A Free Boundary Problem for a Prey-Predator Model with Degenerate Diffusion and Predator-Stage Structure
Abstract
In this paper, we study a free boundary problem for a stage-structured predator-prey model with degenerate diffusion and advection terms. The predator species is classified into immature and mature stages, and the interaction with prey species is modeled using a reaction-diffusion system. Initially, the predator species occupies a bounded region, while the prey population is distributed throughout the spatial domain. The dynamics of the free boundary, which describes the spreading of the predator species, are governed by a Stefan-like condition involving the spatial gradient of the predator density. We establish the existence and uniqueness of classical solutions to the problem using a priori estimates in Holder spaces and the Leray–Schauder fixed-point principle. Furthermore, we analyze the asymptotic behavior of the free boundary and derive the minimal spreading speed for the predator population. Our results demonstrate key phenomena related to spreading and vanishing: If the predator’s initial habitat or movement rate is sufficiently large, it will dominate and successfully spread; In contrast, predators with smaller habitats or slower movement may vanish over time.