A Free Boundary Problem for a Predator-prey Model with Double Free Boundaries

TL;DR

This paper studies a predator-prey model with two free boundaries, analyzing conditions under which the predator successfully spreads or dies out, and characterizing the long-term behavior of the system.

Contribution

It introduces a novel free boundary problem with double boundaries for predator-prey dynamics and establishes a spreading-vanishing dichotomy with criteria for each outcome.

Findings

Predator either spreads to infinity or dies out.

Conditions for successful spreading are identified.

Long-term behavior of prey stabilizes at a positive equilibrium.

Abstract

In this paper we investigate a free boundary problem for a predator-prey model with double free boundaries in one space dimension. This system models the expanding of an invasive or new predator species in which the free boundaries represent expanding fronts of the predator species and are described by Stefan-like condition. We prove a spreading-vanishing dichotomy for this model, namely the predator species either successfully spreads to infinity as $t\to \infty$ at both fronts and survives in the new environment, or it fails to establish and dies out in the long run while the prey species stabilizes at a positive equilibrium state. The long time behavior of solution and criteria for spreading and vanishing are also obtained.