Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems

The present paper is concerned with the spatial spreading speeds and traveling wave solutions of cooperative systems in space-time periodic habitats with nonlocal dispersal. It is assumed that the trivial solution \begin{document}${\bf u} = {\bf 0}$\end{document} of such a system is unstable and the system has a stable space-time periodic positive solution \begin{document}${\bf u^*}(t,x)$\end{document} . We first show that in any direction \begin{document}$ξ∈ \mathbb{S}^{N-1}$\end{document} , such a system has a finite spreading speed interval, and under certain condition, the spreading speed interval is a singleton set, and hence, the system has a single spreading speed \begin{document}$c^{*}(ξ)$\end{document} in the direction of \begin{document}$ξ$\end{document} . Next, we show that for any \begin{document}$c>c^{*}(ξ)$\end{document} , there are space-time periodic traveling wave solutions of the form \begin{document}${\bf{u}}(t,x) = {\bf{Φ}}(x-ctξ,t,ctξ)$\end{document} connecting \begin{document}${\bf u^*}$\end{document} and \begin{document}${\bf 0}$\end{document} , and propagating in the direction of \begin{document}$ξ$\end{document} with speed \begin{document}$c$\end{document} , where \begin{document}$Φ(x,t,y)$\end{document} is periodic in \begin{document}$t$\end{document} and \begin{document}$y$\end{document} , and there is no such solution for \begin{document}$c . We also prove the continuity and uniqueness of space-time periodic traveling wave solutions when the reaction term is strictly sub-homogeneous. Finally, we apply the above results to nonlocal monostable equations and two-species competitive systems with nonlocal dispersal and space-time periodicity.