THE DIFFUSIVE COMPETITION MODEL WITH A FREE BOUNDARY: INVASION OF A SUPERIOR OR INFERIOR COMPETITOR

In this paper we consider the diffusive competition model consisting of an invasive species with density u and a native species with density v, in a radially symmetric setting with free boundary. We assume that v undergoes diffusion and growth in RN, and u exists initially in a ball {r < h(0)}, but invades into the environment with spreading front {r = h(t)}, with h(t) evolving according to the free boundary condition h' (t) = -mu u(r)(t, h(t)), where mu > 0 is a given constant and u(t,h(t)) = 0. Thus the population range of u is the expanding ball {r < h(t)}, while that for v is R-N. In the case that u is a superior competitor (determined by the reaction terms), we show that a spreading-vanishing dichotomy holds, namely, as t -> infinity, either h(t) -> infinity and (u, v) -> (u*, 0), or limt(t -> infinity) h(t) < infinity and (u, v) > (0, v*), where (u*, 0) and (0,v*) are the semitrivial steady-states of the system. Moreover, when spreading of u happens, some rough estimates of the spreading speed are also given. When u is an inferior competitor, we show that (u, v) -> (0, v*) as t ->infinity, so the invasive species u always vanishes in the long run.