Propagation dynamics of a nonlocal reaction-diffusion system

Abstract

This paper is dedicated to investigate the propagation dynamics of a three-species reaction-diffusion system with nonlocal diffusion which includes diffusion kernel functions symmetry and asymmetry. First off, we give the asymptotic spreading speeds by using its partial commonality with the minimum propagation spreadings and construct a set $ \Pi $ to obtain their signs which just shows it is fundamentally different from local diffusion. The following main focus on the asymptotic behaviour with different initial values. The key point for asymmetry kernel is to construct suitable upper and lower solutions and obtain long-time asymptotic behavior by using comparison principle and our improved 'forward-backward spreading' method which first proposed by Xu et al. (J Funct Anal 280(2021)108957). Especially, transforming the square scale, introducing new variables and using linear programming to obtain the existence of variables and other techniques are our improvement of this method which ensures it works for multi-population systems. Accordingly, the asymptotic behavior and some monotone property results with the symmetric kernel are all obtained by comparison principle.