Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 767526, 10 pages

http://dx.doi.org/10.1155/2013/767526

Research Article

Periodicity and Permanence of a Discrete Impulsive Lotka-Volterra Predator-Prey Model Concerning Integrated Pest Management

Chang Tan1,2 and Jun Cao³

¹College of Science, Northeast Forestry University, Harbin 150040, China

²Forestry Engineering Mobile Station, Northeast Forestry University, Harbin 150040, China

³College of Mechanical and Electrical Engineering, Northeast Forestry University, Harbin 150040, China

Received 18 September 2013; Revised 23 November 2013; Accepted 25 November 2013

Academic Editor: Stefan Balint

Copyright © 2013 Chang Tan and Jun Cao. 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.

By piecewise Euler method, a discrete Lotka-Volterra predator-prey model with impulsive effect at fixed moment is proposed and investigated. By using Floquets theorem, we show that a globally asymptotically stable pest-eradication periodic solution exists when the impulsive period is less than some critical value. Further, we prove that the discrete system is permanence if the impulsive period is larger than some critical value. Finally, some numerical experiments are given.

1. Introduction

Impulsive equations are found in almost every domain of applied science, such as population dynamics, ecology, biological systems, and optimal control. In recent years, the theory of impulsive differential equations has been an object of active research (see [1–4] and reference therein) since it is much richer than the corresponding theory of differential equations without impulsive effects.

It is well known that continuous-time dynamic systems play an important role in control theory, population dynamics, and so on. But in applications of continuous-time dynamic systems to some practical problems, such as computer simulation, experimental, or computational purposes, it is usual to formulate a discrete-time system which is a version of the continuous-time system. In some sense, the discrete time model inherits the dynamical characteristics of the continuous-time systems. We refer to [4–16] for related discussions of the importance and the need for discrete-time analogs to reflect the dynamics of their continuous-time counterparts. Nevertheless, the discrete-time version can but not always preserve the dynamics of its initial version because the theory of difference equations is a lot richer than the corresponding theory of differential equations as pointed out in [17, 18]. Therefore, it is important to study the dynamics of its initial version alone.

Due to the above facts, we construct the following discrete impulsive Lotka-Volterra predator-prey model concerning integrated pest management by piecewise Euler method:

where is the intrinsic growth rate of pest, is the coefficient of intraspecific competition, is the per-capita rate of predation of the predator, is the death rate of predator, denotes the product of the per-capita rate of predation and the rate of conversing pest into predator, and is the period of the impulsive effect. represents the fraction of pest (predator) which dies due to the pesticide, and is the release amount of predator at , . That is, we can use a combination of biological (periodic releasing natural enemies) and chemical (spraying pesticide) tactics that eradicates the pest to extinction and show the efficiency of integrated pest management strategy.

System (1) can be regarded as a discrete analogy of the following impulsive Lotka-Volterra predator-prey model concerning integrated pest management:

where , . Liu et al. [19] discussed the dynamical behavior of model (2).

Recently, the studies of discrete impulsive model have received great attention from more scholar (see [5, 15, 16, 20–22]). The main difficulty for dynamical analysis of such equations comes from impulsive effect on the equations since the corresponding theory for impulsive difference equations have not yet been fully developed. The discrete impulsive model (1) gives a new form of describing the impulsive moment. In some papers, authors use to denote impulsive moment (see [20]). It is obvious that describing the impulsive moment of model (1) is easily realized at computer. In addition, some authors use to denote impulsive moment (see [16, 21]). Compared with it, model (1) is a better analogue of the continuous-time dynamic system.

The main aim of this paper is to construct the discrete impulsive model (1) and discuss the dynamical behaviors of the discrete impulsive model (1). We investigate the globally asymptotical stability of pest-eradication periodic solution system (1) and the permanence of system (1).

2. Global Qualitative Analysis for Model (1)

Before our main results, we will give some lemmas which will be useful for our main results. First, we present the Floquent theory for the linear -periodic difference equation

where , . is a real matrix whose entries are functions of satisfying for a positive integer and is nonsingular; that is, det for all  . As usual denotes the monodromy of (3).

Lemma 1 (see [5, 6]). Let be an nonsingular matrix and let be any positive integer. Then there exists some matrix , such that .

Lemma 2 (see [5, 6]). If we write , then there exists a change of variable , with for all , such that (3) becomes

Lemma 3 (see [5]). Let be a solution of the following impulsive inequality:

and let be a solution of the following impulsive inequality

Then , .

Proof. First, we know that . By induction, we assume that , , .

There are two cases.

Case  1. Considering , or , , we have

Case  2. Considering ,  , we have

Therefore, for both cases we have . The proof is complete.

Lemma 4. Suppose is a solution of (1) subject to , ; then for all , , .

2.1. Global Stability of the Pest-Eradication Periodic Solution

Consider the following system:

Clearly, ,  , with , is a positive periodic solution of (9). Since the solution of (9) is , , , we have the following lemma.

Lemma 5. Equation (9) has a positive periodic solution , , , and, for every solution of (9), we have as .

Therefore, system (1) has a pest-eradication periodic solution:

and .

Now we give the conditions which assure the globally stability of the pest-eradication periodic solution .

Theorem 6. Let be any solution of (1); then is globally asymptotically stable provided

Proof. Firstly, we proved the local stability. The local stability of periodic solution may be determined by considering the behavior of small amplitude perturbations of the solution. Defining , , there may be written

Equation (12) can be expanded in a Taylor series: after neglecting higher-order terms, the linearized equations read as

Hence

the fundamental solution matrix is

There is no need to calculate the exact form of as it is not required in the analysis that follows.

