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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 832701, 18 pages

http://dx.doi.org/10.1155/2013/832701

## Stability and Permanence of a Pest Management Model with Impulsive Releasing and Harvesting

Department of Mathematics, Chongqing Normal University, Chongqing 400047, China

Received 16 January 2013; Accepted 27 February 2013

Academic Editor: Chuangxia Huang

Copyright © 2013 Jiangtao Yang and Zhichun Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We formulate a pest management model with periodically releasing infective pests, immature and mature natural enemies, and harvesting pests and crops at two different fixed moments. Sufficient conditions ensuring the locally and globally asymptotical stability of the susceptible pest-eradication period solution are found by means of Floquet theory, small amplitude perturbation techniques, and multicomparison results. Furthermore, the permanence of system is also derived. By numerical analysis, we also show that impulsive releasing and harvesting at two different fixed moments can bring obvious effects on the dynamics of system, which also corroborates our theoretical results.

#### 1. Introduction

As is known to all, pest outbreaks often cause serious ecological and economic problems. Therefore, how to effectively control insects and other arthropods has become an increasingly complex issue. Usually, chemical pesticides were taken as a relatively simple way to solve the pest-related problems, and some mathematical models on pest management with toxin (pesticide) input were studied in [1–4]. However, the overuse of chemical pesticides may create new ecological and sociological harm such as pesticide pollution and pesticide-resistant pest varieties and inflicts harmful effects on humans and so forth. Therefore, nonchemical use instead for pest control has become a hot topic in order to reduce pest density to tolerable levels and minimize the damage caused. For instance, biological control methods by periodically releasing infective pests or their natural enemies are often taken due to their advantage in the aspects of self-sustainable mechanism, lower environmental impact, and cost effectiveness.

Recently, some biocontrol models on pest management described by impulsive differential equation were proposed and the dynamics such as stability, permanence, periodicity, and bifurcation are deeply investigated (see also, e.g., [2–12]). In [5], an impulsive system to model the process of periodic releasing natural enemies and harvesting pest at different fixed time for pest control is considered, and the sufficient conditions on the existence and global stability of the periodic solution are derived for the given model. Georgescu et al. [6, 7] construct an integrated pest management model which relies on the simultaneous periodic release of infective pest individuals and of natural predators with age structure and obtain some sufficient conditions on the local and global stability, permanence, and bifurcation of the systems. However, most of the existing models on pest management scarcely take into account the factor on the relation between pest and its food (e.g., crop). In fact, farmers may harvest crops several times in process of its growth, which should cause a great impact on the density of the pest.

Motived by the above discussion, we construct a model of pest control by periodically releasing infective pests, immature and mature natural enemies, and harvesting pests and crops. To account for the discontinuity of release and harvest at different fixed moments, our model is based on impulsive differential equations. We analyze the dynamical behavior of the system by using the theory of impulsive differential equation introduced in [13–15].

The rest of this paper is organized as follows. A pest management model with impulsive releasing and harvesting is introduced in Section 2 and some useful preliminaries are given in Section 3. Section 4 deals with stability and permanence analysis of system. In this section, two sufficient conditions are deduced including the locally and globally asymptotical stability of the susceptible pest-eradication period solution, the permanence of system is also discussed. A simple example and conclusions are given in Section 5.

#### 2. Model Description

In the following, to establish our pest management model, we rely on the following biological assumptions. The pest population is divided into two classes, the susceptible and infective. The infective pests neither recover nor reproduce and infective pests cannot damage crops. The disease is transmitted from infective pests to susceptible pests and does not propagate to predators. In the absence of susceptible pests, the crops have a logistic growth rate with intrinsic birth rate and carrying capacity . The predators (natural enemies) have an age structure, that is, immature and mature. Only the mature predators have the ability to feed on susceptible pests, but do not prey on infective pests and crops. The functional response of the susceptible pest is described by the abstract function , the functional response of the mature predator is described by the abstract function , and the infection rate is described by the abstract function , where , , and satisfy certain assumptions outlined below.

On the basis of the above assumptions, we establish the following impulsively controlled system: where represents the density of the crop at time , represents the density of the susceptible pest at time , represents the density of the infective pest at time , and represent the density of the immature and mature predator at time , respectively; is the logistic intrinsic growth rate of the crop in the absence of the susceptible pest, is its carrying capacity; , represent the conversion rate at which ingested preys in excess of what is needed for maintenance is translated into predator population increase; is the rate at which the immature predators become the mature predators. , , , are the death rates of the susceptible pest population, infective pest population, and of the immature and mature predator population, respectively; , , , , ; is the period of the impulsive effect; is the harvesting rate of crop population; , , , denote the transfer rate of susceptible pest population, infective pest population, immature and mature predator population at every impulsive period , respectively; , , represents the amount of infective pests, immature and mature predators, respectively, which are released at every impulsive period , respectively; Also, , , , here , and for all .

Some familiar examples of functions in the biological literature include , with ; , with ; , with , where functions and are known as Holling type functional responses (see, [16–26]), and belongs to Ivlev type functional responses (see, [27–30]).

#### 3. Preliminaries

In this section, we will give some definitions and lemmas, which will be useful for our main results. Let and . Denote the map defined by the right hand of the first five equations in system (1). Let , then if (1) is continuous in , and for each , and exist. (2) is locally Lipschitzian .

