#### Abstract

By using pollution model and impulsive delay differential equation, we formulate a pest control model with stage structure for natural enemy in a polluted environment by introducing a constant periodic pollutant input and killing pest at different fixed moments and investigate the dynamics of such a system. We assume only that the natural enemies are affected by pollution, and we choose the method to kill the pest without harming natural enemies. Sufficient conditions for global attractivity of the natural enemy-extinction periodic solution and permanence of the system are obtained. Numerical simulations are presented to confirm our theoretical results.

#### 1. Introduction

Nowadays, the problem of the world’s environmental pollution is serious, which has a frustrating effect on the ecosystem damage in the direct or indirect ways. Pollution leads to the living environmental change and gene mutation. It results in not only birth defects and deformities but also population variability, which decreases the number of the population in the nature and even makes them extinct. In order to assess the risk of the populations exposed to a polluted environment, in recent years, mathematical models concerning this topic have been studied extensively including continuous pollution input and impulsive pollution input [1–11].

As we all know, the predator-prey system can be used to model the process of controlling the pests by spraying pesticides, as well as relying on their natural enemies. However, in a polluted environment, some natural enemies are affected by pollution seriously and pests almost are not affected. For example, frogs are the natural enemies of beetles, locusts, and mole cricket, but some chemical plants discard waste products into rivers for their convenience, which cause severe water contamination, seriously injures frog’s reproductive system, and significantly decreases their fertility. Moreover, water pollution also causes large quantities of the fertilized eggs and tadpoles to die, resulting in the decrease of frogs. It is shown in a Sweden’s new study that male tadpoles can eventually grow into female frogs only in the environment similar to the nature but full of pollutants with estrogen. However, some male frogs have ovaries but no fallopian tubes, and they finally turn into lifelong infertile frogs, which are called “Yin and Yang frog”, and nearly one-third of the world’s frog species may be extinct because of the environmental pollution. People must control the period and quantity of emission of pollution to prevent natural enemy from extinction. In addition, too much pesticide spraying will reduce pests significantly; meanwhile, it also causes serious environmental pollution. Therefore, when controlling pests, we had better choose the method to kill the pests without polluting the environment and harming natural enemies at regular intervals.

The predator-prey models with stage structure for the predator were introduced or investigated by Hastings and Wang [12–14]. Since the immature predator takes (which is called maturation time delay) units of time to mature, the death toll during the juvenile period should be considered, and time delays have important biological meanings in stage structured models. Recently, many models with time delay were extensively studied [15–22].

According to the above biological background, in this paper, we suggest an impulsive predator-prey pollution model with stage structured for predator by introducing a constant periodic pollutant input and proportional killing pest at different fixed moments to model the process of pest control and polluted environment. Recently, there has been quite a lot of literatures on the applications of impulsive differential equations on population [1, 2, 8, 10, 11, 20–31]. To our knowledge, there have been no results on this topic in the literature. The questions that arise here are as follows: how do we control the emission of pollution to prevent the extinction of natural enemies? Under what condition can the system be permanent? How can we control pests effectively?

The organization of this paper is as follows. In the next section, we formulate our model and give several lemmas which are useful for our main results. In Section 3 and Section 4, the sufficient conditions for the global attractivity of the “natural enemy-extinction” periodic solution and permanence of the system are obtained. We give a brief discussion of our results in Section 5. Numerical simulations are presented to illustrate our theoretical results.

#### 2. Model Formulation and Preliminaries

In this paper, we assume only that the natural enemies are affected by pollution and we choose the method to kill the pest without harming natural enemies. Then a pest control model with stage structure for natural enemy in a polluted environment by introducing a constant periodic pollutant input and killing pests at different fixed moment is formulated as follows: where , , , , . , , and represent the densities of prey (pest), immature, and mature predator (natural enemy) at time , respectively; , represent the concentration of pollution in the environment and organism at time , respectively; is intrinsic growth rate of the pests in the absence of natural enemies; is the pest capacity of environment; is the predation rate of natural enemy and represents the conversion rate at which ingested pest in excess of what is needed for maintenance is translated into natural enemy increase; is the saturation which represents that a certain amount of natural enemies can prey on a limited amount of pests, though the pests are numerous; and are the death rate of immature and mature natural enemies, respectively; in addition, we assume that juveniles suffer a mortality rate of (the through-stage death rate) and take units of time to mature. and are the dose-response parameters of species to the pollution in the immature and mature natural enemies, respectively; the exogenous quantity of impulsive input of pollutant into the environment at time is represented by ; () represents a proportional decrease of pest because of being harvested at time . The other parameters can be seen in [1].

The initial conditions for (1) are

Note that the variable does not appear in the first, third, forth, and fifth equations of system (1); hence, we only need to consider the subsystem of (1) as follows:

Lemma 1 (see [32]). *Consider the following delay differential equation:
**
where , , and are all positive constants and for ; we have*(1)*if , then ;*(2)*if , then .*

Lemma 2 (see [1]). *Consider the following subsystem of (1):
**
Then system (5) has a unique positive -periodic solution , which is globally asymptotically stable, where
**
and .*

Lemma 3. * There exists a constant such that , , , , and .*

*Proof. *Define . Since , , in addition, , , thus for large enough. Define ; then, for , ; we have
where . Consider the following impulse differential inequalities:
We have
Hence
so is uniformly ultimately bounded. Therefore, by the definition of , system (1) is uniformly ultimately bounded. The proof is completed.

