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. B 19 (2014), 3105–3132). Precisely, when u is an inferior competitor, we prove that ( u, v ) → (0, V ) as t →∞. When u is a superior competitor, we prove that a spreading–vanishing dichotomy holds, namely, as t →∞, either h ( t )→∞ and ( u, v ) → ( U , 0), or lim t →∞ h ( t ) < ∞ 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.

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