Existence and Uniqueness of Positive Periodic Solutions for a Delayed Predator-Prey Model with Dispersion and Impulses

Zhenguo Luo,Yunhui Zeng,Liping Luo,Liu Yang,Zhenghui Gao

doi:10.1155/2014/592543

Abstract

An impulsive Lotka-Volterra type predator-prey model with prey dispersal in two-patch environments and time delays is investigated, where we assume the model of patches with a barrier only as far as the prey population is concerned, whereas the predator population has no barriers between patches. By applying the continuation theorem of coincidence degree theory and by means of a suitable Lyapunov functional, a set of easily verifiable sufficient conditions are obtained to guarantee the existence, uniqueness, and global stability of positive periodic solutions of the system. Some known results subject to the underlying systems without impulses are improved and generalized. As an application, we also give two examples to illustrate the feasibility of our main results.

Highlights

The aim of this paper is to investigate the existence and uniqueness of the positive periodic solution of the following impulsive Lotka-Volterra type predator-prey model with prey dispersal in two-patch environments and time delays: x1󸀠 (t) = x1 (t) [r1 (t) − a11 (t) x1 (t) − a13 (t) x3 (t)]
Since σ󸀠(t) < 1, t ∈ [0, ω] and t − σ(t) is continuous on R, it follows that t − σ(t) has a unique inverse function μ(t) ∈ C(R, R) on R
According to Theorem 7, we see that model (125) has at least one positive 2π-periodic solution

Summary

Introduction

The aim of this paper is to investigate the existence and uniqueness of the positive periodic solution of the following impulsive Lotka-Volterra type predator-prey model with prey dispersal in two-patch environments and time delays: x1󸀠 (t) = x1 (t) [r1 (t) − a11 (t) x1 (t) − a13 (t) x3 (t)]. Xu et al [12] had considered the following delayed periodic Lotka-Volterra type predator-prey system with prey dispersal in two-patch environments: x1󸀠 (t) = x1 (t) [r1 (t) − a11 (t) x1 (t) − a13 (t) y (t)]. The aim of this paper is to obtain a set of verifiable sufficient conditions to guarantee the existence, uniqueness, and global stability of positive periodic solutions of the system (1) by further developing the analysis technique of [10,11,12,13,14,15].

Existence of Positive Periodic Solutions

Uniqueness and Global Stability

Some Examples