Lemma 4 (see [29]). * If holds, system
**
has a unique positive globally asymptotically stable periodic solution , .*

Therefore, if holds, the system (3) has a natural enemy-extinction periodic solution . In this paper, we assume that always holds.

*Remark 5 (see [1]). * and are the concentration of pollution. To assure and , it is necessary that .

*Remark 6 (see [1]). * According to the biological significance, we assume .

#### 3. Global Attractivity of the “Natural Enemy-Extinction” Periodic Solution

In this section, we discuss under what condition the natural enemies will go extinct.

Denote where

Theorem 7. * If , then the “mature natural enemy-extinction” periodic solution of system (3) is globally attractive.*

*Proof. *Since , we have

By Lemma 2, for sufficiently small enough , there exists a positive constant such that
holds for .

Note that
Then we consider the following comparison system:
According to Lemma 4, we know that
is a unique globally asymptotically stable positive -periodic solution of system (17).

By using comparison theorem of impulsive differential equation, there exist a positive integer and a sufficiently small positive constant such that for all ,
holds. From (15), (19), and the second equation of (3), we obtain that for ,
holds.

Consider the following comparison equation:

By inequality (14), we have that holds; then, according to Lemma 1, we obtain that

By the comparison theorem of delay differential equation, we have .

Without loss of generality, we may assume that ( is sufficiently small positive constant such that ) for all ; by the first equation of system (3), we have

Consider the following comparison equation:

By Lemma 4,
is a unique globally asymptotically stable positive -periodic solution of system (24). By using comparison theorem of impulsive differential equation, for above and large enough, we have

It follows from (19) and (26) that
holds for large enough. Let ; we can get , so
holds for large enough, which implies as . According to Lemma 2, , as . This completes the proof.

#### 4. Permanence

*Definition 8. *System (1) is said to be permanent if there are positive constants , , and a finite time such that for all solutions with initial conditions (2), , , , holds for all , .

Denote
where

Theorem 9. *If , there exists a positive constant such that for any solution of system (3).*

*Proof. *Since , we can choose positive constants , , and such that
and hold, where
The second equation of system (3) can be written as
Define
Calculating the derivative of along the solution of (3), we have
We claim that the inequality cannot hold for all . Otherwise, there is a positive constant such that for all . From the first equation of system (3), we have
Consider the following comparison system:
Then
is a unique globally asymptotically stable positive -periodic solution of system (37). By using comparison theorem of impulsive differential equation, for , there exists a such that for ,
holds.

By Lemma 2, for , there exists a such that for ,
Let , and from (39) and (40), we have
Let
We will show that for all . Otherwise, there exists a nonnegative constant such that for , and . Thus from the second equation of (3), (31), and (41), we easily see that
which is a contradiction. Hence we get that for all . Then we have
which implies as . This is a contradiction to . Therefore, for any positive constant , the inequality cannot hold for all . If holds true for all large enough, then our aim is obtained; otherwise, is oscillatory about . Let
In the following, we will show that . There exist two positive constants and such that
When is large enough, the inequality holds true for . Since is continuous and bounded and not affected by impulses, we conclude that is uniformly continuous. Hence there exists a constant (, and is independent of the choice of ) such that for all . If , our aim is obtained. If , from the second equation of (3), we have that
Then we have for . It is clear that for .

If , by the second equation of (3), then we have that for . The same arguments can be continued, and we can obtain for . Since the interval is arbitrarily chosen, we get that for t large enough. In view of our arguments above, the choice of is independent of the positive solution of (3). This completes the proof.

Theorem 10. * If , the system (1) is permanent.*

*Proof. *Suppose that is any positive solution of system (1) with initial conditions (2). By (39), we have for large enough. By Theorem 9, we have for large enough. From the second equation of system (1), we obtain
So,
Thus for large enough. By Lemma 2, we know for a sufficiently small positive , . Then from (15), Lemma 3, and Definition 8, we have that system (1) is permanent. The proof is completed.

#### 5. Discussion

In this paper, we discuss a pest control model with stage structure for natural enemy in a polluted environment by introducing a constant periodic pollutant input and killing pest at different fixed moments. From Theorems 7, 9, and 10, we can observe that the extinction and permanence of the population are very much dependent on , , and .

To verify the theoretical results obtained in this paper, in the following we will give some numerical simulations and take ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; (see Figure 1), and here we can compute , and from Theorem 10 we know the system (1) is permanent. If we decrease the period of pulsing () or increase the pollution input amount to (), and other parameters are the same with those in Figure 1, the natural enemy will be extinct (see Figures 2 and 3). If we increase the harvesting rate of pests to , and other parameters are the same with those in Figure 1, then , and the natural enemies will also be extinct (see Figure 4). Our results indicate that if impulsive period is short or or is too large, the natural enemy will go extinct, but we wish to protect natural enemy from extinction, so we should harvest the pests reasonably and control the period and quantity of emission of pollution into the environment efficiently. This offers us some reasonable suggestions for pest management.

#### Acknowledgments

This work is supported by National Natural Science Foundation of China (10971001, 11371030) and the Excellent Talents Support Project of Universities and Colleges in Liaoning.