*Definition 1. *Letting , one defines the upper right derivative of with respect to the impulsive differential system (1) at and by

*Definition 2. *The system (1) is said to be permanent if there are positive constants and a finite time such that all solutions of (1) with initial values , , , , , , , , , hold for all , where and are independent of initial value, may depend on initial value.

*Remark 3. *The global existence and uniqueness of system (1) is guaranteed by the smoothness properties of (for details, see [13, 14]).

Lemma 4 (see [15]). * Let satisfy , and assume that
**
where is continuous in and , for each , the limit and exists. is quasimonotone nondecreasing in . is nondecreasing for all . Let be the maximal solution of the following impulsive differential equation on :
**Then for any solution of the system (3), ≤ (≥) implies that for all . *

Lemma 5 (see [13, 15]). *Consider the following system:
**
where and , and are constants. Suppose that ** the sequence satisfies , with ; ** and is left-continuous at . ** Then, for ,
*

Lemma 6. * There exists a constant , such that , , , , for each solution of (1) with large enough. *

*Proof. *Since , then , and , so for large enough. Let us define by and denote . Then, it is obvious that
Since the right-hand side (7) is bounded from above by , it follows that
When and , it is easy to obtain that
Then, by Lemma 5, we can obtain that
where . So it follows that is uniformly bounded on . The proof is completed.

Lemma 7 (see [31]). *Let one consider the following impulsive control subsystem:
**Suppose . Then one has the following results. *(1)*If , then the trivial periodic solution of system (11) is locally asymptotically stable. *(2)*If , then the system (11) has a unique positive periodic solution , which is globally asymptotically stable, where
*

*Remark 8. *From Lemma 7, we have if , then for all ; if , , then the periodic solution can be rewritten in the form

Lemma 9. * Let one consider the following impulsive control subsystem:
**
where is a -periodic function. are the positive real constants and . Then system (14) has a unique T-periodic solution , and for each solution of (14), as , where
*

*Proof. *First, it is easy to obtain that
Since the *T*-periodicity requirement, we have
By (17), we can obtain that
So, we will obtain the *T*-periodic solution of (14):
Let . Substituting into (14), we have
Then, , as . The proof is completed.

#### 4. Main Results

##### 4.1. Local and Global Stability

In this section, we will study the existence and stability of the system (1) susceptible pest-eradication periodic solution . To this purpose, it is seen first that when , system (1) can be rewritten in the form
which describes the dynamics of system (1) in the absence of the susceptible pest population. So, when , we can calculate the *T*-periodic solution of (21) by Lemmas 7 and 9. It is seen that
where
To discuss the locally asymptotical stability of the susceptible pest-eradication periodic solution, we now introduce the Floquet theory for a linear impulsive control system:
under the following conditions: *H*1: and for . *H*2:, , for , and denotes the real identity matrix. *H*3: There is a such that , for .

Let be a fundamental matrix of (29), then there is a unique reversible matrix such that for all , which is called the monodromy matrix of (29) corresponding to . All monodromy matrices of (29) are similar and they have the same eigenvalues , which are called the Floquet multipliers of (29).

Lemma 10 (see [13] (Floquet theory)). *Let the conditions H1–H3 hold. Then system (29) have the following properties *(1)

*stable if and only if all Floquet multipliers of (29) satisfy and moreover, to those for which , there correspond simple elementary divisors;*(2)

*asymptotically stable if and only if all Floquet multipliers of (29) satisfy ;*(3)

*unstable if there is a Floquet multipliers of (29) such that .*

In the following, we present two main results with the locally and globally asymptotical stability of the susceptible pest-eradication periodic solution , , , , .

Theorem 11. *If
**
then the susceptible pest-eradication periodic solution of system (1) is locally asymptotically stable. *

*Proof. * Let be any solution of system (1). We define error . The linearized system of (1) at is
Let be the fundamental matrix of (31), then satisfies Then, a fundamental matrix of (31) iswhere
The resetting impulsive condition of (31) becomes
Then, it is easy to obtain all eigenvalues of
We have , , , and . Since , so . By the condition (30), we have . Therefore, according to Lemma 10, the susceptible pest-eradication periodic solution of system (1) is locally asymptotically stable. The proof is completed.

Theorem 12. * If
**
where is an ultimate boundedness constant for S, then the susceptible pest-eradication periodic solution of system (1) is globally asymptotically stable. *

*Proof. *Since
we can choose an small enough such that
where is defined in the following. According to system (1), we have
From the first equation of system (40), we obtain the following comparison system:
By Lemma 7, system (41) has a positive periodic solution , and for any solution of (41), as large enough, where . Then, according to Lemmas 4 and 7, there exists a positive constant such that for all
Let us define and , where , . Then, we have
Then, the multicomparison system of (43) is
where , and .

By Lemma 9, it is easy to obtain a periodic solution of system (45). Then, according to Lemmas 4 and 9, one may find such that for all
From the fourth equation of system (40), we have , by Lemmas 4 and 9, there exists such that for all
where
Therefore,
for . Let and . Integrating (50) on , we have
Then for . Since , we can easily get , as . In the following, we prove , as . Give small enough , there must exist such that , for . Then, we have
Analyzing (52) with similarity as (40), there exists such that for all
where
Letting , we have . Together with (42), (46), (47), and (53), we get as . Therefore, the susceptible pest-eradication periodic solution , , , , is globally attractive. Since (37) implies (30), it follows from Theorem 11 that , ,