Stability of periodic traveling waves for nonlocal dispersal cooperative systems in space–time periodic habitats

Abstract

The current paper is devoted to the study of the stability of space–time periodic traveling wave solutions and positive space–time periodic entire solutions of nonlocal dispersal cooperative systems in space–time periodic habitats. We first show the existence, uniqueness and stability of positive space–time periodic entire solution $${\mathbf {u}}^{*}(t,x)$$ for such nonlocal dispersal cooperative system. The existence of space–time periodic traveling wave solution connecting $${\mathbf {0}}$$ and positive space–time periodic entire solution $${\mathbf {u}}^{*}(t,x)$$ has been established by Bao, Shen and Shen (Commun. Pure Appl. Anal. 18: 361–396, 2019). In this paper, by using comparison principle and a weight function, we further show that the space–time periodic traveling wave solution for nonlocal dispersal cooperative system is asymptotically stable, as long as the initial value is uniformly bounded in a weighted space.