Proceedings of the Royal Society of Edinburgh: Section A Mathematics



On a non-local equation arising in population dynamics


Jerome Coville a1 and Louis Dupaigne a2
a1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, boîte courrier 187, 75252 Paris Cedex 05, France (coville@ann.jussieu.fr)
a2 Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, Faculté de Mathématiques et d'Informatique, Université Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens Cedex 1, France (dupaigne@math.cnrs.fr)

Article author query
coville j   [Google Scholar] 
dupaigne l   [Google Scholar] 
 

Abstract

We study a one-dimensional non-local variant of Fisher's equation describing the spatial spread of a mutant in a given population, and its generalization to the so-called monostable nonlinearity. The dispersion of the genetic characters is assumed to follow a non-local diffusion law modelled by a convolution operator. We prove that, as in the classical (local) problem, there exist travelling-wave solutions of arbitrary speed beyond a critical value and also characterize the asymptotic behaviour of such solutions at infinity. Our proofs rely on an appropriate version of the maximum principle, qualitative properties of solutions and approximation schemes leading to singular limits.

(Published Online July 23 2007)
(Received July 22 2004)
(Accepted June 28 2006)