The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor

Abstract

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 $\mathbb{R}^N$, 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 $\mathbb{R}^N$. In the case that $u$ is a superior competitor (determined by the reaction terms), we show that a spreading-vanishing dichotomy holds, namely, as $t\to\infty$, either $h(t)\to\infty$ and $(u,v)\to (u^*,0)$, or $\lim_{t\to\infty} h(t)<\infty$ and $(u,v)\to (0,v^*)$, 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 $(u,v)\to (0,v^*)$ as $t\to\infty$, so the invasive species $u$ always vanishes in the long run.