Global Dynamics of a Predator-Prey Model with Stage Structure and Delayed Predator Response

Lili Wang,Rui Xu

doi:10.1155/2013/724325

Abstract

A Holling type II predator-prey model with time delay and stage structure for the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is discussed. The existence of Hopf bifurcations at the coexistence equilibrium is established. By means of the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functionals and LaSalle’s invariance principle, it is shown that the predator-extinction equilibrium is globally asymptotically stable when the coexistence equilibrium is not feasible, and the sufficient conditions are obtained for the global stability of the coexistence equilibrium.

Highlights

The functional response of predator to prey density refers to the change in the density of prey attacked per unit time per predator as the prey density changes [1]
Based on experiments, Holling [2] suggested three different kinds of functional responses for different kinds of species to model the phenomena of predation, which made the standard Lotka-Volterra system more realistic
The most popular functional response used in the modelling of predator-prey systems is Holling type II with φ(x) = fx/(1 + mx) which takes into account the time a predator uses in handing the prey being captured

Summary

Introduction

The functional response of predator to prey density refers to the change in the density of prey attacked per unit time per predator as the prey density changes [1]. Motivated by the work of Wangersky and Cunningham [7] and Tian and Xu [10], we are concerned with the combined effects of the stage structure for the predator and time delay due to the gestation of mature predator on the global dynamics of a predator-prey model with Holling type II functional response. To this end, we consider the following delay differential system: ẋ x.

Local Stability and Hopf Bifurcation

Permanence

Global Stability