Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 380492, 13 pages
http://dx.doi.org/10.1155/2015/380492

Research Article

An Impulsively Controlled Three-Species Prey-Predator Model with Stage Structure and Birth Pulse for Predator

Yanyan Hu,1,2 Mei Yan,² and Zhongyi Xiang1,2

¹Key Laboratory of Biologic Resources Protection and Utilization of Hubei Province, Enshi, Hubei 445000, China
²Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei 445000, China

Received 22 April 2015; Accepted 21 June 2015

Academic Editor: Alicia Cordero

Copyright © 2015 Yanyan Hu et al. 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 investigate the dynamic behaviors of a two-prey one-predator system with stage structure and birth pulse for predator. By using the Floquet theory of linear periodic impulsive equation and small amplitude perturbation method, we show that there exists a globally asymptotically stable two-prey eradication periodic solution when the impulsive period is less than some critical value. Further, we study the permanence of the investigated model. Our results provide valuable strategy for biological economics management. Numerical analysis is also inserted to illustrate the results.

1. Introduction

In the natural world, the predator-prey relationship is one of the important interactions among species, and it has been extensively studied by many authors [1–7] because of its universal existence. The following two-prey, one-predator model was studied by many works [8–10]:

where and are densities of the prey population and the predator population, respectively. are coefficients of decrease of prey due to predation; are intrinsic rates of increase or decrease. , are parameters representing competitive effects between two prey species; is an equal transformation rate of the predator.

As we know, life history often occurs in natural ecological environments which has significant morphological and behavioral differences between immature and mature species; the dynamics of stage-structured prey-predator system has been widely studied [11–13]. Recently, many impulsive differential equation models or hybrid dynamical systems have been proposed to model the introduction of a periodic IPM strategy [14, 15]; Xiang and Song [16] proposed impulsive prey-dependent consumption two-prey one-predator models with stage structure for the predator, which combined the biological control and chemical control:

The biological meanings of the parameters in (2) can be seen in [16]. As far as the population dynamics is concerned, most models often considered that the population reproduces throughout all year. However, many species give birth seasonally or in regular pules. Particular examples of birth pulse type are The model discussed in [17] considered the birth pulse , . A birth pulse type with and are assumed in [18]. Comparatively, we consider birth pulse

In view of birth pulse and impulsive control strategy, we formulate the following two prey-predator models with stage-structure and birth pulse for predator:

where are densities of the prey population and and represent the densities of the immature and the mature predator population, respectively. are coefficients of decrease of prey due to predation, , are parameters representing competitive effects between two prey species, and represent the rate of conversion of nutrients into the predators. The maturity is , which determines the mean length of the juvenile period; is the natural death rate of the predator population. is birth pulse at as intrinsic rate of natural increase and density dependence rate of predator population are denoted by , , respectively. is the carrying capacity of the predator population. , represent the fraction of prey and predators which die due to the pesticide at ,

The organization of this paper is as follows. In the next section, some important lemmas are presented. Sections 3 and 4 give the global asymptotical stability of the prey eradication periodic solution and permanence for system (3). Numerical analysis is displayed in Section 5. Finally, a brief discussion is given to conclude this work.

2. Preliminaries

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

Let , . Thus ; let be the set of all nonnegative integers. Denote , the map defined by the right hand side of the first four equations of system (3). Let ; then is said to belong to class if

(i) is continuous in and for each, , , and existing;(ii) is locally Lipschitzian in .

Definition 1. Consider ; then, for and , the upper right derivative of with respect to the impulsive differential system (3) is defined as .
The solution of system (3), denoted by , is continuously differentiable on and , ,
Obviously the smoothness properties of guarantee the global existence and uniqueness of solutions of system (3); for details see [19]. The following lemma is obvious.

Lemma 2. Suppose is a solution of (3) with ; then for all . And further , , if .
Make a notation as

If , , then system (3) can be rewriten to

It is easy to obtain the analytic solution of system (5) between pulses:

Considering (5), we have the stroboscopic map of system

For convenience, we choose , , and ; the following two equivalence relations are calculated to be

The two fixed points of (5) are obtained as and , where

Lemma 3. (i) If , the fixed point is globally asymptotically stable.
(ii) If , the fixed point is globally asymptotically stable.

