#### Abstract

According to biological strategy for pest control, a mathematical model with periodic releasing virus particles for insect viruses attacking pests is considered. By using Floquet's theorem, small-amplitude perturbation skills and comparison theorem, we prove that all solutions of the system are uniformly ultimately bounded and there exists a globally asymptotically stable pest-eradication periodic solution when the amount of virus particles released is larger than some critical value. When the amount of virus particles released is less than some critical value, the system is shown to be permanent, which implies that the trivial pest-eradication solution loses its stability. Further, the mathematical results are also confirmed by means of numerical simulation.

#### 1. Introduction and Model Formulation

Currently, applications of chemical pesticides to combat pests are still one of the main measures to improve crop yields. Though chemical crop protection plays an important role in modern agricultural practices, it is still viewed as a profit-induced poisoning of the environment. The nondegradable chemical residues, which accumulate to harmful levels, are the root cause of health and environmental hazards and deserve most of the present hostility toward them. Moreover, synthetic pesticides often disrupt the balanced insect communities. This leads to the interest in Biological control methods for insects and plant pests [1, 2].

Biological control is, generally, man's use of a suitably chosen living organism, referred as the biocontrol agent, to control another. Biocontrol agents can be predators, pathogens, or parasites of the organism to be controlled that either kill the harmful organism or interfere with its biological processes [3]. In a large number of biopesticides, the insect virus pesticide because of its high pathogenicity, specificity, and ease production plays an important role in pest biological control. The control of rabbit pests in Australia by the virus disease called “myxomatosis” provides a spectacular example of a virus controlling pest [4]. The insect viruses for the biological control of pests are mainly baculoviruses. Baculoviruses comprise a family of double-stranded DNA viruses which are pathogenic for arthropods, mainly insects. The polyhedral occlusion body (OB) is the characteristic phenotypic appearance of baculoviruses and in case of a nucleopolyhedrovirus (NPV) typically comprised of a proteinaceous matrix with a large number of embedded virus particles. Baculoviruses have a long history as effective and environmentally benign insect control agents in field crops, vegetables, forests, and pastures [5].

Transmission is also key to the persistence of baculoviruses in the environment [6, 7]. Transmission occurs primarily when an NPV-infected larva dies and lyses, releasing a massive number of OBs onto foliage and soil. Susceptible hosts become infected when they ingest OBs while feeding. Defecation and regurgitation by infected larvae have been reported as additional routes of contamination of host plants with viruses [7–9]. Moreover, some studies suggest that cannibalism and predation may also be routes of virus transmission [10]. Environmental factors such as rainfall, wind transport, and contaminated ovipositors of parasitic hymenopterans could contribute to NPV transmission as well [11, 12].

Insight in the epidemiological dynamics, it is necessary to predict optimal timing, frequency, and dosage of virus application and to assess the short and longer term persistence of NPV in insect populations and the environment. Modeling studies can help to obtain preliminary assessments of expected ecological dynamics at the short and longer term. There is a vast amount of literature on the applications of microbial disease to suppress pests [13–15], but there are only a few papers on mathematical models of the dynamics of viral infection in pest control [16, 17]. System with impulsive effects describing evolution processes is characterized by the fact that at certain moments of time they abruptly experience a change of state. Processes of such type are studied in almost every domain of applied science. Impulsive differential equations have been recently used in population dynamics in relation to impulsive vaccination, population ecology, the chemotherapeutic treatment of disease, and the theory of the chemostat [18–22].

In this paper, according to the above description, we should construct a more realistic model by introducing additional virus particles (i.e., using viral pesticide) to investigate the dynamical behavior of viruses attacking pests, which is described as follows: where , and denote the density of susceptible pests, infected pests, and virus particles at time , respectively. is the impulsive period, , is the release amount of virus particles, , and . The assumptions in the model are as follows.

