On competition models under allee effect: Asymptotic behavior and traveling waves

Abstract

<p style='text-indent:20px;'>In this article, we study a reaction-diffusion model on infinite spatial domain for two competing biological species (<inline-formula><tex-math id="M1">\begin{document}$ u $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ v $\end{document}</tex-math></inline-formula>). Under one-side Allee effect on <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula>-species, the model demonstrates complexity on its coexistence and <inline-formula><tex-math id="M4">\begin{document}$ u $\end{document}</tex-math></inline-formula>-dominance steady states. The conditions for persistence, permanence and competitive exclusion of the species are obtained through analysis on asymptotic behavior of the solutions and stability of the steady states, including the attraction regions and convergent rates depending on the biological parameters. When the Allee effect constant <inline-formula><tex-math id="M5">\begin{document}$ K $\end{document}</tex-math></inline-formula> is large relative to other biological parameters, the asymptotic stability of the <inline-formula><tex-math id="M6">\begin{document}$ v $\end{document}</tex-math></inline-formula>-dominance state <inline-formula><tex-math id="M7">\begin{document}$ (0,\:1) $\end{document}</tex-math></inline-formula> indicates the competitive exclusion of the <inline-formula><tex-math id="M8">\begin{document}$ u $\end{document}</tex-math></inline-formula>-species. Applying upper-lower solution method, we further prove that for a family of wave speeds with specific minimum wave speed determined by several biological parameters (including the magnitude of the <inline-formula><tex-math id="M9">\begin{document}$ u $\end{document}</tex-math></inline-formula>-dominance states), there exist traveling wave solutions flowing from the <inline-formula><tex-math id="M10">\begin{document}$ u $\end{document}</tex-math></inline-formula>-dominance states to the <inline-formula><tex-math id="M11">\begin{document}$ v $\end{document}</tex-math></inline-formula>-dominance state. The asymptotic rates of the traveling waves at <inline-formula><tex-math id="M12">\begin{document}$ \xi \rightarrow \mp \infty $\end{document}</tex-math></inline-formula> are also explicitly calculated. Finally, numerical simulations are presented to illustrate the theoretical results and population dynamics of coexistence or dominance-shifting.