Proof. For convenience, we make a notation as Linear form (5) can be written as

Obviously, the near dynamics of and are determined by linear system (10). The stabiles of and are determined by the eigenvalue of less than 1. If satisfies the Jury criteria [20], we can know that the eigenvalue of is less than 1:

where

(i) If , then is the unique fixed point of system (5); we have

Calculating , and from the Jury criteria is locally stable, and then it is globally asymptotically stable.
(ii) If , then is unstable. For exists, we have

Consider . From the Jury criteria is locally stable, and then it is globally asymptotically stable. The proof is completed.

Summarizing the above results, we have the following theorems.

Theorem 4. The triviality periodic solution is stable if ; otherwise, if , the triviality periodic solution is unstable, and the periodic solution is globally asymptotically stable.
Correspondingly, system (5) has a globally asymptotically stable positive periodic solution , where

Lemma 5. There exists a constant such that , , for each positive solution of (5) with all large enough.

Proof. Define a function such that , where When ,  , we have

When ,

and when ,

It follows from the comparison theorem of impulsive differential equations (see Lemma [21], page 23) that, for , we have

So is ultimately bounded. Hence, by the definition of , there exists a constant , such that ,  ,  , for large enough. The proof is complete.

Definition 6. System (3) is said to be permanent if there are constants and a finite time such that all solutions with all initial values , , , and hold for all .

3. Extinction

In this section, we investigate the stability of the two-pest prey eradication periodic solution as a solution of system (3). We give the condition which assures the asymptotical stability of the prey eradication periodic solution .

Theorem 7. Let be any solution of (3); then is globally asymptotically stable if

hold true.

Proof. Firstly, we prove the local stability of -period solution which may be determined by considering the behavior of small-amplitude perturbations of the solution.
Define

where , , , and are small perturbations; they may be written as

where satisfy

with , where is the identity matrix.
Hence the fundamental solution matrix is

where and . There is no need to calculate the exact form of as it is not required in the analysis that follows. The resetting impulsive conditions of (3) become

Hence, the stability of the periodic solution is determined by the eigenvalues of

which are

According to the Floquet theory (see [21]), the two-pest eradication solution is locally asymptotically stable.
In the following, we prove the global attractivity. Choose such that

Noting that , , consider the following impulsive differential equation:

It follows from impulsive differential equation (30) that , and , as This is

for all large enough. For convenience, we may assume that (31) holds. From (3) and (31) we have

holding for all large enough. Integrate (32) on , which yields Thus and as . Therefore as , since for . By the same method we can prove , as , so we omit it.
Next, we will prove that , as If and , for , there exists such that and , for all ; then we have

From the left hand inequality of (33), it follows from impulsive differential equation (5) that and as For the right hand inequality, we consider the following impulsive differential equation:

System (34) has a globally asymptotically stable positive periodic solution , where

where

and , , and
Set

where and
Therefore, there exists such that

for large enough; let ; then we get , . Hence as . Similarly, we can prove as , so we omit it. This proof is complete.

4. Permanence

In this section, we will investigate the permanence of system (3). In biological terms, the permanence implies that preys and predators will coexist, none of them facing extinction or growing indefinitely.

Theorem 8. System (3) is permanent if , and ,

Proof. Let us suppose a solution of (3) with initial value . From Lemma 5, we have proved that there exists a constant such that , , and for . Note that

Consider the following comparison equation:

We have and as ; thus there exists such that for large enough. Without loss of generality, we may assume for . Similarly, we may set . From (31), we know for all large enough ( is enough small), so that for large enough. Thus we only need to find such that for large enough. We will do it in the following two steps.
Step 1. Denote

and We will prove that there exist , such that and . Otherwise there will be three cases:

(i)There exists , such that and for all .(ii)There exists , such that and for all .(iii)Consider , , for all .

