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

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. 1. Introduction In population dynamics, 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 which takes into account the time a predator uses in handing the prey being captured. There has been a large body of work about predator-prey systems with Holling type II functional responses, and many good results have been obtained (see, e.g., [1, 3, 4]). Time delays of one type or another have been incorporated into biological models by many researchers. We refer to the monographs of Gopalsamy [5], Kuang [6], and Wangersky and Cunningham [7] on delayed predator-prey systems. In these research works, it is shown that a time delay could cause a stable equilibrium to become unstable and cause the population to fluctuate. Hence, delay differential equations exhibit more complex dynamics than ordinary differential equations. Time delay due to gestation is a common example, since generally the consumption of prey by the predator throughout its past history decides the present birth rate of the predator. In [7], Wangersky and Cunningham proposed and studied the following non-Kolmogorov-type predator-prey model: In this model, it is assumed that a duration of time units elapses when an individual prey is killed and the moment when the corresponding addition is made to the predator population. In natural world, there are many species whose individuals have a history that