# American Institute of Mathematical Sciences

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January 2019, 18(1): 361-396. doi: 10.3934/cpaa.2019019

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

 1 School of Science, Chang'an University, Xi'an, Shaanxi 710064, China 2 Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA 3 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada

* Corresponding author

Received  January 2018 Revised  January 2018 Published  August 2018

Fund Project: X. Bao was partially supported by Natural Science Basic Research Plan in Shaanxi Province of China (2017JQ1014) and NSF of China (11701041). Z. Shen was supported by a start-up grant from the University of Alberta.

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 ${\bf u} = {\bf 0}$ of such a system is unstable and the system has a stable space-time periodic positive solution ${\bf u^*}(t,x)$. We first show that in any direction $ξ∈ \mathbb{S}^{N-1}$, 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 $c^{*}(ξ)$ in the direction of $ξ$. Next, we show that for any $c>c^{*}(ξ)$, there are space-time periodic traveling wave solutions of the form ${\bf{u}}(t,x) = {\bf{Φ}}(x-ctξ,t,ctξ)$ connecting ${\bf u^*}$ and ${\bf 0}$, and propagating in the direction of $ξ$ with speed $c$, where $Φ(x,t,y)$ is periodic in $t$ and $y$, and there is no such solution for $c<c^{*}(ξ)$. 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.

Citation: Xiongxiong Bao, Wenxian Shen, Zhongwei Shen. Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems. Communications on Pure & Applied Analysis, 2019, 18 (1) : 361-396. doi: 10.3934/cpaa.2019019
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