We assume that only susceptible pests are capable of reproducing with logistic law; that is, the infected pests are removed by lysis before having the possibility of reproducing. However, they still contribute with to population growth toward the carrying capacity. is intrinsic birth rate and is carrying capacity.The term denotes that the susceptible pests become infected as they ingest foods contaminated with virus particles, in which is positive constant and represents the “effective per pest contact rate with viruses.” And denotes that the susceptible pests become infected by the transmission of infective pests according to other ways; perhaps is very small and close to zero.An infective pest has a latent period, which is the period between the instant of infection and that of lysis, during which the virus reproduces inside the pest. The lysis death rate constant . gives a measure of such a latency period being . The lysis of infected pests, on the average, produces virus particles (). is the virus replication factor.The virus particles have a natural death rate due to all kinds of possible mortality of viruses such as enzymatic attack, pH dependence, temperature changes, UV radiation, and photooxidation.The paper is organized as follows: in Section 2, some auxiliary results which establish the a priori boundedness of the solutions, together with the asymptotic properties of certain reduced systems which are used throughout the paper as a basis of several comparison arguments, are stated. In Section 3, by using Floquet's theory for impulsive differential equations, small-amplitude perturbation methods, and comparison techniques, we provide the sufficient conditions for the local and global stability of the pest-eradication periodic solution and the conditions for the permanence of the system. Finally, a brief discussion and numerical examples are given. We also point out some future research directions.

#### 2. Preliminary

We give some definitions, notations, and lemmas which will be useful for stating and proving our main results. Let . Denote by the map defined by the right hand of the first three equations in system (1.1). Let , then if

(i) is continuous in and for each exists;(ii) is locally Lipschitzian in .*Definition 2.1. *, then for , the upper right derivative of with respect to system (1.1) is defined as
The solution of (1.1), denoted by , is a piecewise continuous function : , is continuous on , and exists. Obviously, the existence and uniqueness of the solution of (1.1) is guaranteed by the smoothness properties of (for more details see [18]).

Lemma 2.2. *Suppose that is a solution of (1.1) with , then for all . Moreover, if , then for all .*

Lemma 2.3. *Let and . Assume that
**
where is continuous in and for each **
exists, and is nondecreasing. Let be the maximal(minimal) solution of the scalar impulsive differential equation
**
existing on . Then implies that
**
where is any solution of (1.1) existing on .**Note that if one has some smoothness conditions of to guarantee the existence and uniqueness of solutions for (2.5), then is exactly the unique solution of (2.5).*

Lemma 2.4. *There exists a constant such that for each positive solution of (1.1) with being large enough.*

*Proof. *Define a function such that . Then we have
Obviously, the right hand of the above equality is bounded; thus, there exists such that . It follows that . Therefore, by the definition of we obtain that there exists a constant such that From the third and sixth equations of system (1.1), we have
According to Lemma 2.3 in [18] we derive
Therefore, there exists a constant such that The proof is complete.

Next, we give some basic property of the following subsystem:

Lemma 2.5. *System (2.10) has a positive periodic solution and for every positive solution of system (2.10), as , where and *

Lemma 2.6. *System (2.11) has a positive periodic solution and for every positive solution of system (2.11), as , where and .*

#### 3. Main Results

When , from the second and fifth equations of system (1.1), we have . Further, from the third and sixth equations of system (1.1), we have

By Lemma (2.10), we can obtain the unique positive periodic solution of system (3.1): , with initial value . Thus the pest-eradication solution is explicitly shown. That is, system (1.1) has a so-called pest-eradication periodic solution . Next, we shall give the condition to assure its global asymptotic stability.

Theorem 3.1. *Let be any solution of system (1.1) with positive initial values. Then the pest-eradication periodic solution is locally asymptotically stable provided that
*

*Proof. *The local stability of periodic solution may be determined by considering the behavior of small amplitude perturbation of the solution. Let . The corresponding linearized system of (1.1) at is
Let be the fundamental matrix of (3.3), then satisfies
and (unit matrix). Hence, the fundamental solution matrix is
where the exact expressions of are omitted, since they are not used subsequently. The resetting impulsive condition of (3.3) becomes
Hence, if all the eigenvalues of
have absolute values less than one, then the periodic solution is locally stable. Since the eigenvalues of are
and if and only if (3.2) holds, according to Floquet's theory of impulsive differential equation, the pest-eradication periodic solution is locally stable.

In fact, for condition (3.2), represents the normalized gain of the pest in a period, while represents the normalized loss of the pest in a period due to viral disease. That is, this condition is a balance condition for the pest near the pest-eradication periodic solution, which asserts the fact that in a vicinity of this solution the pest is depleted faster than they can recover and consequently the pest is condemned to extinction.

Theorem 3.2. *Let be any solution of system (1.1) with positive initial values. Then the pest-eradication periodic solution is globally asymptotically stable provided that
*

*Proof. *From (3.9), we know that (3.2) also holds. By Theorem 3.1, we know that is locally stable. Therefore, we only need to prove its global attractivity. Since , we can choose an small enough such that
From the first equation of system (1.1), we obtain Consider the comparison equation , then we have and as Thus, there exists an such that for being large enough. Without loss of generality, we assume for all .

