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We study the diffusive logistic equation with a free boundary in timeperiodicenvironment. To understand the effect of the dispersal rate $d$, the originalhabitat 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 intotwo cases: favorable habitat and unfavorable habitat. By choosing $d$, $h_0$,$\mu$ and $u_0$ as variable parameters, we obtain a spreading-vanishingdichotomy and sharp criteria for the spreading and vanishing in time-periodicenvironment. We introduce the principal eigenvalue $\lambda_1(d, \alpha,\gamma, h(t), T)$ to determine the spreading and vanishing of the invasivespecies. We prove that if $\lambda_1(d, \alpha, \gamma, h_0, T)\leq 0$, thespreading must happen; while if $\lambda_1(d, \alpha, \gamma, h_0, T)> 0$, thespreading is also possible. Our results show that the species in the favorablehabitat can establish itself if the dispersal rate is slow or the occupyinghabitat is large. In an unfavorable habitat, the species vanishes if theinitial density of the species is small, while survive successfully if theinitial value is big. Moreover, when spreading occurs, the asymptotic spreadingspeed of the free boundary is determined.

A diffusive logistic problem with a free boundary in time-periodic environment: favorable habitat or unfavorable habitat