A New Global Stable Conclusion on a Diffusive Leslie-Gower Predator-Prey System with Additive Allee Effect

Liu Yang

doi:10.1155/2019/7956015

Abstract

In this work, a diffusive Leslie-Gower predator-prey model with additive Allee effect on prey under a homogeneous Neumann boundary condition is reconsidered. We establish new sufficient conditions for the global stability of the unique positive equilibrium point of the system by using the comparison method rather than the Lyapunov function method. It is shown that our result supplements and complements one of the main results of Yang and Zhong, 2015. Furthermore, numerical simulations are performed to consolidate the analytic finding.

Highlights

Taking into account the inhomogeneous distribution of the predators and their preys in different spatial locations, the authors [1] established the following diffusive Leslie-Gower predator-prey model with additive Allee effect:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨zu zt zv zt d1Δu + u r1 − d2Δv + v􏼔r2 −a1u􏼁 − c2v 􏼕, u+k mu u+b − c1uv, ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩zu zv zn zn 0, u(x, 0) u0(x), v(x, 0)v0(x), t > 0, x ∈ Ω, t > 0, x ∈ Ω, t > 0, x ∈ zΩ, x ∈ Ω, (1)where u(x, t) and v(x, t) denote the densities of prey and predator at time t and position x, respectively. m and b are constants that indicate the severity of the Allee effect that has been modeled
Where u(x, t) and v(x, t) denote the densities of prey and predator at time t and position x, respectively. m and b are constants that indicate the severity of the Allee effect that has been modeled
The Allee effect is induced by predation

Summary

Introduction

Where u(x, t) and v(x, t) denote the densities of prey and predator at time t and position x, respectively. For the global stability of the diffusive system (1), the authors established the sufficient conditions with the Lyapunov function method in [1], which was used in most studies [1,2,3,4]. We will obtain a new global stability conclusion by the comparison method, which was used in [5, 6].

Results

Conclusion