Firstly, consider case (i). Let be small enough, so that

where

According to the above assumption, it is easy to get that and Then we consider the following impulsive differential equation:

where

and , , , and

Therefore, there exists such that

Thus we can easily get

for . Set , . Integrating (49) on ,  , then we obtain

consequently as , which contradicts to the boundedness of .
Similarly, cases (ii) and (iii) can be analyzed as in case (i). Here we omit it. From the above three cases, we conclude that there exist ,   such that ,  .
Step 2. If for all , then our aim is obtained. Otherwise, for any . Let ; there are two possible cases for .
Case 1. Consider ,   Then for and . Choose such that

Set ; we claim that there must exist such that . In view of (45), with , , we have

And . Then

thus and for , which implies that (49) holds for . So as in step , we get

From system (3), we get

Integrating (55) on , we have

Therefore, we deduce that

which is a contradiction.
Let ; then for , and . For , set , , ; from (55), we have

Let , so we have for . For , the same argument can be continued since .
Case 2. Consider , . Then for and . Let , . There are two possible cases for .
If for all , there must be such that . Here, we omit it.
Let ; then for and . For , let ,  ,  ; from (55), we have

Let , so we have for . For , the same argument can be continued since .
If there exists such that , let ; then for and . For , (55) holds; integrating (55) on , we have

Since for , the same argument can be continued. Hence, we have for all . Similarly, we can prove for all . The proof is complete.

From the proof of Theorems 7 and 8, we can derive the following results.

Corollary 9. Let be any solution of (3); then , , are permanent; as provided, , ,

Corollary 10. Let be any solution of (3); then and , , are permanent; as provided, , ,

5. Numerical Analysis

In this section, we carry out numerical simulations of system (3) to illustrate and support our analytical results.

(1) Consider the following choice of parametric values: , , , , , , , , , , , , , , , , , , , , , , and

Correspondingly, and According to Theorem 7, we know that the prey eradication periodic solution of system (3) is globally asymptotically stable for this case (see Figure 1).

(a)

(b)

(c)

(d)

Figure 1: Globally asymptotically stable prey eradication solution of system (3). (a, b) Time series of prey populations , and (c, d) time series of predator populations , with initial values , , , and .

(2) Consider the following set of parameters: , , , , , , , , , , , , , , , , , , , , , , and

Since conditions of Theorem 8 hold for this choice, system (3) is permanent (see Figure 2).

(a)

(b)

(c)

(d)

Figure 2: The permanence of system (3). (a, b) Time series of prey populations , and (c, d) time series of predator populations , with initial values , , , and .

6. Discussion

In this paper, we study an impulsively controlled three-species prey-predator model with stage structure and birth pulse for predator. Using the comparison theorems, we have shown that there exists a globally asymptotically stable two-pest eradication periodic solution when the impulsive period In biological terms, the pest population is eradicated totally and the predator population will tend to a stable level. However, the prey population is eradicated totally, which is not most desired for us from the biological point and saving resources; we hope to control the species under the economic threshold level or drive the target prey population to extinction and let the nontarget pest (or harmless) be permanent. Therefore, we give the sufficient conditions for the permanence of the system. What is more, the sufficient conditions for the extinction of one of the two prey species and permanence of the remaining other preys and predators are given in Corollaries 9 and 10.

Compared to earlier modeling studies on the prey-predator concerning chemical control for pest control at different fixed time, our model considers the predator given birth in regular pules at , To avoid the fact that pesticides impact on the newborn predator, pesticide applications and birth pulse in predator occur at different time in this paper. All these results show that the dynamical behaviors of system (3) become more complex under different time periodically impulsive effect. Our results suggest a new approach in pest control. We are more interested in the following: how different patterns of insecticide applications affect the results, for instance, applying pesticide after predator is born and applying pesticide when predator is born. We also can study the results of two cases and find the better control way. These works will be done in our next consideration.

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 Key Laboratory of Biologic Resources Protection and Utilization of Hubei Province (PKLHB1524), the innovation projects for undergraduates of Hubei province (201310517014), the soft science research project of Hubei province (2012GDA01309), and key discipline of Hubei province, Forestry.

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