An evolutional free-boundary problem of a reaction-diffusion-advection system

In this paper we consider a system of reaction-diffusion-advection equations with a free boundary, which arises in a competition ecological model in heterogeneous environment. The evolution of the free-boundary problem is discussed, which is an extension of the results of Du and Lin (Discrete Contin. Dynam. Syst. B19 (2014), 3105-3132). Precisely, when u is an inferior competitor, we prove that (u, v) -> (0, V) as t -> infinity. When u is a superior competitor, we prove that a spreading-vanishing dichotomy holds, namely, as t -> infinity, either h(t) -> infinity and (u, v) -> (U, 0), or lim(t ->infinity) h(t) < infinity and (u, v) -> (0, V). Moreover, in a weak competition case, we prove that two competing species coexist in the long run, while in a strong competition case, two species spatially segregate as the competition rates become large. Furthermore, when spreading occurs, we obtain some rough estimates of the asymptotic spreading speed.