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.

本文致力于研究时空生境中非局部扩散合作系统的时空周期行波解和时空周期整体正解的稳定性。我们首先证明了这种非局部分散合作系统的正时空周期整体解\（{\ mathbf {u}} ^ {*}（t，x）\）的存在，唯一性和稳定性。连接\（{{mathbf {0}} \）和正时空周期整体解\（{\ mathbf {u}} ^ {*}（t，x）\）的时空周期行波解的存在由Bao，Shen和Shen建立（Commun。Pure Appl。Anal。18：361–396，2019）。本文利用比较原理和权函数，进一步证明了只要初始值在加权空间内均匀有界，则非局部色散合作系统的时空周期行波解是渐近稳定的。