Communications in Mathematical Sciences

Volume 13 (2015)

Number 6

A Hamilton–Jacobi approach for a model of population structured by space and trait

Pages: 1431 – 1452



Emeric Bouin (UMR CNRS 5669 ‘UMPA’ and INRIA project ‘NUMED’, École Normale Supérieure de Lyon, France)

Sepideh Mirrahimi (CNRS, Institut de Mathématiques de Toulouse UMR 5219, Toulouse, France)


We study a non-local parabolic Lotka–Volterra type equation describing a population structured by a space variable $x \in \mathbb{R}^d$ and a phenotypical trait $\theta \in \Theta$. Considering diffusion, mutations, and space-local competition between the individuals, we analyze the asymptotic (long-time/long-range in the $x$ variable) exponential behavior of the solutions. Using some kind of real phase WKB ansatz, we prove that the propagation of the population in space can be described by a Hamilton–Jacobi equation with obstacle which is independent of $\theta$. The effective Hamiltonian is derived from an eigenvalue problem.

The main difficulties are the lack of regularity estimates in the space variable, and the lack of comparison principle due to the non-local term.


structured populations, asymptotic analysis, Hamilton–Jacobi equation, spectral problem, front propagation

2010 Mathematics Subject Classification

35B25, 35F21, 45K05, 49L25, 92D15

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Published 13 May 2015