Zuhan Liu

A free boundary problem of a predator-prey model with advection in heterogeneous environment

This paper is concerned with a system of reaction-diffusion-advection equations with a free boundary, which arises in a predator-prey ecological model in heterogeneous environment. The evolution of the free boundary problem is discussed. Precisely, we prove a spreading-vanishing dichotomy, namely both prey and predator either survive and establish themselves successfully in the new environment, or they fail to establish and vanishes eventually. Furthermore, when spreading occurs, we obtain an upper bound of the asymptotic spreading speed, which is smaller than the minimal speed of the corresponding traveling wave problem.

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 ) 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.