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Spatial heterogeneity and habitat characteristic are shown to determine theasymptotic profile of the solution to a reaction-diffusion model with freeboundary, which describes the moving front of the invasive species. A thresholdvalue $R_0^{Fr}(D,t)$ is introduced to determine the spreading and vanishing ofthe invasive species. We prove that if $R_0^{Fr}(D,t_0)\geq 1$ for some$t_0\geq 0$, the spreading must happen; while if $R_0^{Fr}(D,0)<1$, thespreading is also possible. Our results show that the species in the favorablehabitat can establish itself if the diffusion is slow or the occupying habitatis large. In an unfavorable habitat, the species dies out if the initial valueof the species is small. However, big initial number of the species is benefitfor the species to survive. When the species spreads in the whole habitat, theasymptotic spreading speed is given. Some implications of these theoreticalresults are also discussed.

The spreading front of invasive species in favorable habitat or unfavorable habitat