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On a non-local equation arising in population dynamics

Published online by Cambridge University Press:  23 July 2007

Jerome Coville  and
Louis Dupaigne
Jerome Coville
Affiliation:
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, boîte courrier 187, 75252 Paris Cedex 05, France ([email protected])
Louis Dupaigne
Affiliation:
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 ([email protected])
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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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 137 , Issue 4 , August 2007 , pp. 727 - 755
Copyright
2007 Royal Society of Edinburgh

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