A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment

Abstract

<p style='text-indent:20px;'>In this paper, we investigate a reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. The primary aim is to study the impact of small advection terms and heterogeneous environment, which is on two species' dynamics via a free boundary. The function <inline-formula><tex-math id="M1">\begin{document}$ m(x) $\end{document}</tex-math></inline-formula> represents heterogeneous environment, and it can satisfy positive everywhere condition or changeable sign condition. Firstly, on one hand, we provide long time behaviors of the solution in vanishing case when <inline-formula><tex-math id="M2">\begin{document}$ m(x) $\end{document}</tex-math></inline-formula> satisfies both conditions above; on the other hand, long time behaviors of the solution in spreading case are got when <inline-formula><tex-math id="M3">\begin{document}$ m(x) $\end{document}</tex-math></inline-formula> satisfies positive everywhere condition. Secondly, a spreading-vanishing dichotomy and several sufficient conditions through the initial data and the moving parameters are obtained to determine whether spreading or vanishing of two species happens when <inline-formula><tex-math id="M4">\begin{document}$ m(x) $\end{document}</tex-math></inline-formula> satisfies both conditions above. Furthermore, we derive estimates of spreading speed of the free boundary when <inline-formula><tex-math id="M5">\begin{document}$ m(x) $\end{document}</tex-math></inline-formula> satisfies positive everywhere condition and two species spreading occurs.</p>