Traveling waves for nonlocal Lotka-Volterra competition systems

In this paper, we study the traveling wave solutions of a Lotka-Volterra diffusion competition system with nonlocal terms. We prove that there exists traveling wave solutions of the system connecting equilibrium \begin{document}$ (0, 0) $\end{document} to some unknown positive steady state for wave speed \begin{document}$ c>c^* = \max\left\{2, 2\sqrt{dr}\right\} $\end{document} and there is no such traveling wave solutions for \begin{document}$ c , where \begin{document}$ d $\end{document} and \begin{document}$ r $\end{document} respectively corresponds to the diffusion coefficients and intrinsic rate of an competition species. Furthermore, we also demonstrate the unknown steady state just is the positive equilibrium of the system when the nonlocal delays only appears in the interspecific competition term, which implies that the nonlocal delay appearing in the interspecific competition terms does not affect the existence of traveling wave solutions. Finally, for a specific kernel function, some numerical simulations are given to show that the traveling wave solutions may connect the zero equilibrium to a periodic steady state.