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.


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 () = /(1 + ) 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 predatorprey model: ẋ () =  () ( 1 −  () −  1  ()) , ẏ () =  2  ( − )  ( − ) −  2  () . ( 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 can be divided into two stages immature and mature.Usually the dynamics-eating habits of predator are often quite different in different stages.Generally speaking, the immature predators are raised by their parents and do not have the ability to attack prey, so the rate at which they attack prey and the reproductive rate can be ignored.Hence, it is of ecological importance to investigate predator-prey models with stage structure.In recent years, the predator-prey population models with stage structure have received much attraction (see, e.g., [8][9][10]).In [10], it was assumed that feeding on prey can only make contribution to the increasing of the physique of the predator and does not make contribution to the reproductive ability, and the following strengthen type predator-prey model with stage structure was studied: where () represents the density of the prey at time  and  1 () and  2 () represent the densities of the immature and the mature predator at time , respectively. 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: where the meanings of the variables ,  1 ,  2 , and the parameters , , ,  1 ,  2 ,  1 ,  2 , , are the same as those of system (2) and the constant  ≥ 0 denotes the time delay due to the gestation of the mature predator.This is based on the assumption that the change rate of predators depends on the number of prey and of mature predators present at some previous time.
The initial conditions for system (3) take the form where By the fundamental theory of functional differential equations [11], it is well known that system (3) has a unique solution ((),  1 (),  2 ()) satisfying initial conditions (4).Further, it is easy to show that all solutions of system (3) are defined on [0, +∞) and remain positive for all  ≥ 0.
The paper is organized as follows.In the next section, by analyzing the corresponding characteristic equations, the local stability of each of nonnegative equilibria of system (3) is discussed and the existence of Hopf bifurcations at the coexistence equilibrium is established.In Section 3, permanence of the system (3) is proved by means of the persistence theory on infinite dimensional systems.In Section 4, by using Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are received for the global asymptotic stability of the predator-extinction equilibrium and the coexistence equilibrium.

Local Stability and Hopf Bifurcation
In this section, we discuss the local stability of each of feasible equilibria of system (3) and the existence of Hopf bifurcations at the coexistence equilibrium.
Concluding the above discussions, we obtain the following results.
Theorem 1.For system (3), one has the following.
In the following, we discuss the local stability of the coexistence equilibrium  * = ( * ,  * 1 ,  * 2 ) and the existence of Hopf bifurcations at  * .

Lemma 2. Suppose that (H1) is satisfied and 𝑧
Denote then ±  is a pair of purely imaginary roots of ( 14) with  =  [12], the following result can be obtained.Lemma 3. Suppose that (H1) and (H2) are satisfied.
Applying Lemmas 3 and 4, we obtain the following results.
Next, we use the persistence theory on infinite dimensional systems introduced by Hale and Waltman in [13] to prove the permanence of system (3).
Proof.We need only to show that the boundaries of  3 +0 repel positive solutions of system (3) uniformly.
Let  + ([−, 0],  3 +0 ) denote the space of continuous functions mapping [−, 0] into  3 +0 .Define ).Now, we verify that the conditions in Lemma 9 are satisfied.According to the definition of  0 and  0 , it is easy to know that  0 and  0 are positively invariant, so condition (ii) in Lemma 9 is satisfied.The solution of system ( 3) is ultimately bounded if (H1) holds by Lemma 8. Thus, by the smoothing property of solutions of delay differential equations introduced in [6, Theorem 2.2.8], condition (i) is satisfied.
By Lemma 9, we conclude that  0 repels positive solutions of system (3) uniformly.Hence, system (3) is permanent.This proof is complete.