A free boundary problem with a Stefan condition for a ratio-dependent predator-prey model

<abstract><p>In this paper we study a ratio-dependent predator-prey model with a free boundary caused by predator-prey interaction over a one dimensional habitat. We study the long time behaviors of the two species and prove a spreading-vanishing dichotomy; namely, as $ t $ goes to infinity, both prey and predator successfully spread to the whole space and survive in the new environment, or they spread within a bounded area and eventually die out. The criteria governing spreading and vanishing are obtained. Finally, when spreading occurs we provide some estimates to the asymptotic spreading speed of the moving boundary $ h(t) $.</p></abstract>

Dynamics of a three species ratio-dependent food chain model with diffusion and double free boundaries

This paper focuses on the dynamics of a three species ratio-dependent food chain model with diffusion and double free boundaries in one dimensional space, in which the free boundaries represent expanding fronts of top predator species. The existence, uniqueness and estimates of the global solution are discussed firstly. Then we prove a spreading–vanishing dichotomy, specifically, the top predator species either successfully spreads to the entire space as time t goes to infinity and survives in the new environment, or fails to establish and dies out in the long run. The long time behavior of the three species and criteria for spreading and vanishing are also obtained. Besides, our simulations illustrate the impacts of initial occupying area and expanding capability on the dynamics of top predator for free boundaries.