Note that ; by Lemmas 2.3 and 2.5, there exists a such that for all
where is the solution of
Thus we have
Integrating the above inequality on , yields
Since , we can easily get as . For small enough being (), there must exist an such that for ; then from the second equation of system (1.1), we have , so .

In the following, we prove as From system (1.1), we have
and by Lemmas 2.3, 2.5, and 2.6, there exists an such that
where Let , we have . Therefore, is globally attractive. This completes the proof.

Corollary 3.3. *If or , then the pest-eradication periodic solution is globally asymptotically stable.*

We have proved that, if or , the pest-eradication periodic solution is globally asymptotically stable; that is, the pest population is eradicated totally. But in practice, from the view point of keeping ecosystem balance and preserving biological resources, it is not necessary to eradicate the pest population. Next we focus our attention on the permanence of system (1.1). Before starting our result, we give the definition of permanence.

*Definition 3.4. *System (1.1) is said to be permanent if there are constants (independent of initial value) and a finite time such that all solutions with initial values , hold for all . Here may depend on the initial values

Theorem 3.5. *Let be any positive solution of (1.1) with positive initial values . Then system (1.1) is permanent provided that
*

*Proof. *Suppose that is a solution of system (1.1) with initial values . By Lemma 2.4, there exists positive constants such that , and for being large enough. We may assume for all . From (3.11), we know that
for being large enough. Thus we only need to find such that , for being large enough. We shall do it in two steps.*Step 1. *Since , that is , we can select small enough such that
We shall prove that cannot hold for all . Otherwise, we have that for all . Then from the third equation of system (1.1), we get
Then and as , where is the solution of
Therefore, there exists a such that
for From the second equation of system (1.1), we have Thus, for being large enough. Therefore, there exists such that
for all . Let such that . Integrating the above inequality on , we have
Then as which is a contradiction to the boundedness of Thus, there exists a such that *Step 2. *If for all , then our aim is obtained. Hence we need only to consider the situation that is not always true for , and we denote . Then for and , since is continuous. Suppose . Select such that
where Let . We claim that there must be a such that . Otherwise Consider (3.21) with We have
Thus
Thus, we have
for As in Step 1, we have
On the interval , we have
Thus , which is a contradiction. Let , then for and we have . For , the same arguments can be continued, since , and are -independent. Hence for all . In the following, we shall prove that there exists such that for being large enough. From the third equation of system (1.1), we have
Then and as , where is the solution of
Therefore, there exists a such that
for Then from the second equation of system (1.1), we have
and then we have Therefore for being large enough. The proof is complete.

Corollary 3.6. *If or , then system (1.1) is permanent.*

*Example 3.7. *Let us consider the following system:
According to Corollaries 3.3, and 3.6, we know that if , then is globally asymptotically stable (see Figure 1), and if , then the system is permanent (see Figure 2).

**(a)**

**(b)**

**(c)**

**(a)**

**(b)**

**(c)**

#### 4. Numerical Simulations and Discussion

In this paper, we have investigated the dynamical behavior of a pest management model with periodic releasing virus particles at a fixed time. By using Floquet's theorem, small-amplitude perturbation skills and comparison theorem, we establish the sufficient conditions for the global asymptotical stability of the pest-eradication periodic solution as well as the permanence of the system (1.1). It is clear that the conditions for the global stability and permanence of the system depend on the parameters , which implies that the parameters play a very important role on the model.

From Corollary 3.3, we know that the pest-eradication periodic solution is globally asymptotically stable when or . In order to drive the pests to extinction, we can determine the impulsive release amount such that or the impulsive period such that . If we choose parameters as and , then we have ; so we can make the impulsive period smaller than in order to eradicate the pests (see Figure 1). In the same time , so we can make the impulsive period larger than in order to maintain the system permanent (see Figure 2). Similarly, we can fix and change in order to achieve the same purpose. However, from a pest control point of view, our aim is to keep pests at acceptably low levels, not to eradicate them, only to control their population size. With regard to this, the optimal control strategy in the management of a pest population is to drive the pest population below a given level and to do so in a manner which minimizes the cost of using the control and the time it takes to drive the system to the target. We hope that our results provide an insight to practical pest management. However, in the real world, for the seasonal damages of pests, should we consider impulsive releasing virus particles on a finite interval? Such work will be beneficial to pest management, and it is reasonable. We leave it as a future work.

#### Acknowledgment

This work was supported by National Natural Science Foundation of China (10771179).