The stability of the periodic solution is determined by the eigenvalues of .

Let be eigenvalues of matrix . Then

according to Lemmas 1 and 2, is locally stable if and . Obviously, . So, is locally stable if . That is,

In the following we prove the global attractivity. Choose such that

Noting that , consider the following impulsive equation:

By Lemmas 3 and 5, we have

for large enough, so

which leads to

Therefore, and as . Moreover, , so we have as .

Next, we prove that as if . For , there exists an such that for all , and ; then from (1) we have

By Lemmas 3 and 5, we obtain

where and are solutions of

respectively, and ,  .

Therefore, for any there exists a such that

Let ; we have

which implies as . This completes the proof.

2.2. Permanence

Now we investigate the permanence of system (1). For convenience, let . Firstly, we give the following definition.

Definition 7. System (1) is said to be permanent if there are constants (independent of initial value) and a finite time such that for all solutions with all initial values ,

hold for all . Here may depend on the initial values .

Lemma 8. There exist two positive constants and , such that for every solution of system (1), we have

Proof. To prove (30), we have two cases.

Case I. , . For any , we have

here we used

Case II. , , we have

This completes the proof.

In the following, without loss of generality, we assume is large enough.

Lemma 9. There exists a constant such that, for every solution of (1), we have

Proof. We first prove that there exists a such that

There exist , , such that

where , . Now we claim that (35) holds. Otherwise, there would exist and , such that for . By Lemma 8, there exists an such that

Then if

There are two cases as follows.

(1)If ,

 We claim that . Otherwise, , so

 hence

 which is a contradiction.(2)If ,

Therefore, for both cases, we have , so , which is a contradiction.

For given by (37), in the following we will prove that (34) holds.

Otherwise,

By (35) and (44), there exist

such that

(i)If are at the same region , then

 so

(ii)If are not at the same region, we assume and . Then

So

Therefore, the set is nonempty. So if , then

If , then

There are three case as follows.

(1)If , and ,

(2)If and ,

(3)Other cases are as follows:

 which is a contradiction. This completes the proof.

Theorem 10. Equation (1) is permanence provided that

holds true.

Proof. Suppose is a solution of (1) with . By Lemmas 8 and 9, we have proved there exists a constant such that for large enough.

From (20), we know for all large enough and some , so for large enough.

Thus, we only need to find such that for large enough.

We will do it in the following two steps.

Step I. From condition (56), let , be small enough such that ; we will prove that cannot hold for all . Otherwise,

So we have and ,  , where is the solution of

and ,  , with , is a positive periodic solution of (58).

Therefore, there exists a such that

for . Therefore there exists a , such that if

then as , which is a contradiction to the boundedness of . Hence there exists a such that .

Step II. If for all , then our aim is achieved. Otherwise, for some . Set . We have for .

Assume that . It is easy to see that , .

There are two possible cases for . Here, for large enough, we have and .

Case (a). There exists a , such that . Let ; then for and .

For ,

If , then

If , then .

Case (1). Consider ,

Case (2). Consider ,

Let . So we have for . For the same arguments can be continued since .

Case (b). There does not exist a , such that ; namely, for .

Choose such that

Let . We claim that there must be a , such that . Otherwise, , . Consider (58) with ; we have

for and . Then for ,

which implies that (60) holds for .

So as in Step I, we have

Since , then for all .

So,

If , then

If , then

So,

which is a contradiction.

Let . Then , . For , (70) holds. Suppose , .

We have

If , then

If , then

Let . So for . For , the same arguments can be continued since .

The proof is completed.

Remark 11. Let

Since , as and

so has a unique positive root, denoted by . From Theorem 6 we know that the pest-eradication periodic solution is globally stable when . From Theorem 10, system (1) is permanent if .

3. Numerical Experiments

In Figure 1, we choose parameters of system (1) as , , , , , , , , , and initial value , . It is easy to verify that condition (11) holds, and, by Theorem 6, the pest-eradication periodic solution of (1) is the global stability.

(a) Time-series of the pest population

(b) Time-series of the predator population

Figure 1: Dynamical behavior of the pest-predator system of (1).

In Figure 2, we choose parameters of system (1) as , , , , , , , , , and initial value , . It is easy to verify that condition (56) holds, and, by Theorem 10, (1) is permanence.

(a) Time-series of the pest population

(b) Time-series of the predator population

Figure 2: Dynamical behavior of the pest-predator system of (1).

4. Conclusion

In this paper, by piecewise Euler method, we construct a discrete impulsive Lotka-Volterra predator-prey model concerning integrated pest management. The discrete impulsive model gives a new form of describing the impulsive moment. On the other hand, model (1) is a better analogue of the continuous-time impulsive dynamic system. By using Floquets theorem, we show that a globally asymptotically stable pest-eradication periodic solution exists when the impulsive period is less than some critical value and the discrete system is permanence if the impulsive period is larger than some critical value. By (78), the impulsive period critical value can be obtained. Since is a direct function with respect to , , and , in order to obtain the object of integrated pest management, we can determine the impulsive period according to effect of the chemical pesticides on the populations and cost of the releasing natural enemies.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities (DL12BB23) and the China Postdoctoral Science Foundation (415220).

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