Global asymptotic stability of constant equilibrium in a nonlocal diffusion competition model with free boundaries

Abstract

<p style='text-indent:20px;'>In this paper we give a classification of the global asymptotic stability for a nonlocal diffusion competition model with free boundaries consisting of an invasive species with density <inline-formula><tex-math id="M1">\begin{document}$ u $\end{document}</tex-math></inline-formula> and a native species with density <inline-formula><tex-math id="M2">\begin{document}$ v $\end{document}</tex-math></inline-formula>. We not only prove that such nonlocal diffusion problem has a unique global solution and also determine the long-time asymptotic behavior of the solution for three competition cases : (<b>I</b>) <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula> is an inferior competitor, (<b>II</b>) <inline-formula><tex-math id="M4">\begin{document}$ u $\end{document}</tex-math></inline-formula> is a superior competitor and (<b>III</b>) the weak competition case. Especially, in case (<b>II</b>), under some additional conditions, we determine the long-time asymptotic behavior of the solution when vanishing happens. Moreover, the criteria for spreading and vanishing are obtained.</p>