  We study the diffusive logistic equation with a free boundary in timeperiodic
environment. To understand the effect of the dispersal rate $d$, the original
habitat radius $h_0$, the spreading capability $\mu$, and the initial density
$u_0$ on the dynamics of the problem, we divide the time-periodic habitat into
two cases: favorable habitat and unfavorable habitat. By choosing $d$, $h_0$,
$\mu$ and $u_0$ as variable parameters, we obtain a spreading-vanishing
dichotomy and sharp criteria for the spreading and vanishing in time-periodic
environment. We introduce the principal eigenvalue $\lambda_1(d, \alpha,
\gamma, h(t), T)$ to determine the spreading and vanishing of the invasive
species. We prove that if $\lambda_1(d, \alpha, \gamma, h_0, T)\leq 0$, the
spreading must happen; while if $\lambda_1(d, \alpha, \gamma, h_0, T)> 0$, the
spreading is also possible. Our results show that the species in the favorable
habitat can establish itself if the dispersal rate is slow or the occupying
habitat is large. In an unfavorable habitat, the species vanishes if the
initial density of the species is small, while survive successfully if the
initial value is big. Moreover, when spreading occurs, the asymptotic spreading
speed of the free boundary is determined.
