Stability of Traveling Wave Fronts for a Three Species Predator-Prey Model with Nonlocal Dispersals

Dongmei Yuan,Yuzhen Bai

doi:10.1155/2019/8742958

Abstract

In this paper, we consider a predator-prey model with nonlocal dispersals of two cooperative preys and one predator. We prove that the traveling wave fronts with the relatively large wave speed are exponentially stable as perturbation in some exponentially weighted spaces, when the difference between initial data and traveling wave fronts decay exponentially at negative infinity, but in other locations, the initial data can be very large. The adopted method is to use the weighted energy method and the squeezing technique with some new flavors to handle the nonlocal dispersals.

Highlights

We investigate the stability of traveling wave fronts for a three species predator-prey model:
Yu and Pei [5] studied the stability of traveling wave fronts for the cooperative system with nonlocal dispersals:
E authors adopt the weighted energy method and the squeezing technique to prove the stability of the traveling wave fronts. e weighted energy method for treating timedelayed reaction diffusion equations was firstly introduced by Mei et al [6]. en, by combining the squeezing argument, it was developed for proving the global stability of wavefronts by [7,8,9]

Summary

Introduction

We investigate the stability of traveling wave fronts for a three species predator-prey model: zu(x, t) zt zv(x, t) zt zz(x, t) zt. Yu and Pei [5] studied the stability of traveling wave fronts for the cooperative system with nonlocal dispersals:. E authors adopt the weighted energy method and the squeezing technique to prove the stability of the traveling wave fronts. For the nonlocal model using a single integrodifferential equation, the existence, uniqueness, and stability of traveling waves have been widely studied in [10,11,12,13,14]. Many researchers are widely focused on the complex dynamics of biological systems such as stochastic delay population system [18] and many researchers have studied the Lotka–Volterra time delay models with two competitive preys and one predator [19]. We denote by C0([0, T]; B) the space of the B-valued continuous functions on [0, T], and L2([0, T]; B) as the space of the B-valued L2 functions on [0, T]. e corresponding spaces of the B-valued functions on [0, ∞) are defined

Preliminaries and Main Result

Stability

Conclusions