Abstract
In this paper we study a predator–prey system with free boundary in a one-dimensional environment. The predator v is the invader which exists initially in a sub-interval [0, s_{0}] of [0,L] and has the Leslie–Gower terms that measure the loss in the predator population due to rarity of the prey. The prey u (the native species) is initially distributed over the whole region [0,L]. Our primary goal is to understand how the success or failure of the predator’s invasion is affected by the initial datum v_{0}. We derive a spreading–vanishing dichotomy and give sharp criteria for spreading and vanishing in this model.
Highlights
In this paper we study a predator–prey system with free boundary in a one-dimensional environment
Inspired by former work (Chen and Shi [5] for instance) that studies the nonlinear evolution of two species on an unbounded spatial domain, we focus on the case where indigenous population undergoes diffusion and growth in a bounded domain [0, L] to be more realistic
We have studied a Leslie–Gower and Holling-type II predator–prey model in one-dimensional environment
Summary
In this paper we study a predator–prey system with free boundary in a one-dimensional environment. 1.2 Global existence of smooth solutions Theorem 1.1 Assume that (u0, v0) satisfies (3). The following lemma is essential in proving the existence of a global-in-time solution to the free-boundary problem (1)–(2). From Theorem 1.1 and Lemma 1.1, we get the following global existence result.
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