A Free Boundary Problem with a Stefan Condition for a Ratio-dependent Predator-prey Model

TL;DR

This paper investigates a ratio-dependent predator-prey model with a free boundary, analyzing long-term behaviors, spreading-vanishing outcomes, and asymptotic spreading speeds in a one-dimensional habitat.

Contribution

It introduces a free boundary problem with a Stefan condition for a predator-prey model and establishes criteria for spreading and vanishing behaviors.

Findings

Proves a spreading-vanishing dichotomy for the model.

Derives criteria for successful spreading versus extinction.

Provides estimates for the asymptotic spreading speed when spreading occurs.

Abstract

In this paper we study a ratio-dependent predator-prey model with a free boundary causing by both prey and predator 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 die out eventually. Then the criteria governing spreading and vanishing are obtained. Finally, when spreading occurs, we provide some estimates to the asymptotic spreading speed of h(t).