Dynamical Behaviors of a Stage-Structured Predator-Prey Model with Harvesting Effort and Impulsive Diffusion

Lingzhi Huang,Zhichun Yang

doi:10.1155/2015/371852

Abstract

We consider a delayed predator-prey model with harvesting effort and impulsive diffusion between two patches. By the impulsive comparison theorem and the discrete dynamical system determined by the stroboscopic map, we obtain some sufficient conditions on the existence and global attractiveness of predator-eradicated periodic solution for the system. Furthermore, the permanence of the system is derived. The obtained results will modify and improve the ones in some existing publications and give the estimate for the ultimately low and upper boundedness of the systems.

Highlights

The effect of spatial factors in population dynamics is an interesting topic since dispersal often occurs between patches in ecological environment [1, 2]
Since the short-time diffusion is often assumed to be in the form of impulses in the modeling process, some mathematical models on impulsive diffusion have been studied by impulsive differential equations
In [5], a single species model with impulsive diffusion was initially formulated by x1󸀠 (t) = x1 (t) (a1 − b1x1 (t)), x2󸀠 (t) = x2 (t) (a2 − b2x2 (t)), t ≠ nτ, (1)

Summary

Introduction

The effect of spatial factors in population dynamics is an interesting topic since dispersal often occurs between patches in ecological environment [1, 2]. To manage effectively the species, we should know or estimate how many members the population has at large time in every patch when a species is uniformly permanent This corresponds to the low and upper boundedness of solutions of the systems. Motivated by the above discussion, we propose the following stage-structured predator-prey model with generalized functional response, the harvesting effort of the mature predator, and impulsive diffusion between two predators’ territories: x1󸀠 (t) = x1 (t) (a1 − b1x1 (t)) − p1 (x1 (t)) z1 (t) , y1󸀠 (t) = α1z1 (t) − α1e−ω1τ1 z1 (t − τ1) − ω1y1 (t) , z1󸀠 (t) = α1e−ω1τ1 z1 (t − τ1) + β1p1 (x1 (t)) z1 (t). Some examples and their simulations are given to illustrate the effectiveness of our results

Preliminaries

Extinction of the Predator-Eradicated Periodic Solution

Permanence

Examples and Their Simulations