The diffusive competition problem with a free boundary in heterogeneous time-periodic environment

In this paper, we consider the diffusive competition problem with a free boundary and sign-changing intrinsic growth rate in heterogeneous time-periodic environment, consisting of an invasive species with density $u$ and a native species with density $v$. We assume that $v$ undergoes diffusion and growth in $R^{N}$ , and $u$ exists initially in a ball $B_{h_0}(0)$, but invades into the environment with spreading front $\{r = h(t)\}$. The effect of the dispersal rate $d_1$, the initial occupying habitat $h_0$, the initial density $u_0$ of invasive species $u$, and the parameter $\mu$ (see (1.3)) on the dynamics of this free boundary problem are studied. A spreading-vanishing dichotomy is obtained and some sufficient conditions for the invasive species spreading and vanishing are provided. Moreover, when spreading of $u$ happens, some rough estimates of the spreading speed are also given.