The diffusive competition problem with a free boundary in strong heterogeneous environment and weak heterogeneous environment

In this paper, we consider the diffusive competition problem consisting of an invasive species with density $u$ and a native species with density $v$. We assume that $v$ undergoes diffusion and growth in $[0, \infty)$, and $u$ exists initially in $[0, h_0)$, but invades into the environment with spreading front ${x=h(t)}$. To understand the effect of the dispersal rate $d_1$, the initial occupying habitat $h_0$, the initial density $u_{0}(x)$ of invasive species $(u)$, and the parameter $\mu$ (the ratio of the invasion speed of the free boundary and the invasive species gradient at the expanding front) on the dynamics of this free boundary problem, we divide the heterogeneous environment into two cases: strong heterogeneous environment and weak heterogeneous environment. A spreading-vanishing dichotomy is obtained and some sufficient conditions for the invasive species spreading and vanishing is provided both in the strong heterogenous environment and weak heterogenous environment. Moreover, when spreading of $u$ happens, some rough estimates of the spreading speed are also given.