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In this paper, we consider the diffusive competition problem with a freeboundary and sign-changing intrinsic growth rate in heterogeneous time-periodicenvironment, consisting of an invasive species with density $u$ and a nativespecies 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 theenvironment with spreading front $\{r = h(t)\}$. The effect of the dispersalrate $d_1$, the initial occupying habitat $h_0$, the initial density $u_0$ ofinvasive species $u$, and the parameter $\mu$ (see (1.3)) on the dynamics ofthis free boundary problem are studied. A spreading-vanishing dichotomy isobtained and some sufficient conditions for the invasive species spreading andvanishing are provided. Moreover, when spreading of $u$ happens, some roughestimates of the spreading speed are also given.

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