Travelling waves for integro-differential equations in populationdynamics

Narcisa Apreutesei; Arnaud Ducrot; Vitaly Volpert

doi:10.3934/dcdsb.2009.11.541

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2009, Volume 11, Issue 3: 541-561. Doi: 10.3934/dcdsb.2009.11.541

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Travelling waves for integro-differential equations in population dynamics

1.
Department of Mathematics, Technical University of Iasi, Iasi
2.
IMB & INRIA sud-ouest Anubis, Université de Bordeaux
3.
Institute of Mathematics, University Lyon 1, 69622 Villeurbann

Received: December 2007

Revised: October 2008

Published: May 2009

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Abstract

The paper is devoted to integro-differential equations arising in population dynamics. The integral term describes the nonlocal consumption of resources. We study the Fredholm property of the corresponding linear operators and use it to prove the existence of travelling waves when the support of the integral is sufficiently small. In this case, the integro-differential operator is close to the differential operator and we can use the implicit function theorem. We carry out numerical simulations in order to study the case where the support of the integral is not small. We observe various regimes of wave propagation. Some of them, in particular periodic waves do not exist for the usual reaction-diffusion equation.

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Mathematics Subject Classification: Primary: 35K57, 35R10, 47A53; secondary 92D25.

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