| Abstract: |
| This paper is concerned with space
periodic traveling wave solutions of the following Lotka-Volterra competition system with nonlocal dispersal and space periodic dependence,
\[
\begin{cases}
\frac{\partial{u_{1}}}{\partial{t}}=\int_{\Bbb{R}^{N}}\kappa(y-x)u_{1}(t,y)dy-u_{1}(t,x)+u_{1}(a_{1}(x)-b_{1}(x)u_{1}-c_{1}(x)u_{1}),\,\, x\in\RR^N\
\frac{\partial{u_{2}}}{\partial{t}}=\int_{\Bbb{R}^{N}}\kappa(y-x)u_{2}(t,y)dy-u_{2}(t,x)+u_{2}(a_{2}(x)-b_{2}(x)u_{1}-c_{2}(x)u_{2}),\,\, x\in\RR^N.
\end{cases}
\]
Under suitable assumptions, the system admits two semitrivial space periodic equilibria $(u^{*}_{1}(x),0)$ and $(0,u^{*}_{2}(x))$, where $(u^{*}_{1}(x),0)$ is linearly and globally stable and
$(0,u_{2}^{*}(x))$ is linearly unstable with respect to space periodic perturbations. By sub- and supersolution techniques and comparison principals, we show that, for any given $\xi\in S^{N-1}$, there exists a continuous periodic traveling wave solution of the form $(u_1(t,x),u_2(t,x))=\left(\Phi_{1}(x-ct\xi,ct\xi),\Phi_{2}(x-ct\xi,ct\xi)\right)$ connecting $(u^{*}_{1}(\cdot),0)$ and $(0,u^{*}_{2}(\cdot))$ and propagating in the direction of $\xi$ with speed $c>c^{*}(\xi)$, where $c^{*}(\xi)$ is the spreading speed of the system in the direction of $\xi$. Moreover, for $cc^{*}(\xi)$, we also prove the asymptotic stability and uniqueness of traveling wave solution using squeezing techniques. |
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