On a free boundary problem for a nonlocal reaction-diffusion model

Abstract

This paper is concerned with the spreading or vanishing dichotomy of a species which is characterized by a reaction-diffusion Volterra model with nonlocal spatial convolution and double free boundaries. Compared with classical reaction-diffusion equations, the main difficulty here is the lack of a comparison principle in nonlocal reaction-diffusion equations. By establishing some suitable comparison principles over some different parabolic regions, we get the sufficient conditions that ensure the species spreading or vanishing, as well as the estimates of the spreading speed if species spreading happens. Particularly, we establish the global attractivity of the unique positive equilibrium by a method of successive improvement of lower and upper solutions.