#### Abstract

We investigate a predator-prey model with stage structure for the predator and periodic constant impulsive perturbations. Conditions for extinction of prey and immature predator are given. By using the Floquet theory and small amplitude perturbation skills, we consider the local stability of prey, immature predator eradication periodic solution. Furthermore, by using the method of numerical simulation, the influence of the impulsive control strategy on the inherent oscillation is investigated, which shows rich complex dynamic (such as periodic doubling bifurcation, periodic halving bifurcation, nonunique attractors, chaos, and periodic windows).

#### 1. Introduction

Stage structure model has received much attention in the recent years [1–3]. In this paper, based on the model of [1], we study the dynamic complexities of a predator-prey system with stage structure for the predator and periodic constant impulsive perturbations. The model takes the form

where is the density of prey at time are the densities of immature and mature predators at time , respectively. All parameters are positive constants. is the response function of the mature predator. is the intrinsic growth rate of the prey, is the death rate of the immature (mature) predator, constant denotes the coefficient in converting prey into a new immature predator, and the constant denotes the rate of immature predator becoming mature predator. , represents the fraction of prey (immature predator, mature predator) which dies at , is the period of the impulsive effect, and is the release amount of mature predator at .

Recently, it is of great interests to investigate chaotic impulsive differential equations about biological control [4–6]. Gao and Chen [7] and Tang and Chen [8] investigated dynamic complexities in a single-species model with stage structure and birth pulses.

#### 2. Preliminaries

In this section, we give some definitions and lemmas which will be useful to our results.

Let . Denote by the map defined by the right hand of the first, second, and third equations of system (1.1). Let , and let be said to belong to class if

(1) is continuous at for each , and exist;(2) is locally Lipschitzian in .*Definition 2.1. *Let , then for , the upper right derivative of with respect to the impulsive differential system (1.1) is defined as

Lemma 2.2. *Let be a solution of system (1.1) with , then for all . And further if .*

Lemma 2.3. *Suppose . Assume that
**
where is continuous in and for , then and exist, is nondecreasing. Let be maximal solution of the scalar impulsive differential equation
**
existing on , then , implies that , where is any solution of system (1.1).*

In the following, we give some basic properties about the following subsystem (2.4) and (2.7) of system (1.1):

Clearly

is a positive periodic solution of system (2.4). Since

is the solution of system (2.4) with initial value , where , we get the following.

Lemma 2.4. *Let be a positive periodic of system (2.4), and for every solution of system (2.4) with , one has , when ,
*

Lemma 2.5. *In subsystem (2.7) exists a equilibrium .*

Therefore, we obtain a prey, immature predator eradication periodic solution of system (1.1).

#### 3. Extinction and Boundary

In this section, we study the stability of prey, immature predator eradication periodic solution of system (1.1), and we investigate the boundary of system (1.1).

Theorem 3.1. *Let be any solution of system (1.1), then is locally asymptotically stable if
*

*Proof. *The local stability of periodic solution may be determined by considering the behavior of small amplitude perturbations of the solution. Define , then
where satisfies
and , the identity matrix. The impulsive perturbations of system (1.1) become
The stability of the periodic solution is determined by the eigenvalues of
where according to Floquet theory, the solution is locally stable if , this is to say, .

Theorem 3.2. *There exists a positive constant such that for each solution of system (1.1) with all large enough.*

*Proof. *Let be any solution of system (1.1). Define function , when , we select , then
where . When .

According to Lemma 2.3, for we have

Therefore is ultimately bounded, and we obtain that each positive solution of system (1.1) is uniformly ultimately bounded.

#### 4. Numerical Analysis

In this section, we study the influence of impulsive period and impulsive perturbation on complexities of the system (1.1).

Let . Since the corresponding continuous system (1.1) cannot be solved explicitly and system (1.1) cannot be rewritten as equivalent difference equations, it is difficult to study them analytically. So we have to study system (1.1) numerically integrated by stroboscopically sampling one of the variables over a range of values. The bifurcation diagram provides a summary of essential dynamical behavior of system.

From Theorem 3.1, we know that the prey, immature predator eradication periodic solution is locally stable if , which is shown in Figure 1. when , we find that the variable oscillates in a stable cycle. In contrast, the variables rapidly decrease to zero.

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The bifurcation diagram (Figure 2) shows that with increasing from 12.5 to 16, system (1.1) has rich dynamics including periodic doubling bifurcation, chaos, periodic windows, and nonunique attractors. When , system (1.1) has a stable -periodic solution. When , it is unstable and there is a cascade of periodic doubling bifurcations leading to chaos (Figure 3) (chaotic area with periodic windows). Figure 4 shows the phenomena of non-unique attractors [9]: different attractors can coexist, which one of the attractors is reached depends on the initial values.

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The bifurcation diagram (Figure 6) shows that with increasing from 0.01 to 0.8, system (1.1) has rich dynamics including periodic doubling bifurcation, chaos, periodic windows, and periodic halving bifurcation. When , system (1.1) has a stable -periodic solution. When , it is unstable and there is a cascade of periodic doubling bifurcations leading to chaos with periodic windows (Figure 8) which is followed by a cascade of periodic halving bifurcation from chaos to periodic solution.

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#### 5. Conclusion

In this paper, we investigated the dynamic complexities of a predator-prey system with impulsive perturbations and stage structure for the predator. Conditions for the local asymptotical stability of prey, immature predator eradication periodic solution were given by using the Floquet theory and small amplitude perturbation skills. Using the method of numerical simulation, the influence of impulsive control strategy on inherent oscillation showed that there exists complexity for system (1.1) including periodic doubling bifurcation, periodic halving bifurcation, nonunique attractors, chaos, and periodic windows. All these results showed that dynamical behavior of system (1.1) becomes more complex under periodically impulsive control strategy.