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

  • * Corresponding author

    * Corresponding author 
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.
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  • 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.

    Mathematics Subject Classification: 35C07, 45C05, 45G15, 45M20, 47G20, 92D25.


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