Dynamics for the diffusive Leslie-Gower model with double free boundaries

TL;DR

This paper studies a free boundary problem for a diffusive predator-prey model, establishing existence, uniqueness, and conditions for the predator's spreading or vanishing in a one-dimensional environment.

Contribution

It introduces a novel analysis of the Leslie-Gower model with double free boundaries, providing criteria for spreading and vanishing and addressing the challenge of unbounded terms.

Findings

Existence and uniqueness of global solutions.

Spreading-vanishing dichotomy established.

Criteria for long-term survival or extinction.

Abstract

In this paper we investigate a free boundary problem for the diffusive Leslie-Gower prey-predator 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. We first prove the existence, uniqueness and regularity of global solution. Then provide a spreading-vanishing dichotomy, namely the predator species either successfully spreads to infinity as $t\to\infty$ at both fronts and survives in the new environment, or it spreads within a bounded area and dies out in the long run. The long time behavior of $(u,v)$ and criteria for spreading and vanishing are also obtained. Because the term $v/u$ (which appears in the second equation) may be unbounded when $u$ nears zero, it will bring some difficulties for our study.