The Existence and Uniqueness of Positive Periodic Solutions for a Class of Nicholson-Type Systems with Impulses and Delays

Ruojun Zhang,Fuyun Lian

doi:10.1155/2013/980935

Abstract

By establishing the equivalence, respectively, to the existence and uniqueness of positive periodic solutions for corresponding delay Nicholson-type systems without impulses, some criteria for the existence and uniqueness of positive periodic solutions for a class of Nicholson-type systems with impulses and delays are established. The results of this paper extend some earlier works reported in the literature.

Highlights

As we know, in order to describe the dynamics of Nicholson’s blowflies, Gurney et al [1] proposed a mathematical modelN󸀠 (t) = −δN (t) + PN (t − τ) e−aN(t−τ), (1)where N(t) is the size of the population at time t, P is the maximum per capita daily egg production, 1/a is the size at which the population reproduces at its maximum rate, δ is the per capita daily adult death rate, and τ is the generation time
We study the existence of positive periodic solutions for the corresponding equation without impulses by coincidence degree theory, and we study the uniqueness of positive periodic solutions for the corresponding equation without impulses by the Lyapunov function
It suffices to prove that the set of all possible positive ω-periodic solution of (6) is bounded

Summary

Introduction

In order to describe the dynamics of Nicholson’s blowflies, Gurney et al [1] proposed a mathematical model. There have been some results in the literature of the problem of the existence of positive periodic solutions for Nicholson’s blowflies equation [7,8,9,10,11]. Liu [14] established some criteria for existence and uniqueness of positive periodic solutions of system (2) by applying the method of the Lyapunov functional. It is necessary and reasonable to consider impulsive effects on the existence and uniqueness of positive periodic solutions for Nicholson-type delay systems (2). Techniques and methods of existence and uniqueness of positive periodic solutions for system (2) with impulsive effects should be developed and explored.

Preliminaries

Main Results

Conclusion