  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.
