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

Spatial heterogeneity and habitat characteristic are shown to determine the asymptotic profile of the solution to a reaction–diffusion model with free boundary, which describes the moving front of the invasive species. A threshold value R0Fr(D,t) is introduced to determine the spreading and vanishing of the invasive species. We prove that if R0Fr(D,t0)⩾1 for some t0⩾0t0⩾0, the spreading must happen; while if R0Fr(D,0)<1, the spreading is also possible. Our results show that the species in the favorable habitat can establish itself if the diffusion is slow or the occupying habitat is large. In an unfavorable habitat, the species dies out if the initial value of the species is small. However, big initial number of the species is benefit for the species to survive. When the species spreads in the whole habitat, the asymptotic spreading speed is given. Some implications of these theoretical results are also discussed.