that the solution converges to a sum of moving Dirac masses. This limit is described by a constrained Hamilton-Jacobi equation. This was already proved in Dirac concentrations in "Lotka-Volterra parabolic PDEs", Indiana Univ. Math. J., 57:7 (2008), pp. 3275–3301, for the case with a Laplace term. Here we generalize the assumptions on the initial data and prove the same result for the integro-differential equation.

Article information

Methods Appl. Anal., Volume 16, Number 3 (2009), 321-340.

First available in Project Euclid: 4 May 2010

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Zentralblatt MATH identifier

Primary: 35B25: Singular perturbations 35K57: Reaction-diffusion equations 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx] 49L25: Viscosity solutions 92D15: Problems related to evolution

Adaptive evolution Lotka-Volterra equation Hamilton-Jacobi equation viscosity solutions Dirac concentrations


Barles, Guy; Mirrahimi, Sepideh; Perthame, Benoît. Concentration in Lotka-Volterra Parabolic or Integral Equations: A General Convergence Result. Methods Appl. Anal. 16 (2009), no. 3, 321--340.

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