Travelling waves for integro-differential equations in populationdynamics

Abstract

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