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Abstract and Applied Analysis
Volume 2015, Article ID 183526, 8 pages
http://dx.doi.org/10.1155/2015/183526
Research Article

The Dynamics of an Impulsive Predator-Prey System with Stage Structure and Holling Type III Functional Response

1School of Science, Guilin University of Technology, Guilin, Guangxi 541004, China
2Institute of Information Science and Technology, Guilin University of Technology, Guilin, Guangxi 541004, China

Received 7 May 2014; Revised 12 September 2014; Accepted 14 September 2014

Academic Editor: Yonghui Xia

Copyright © 2015 Zhixiang Ju 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

Based on the biological resource management of natural resources, a stage-structured predator-prey model with Holling type III functional response, birth pulse, and impulsive harvesting at different moments is proposed in this paper. By applying comparison theorem and some analysis techniques, the global attractivity of predator-extinction periodic solution and the permanence of this system are studied. At last, examples and numerical simulations are given to verify the validity of the main results.

1. Introduction

In recent decades, with the increase of population and the development of science and technology, human accelerated the exploitation of natural resources. Many harvesting predator-prey models have been studied. The dynamic relationship between predators and their preys has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance; see [16] and the references cited therein. On the other hand, in the real world, many species usually go through two or even more life stages as they proceed from birth to death. Thus, it is practical to introduce the stage structure into predator-prey models; see [712]. For example, Wei and Wang [13] considered the following predator-prey system with stage structure:where ,  , and denote the densities of prey population and immature and mature individual predators at time , respectively. The meanings of all parameters may refer to [13]. Authors obtained the sufficient conditions of the persistence for system (1).

However, it is well known that many evolution processes are characterized by the fact that at certain moments their stage changes abruptly. For example, for IPM strategy on ecosystem, the predators are released periodically every time , and periodic catching or spraying pesticide is also applied. Hence, the predator and prey experience a change of state abruptly. Consequently, it is natural to assume that these processes act in the form of impulse. Impulsive methods have been applied in almost every field of applied sciences. For example, many population models assume that the populations are born throughout the year, whereas it is often the case that many species give birth only during a single period of the year; that is, births occur in regular pulses. Hence, the authors Z. Xiang, D. Long, and X. Y. Song gave a single population logistic model with birth pulse and impulsive harvesting at different moments as follows:where represents the density of the resource population at time . Parameter is the intrinsic growth rate, the positive constant is referred to as the environmental carrying capacity, and parameter is the death rate of resource population. Parameter denotes the harvest rate of resource population. For more details of the biological meaning of system (2), we can refer to [1417].

In addition, in the description of dynamical interactions on predators and their preys, a crucial element of all models is the classic definition of a predator’s functional response. A functional response of the predator to the prey density refers to the change in the density of prey attached per unit time per predator as the prey density changes. For example, Zhang et al. [18] suggested a functioncalled Holling type III. It is monotonic in the first quadrant, that is, if the prey population increases, then the consumption rate of prey per predator will increase too. And is the half-saturation constant. The field of research on the dynamics of impulsive predator-prey model with functional response seems to be a new increasingly interesting area, which draws many scholars’ attention.

Moreover, by the picture of ocean’s food-chain, we know that small fish can prey on fish larvae as a predator; also it can be eaten by the higher predator as a prey (see [19]). Hence, according to the nature of biological resource management, it is interesting to investigate impulsive harvesting on prey and mature predator (e.g., the small fish and higher predator in the ocean’s food-chain) simultaneously at some fixed time.

Based on the above discussion, we consider the stage-structured predator-prey model with Holling type III functional response, birth pulse, and impulsive harvesting at different moments as follows:where denotes the density of the prey and ,   represent the immature and mature predator densities, respectively. Parameters , , , , , , , and are positive constants, where is the intrinsic growth rate of the prey, denotes the capacity rate, concerned with the maintaining of the evolution of the population, represents the predation rate of predator, is the conversion rate that translated into predator population increase, , , and denote the death rate of prey, immature predator, and mature predator, respectively, and is the intrinsic-specific competition rate of the mature predator. Parameter represents a constant time from immaturity to maturity. Parameters , denote the harvesting rates of prey and mature predator at ,  , respectively, , and is the period of the impulsive effect.

It is well known that, in the sustainable development of natural resources, it is very important to study the sustainable survival of species. So, in this paper, we aim to investigate the global attractivity of predator-extinction periodic solution and the permanence of system (4). From the biological point of view, we only consider (4) in the biological meaning region

This paper is organized as follows. Firstly, some preliminaries are given in Section 2. In Section 3, the sufficient conditions for the global attractivity of predator-extinction periodic solution are obtained. The permanence of system (4) is investigated in Section 4. In Section 5, we present some examples and simulations to illustrate our results. At last, a brief conclusion is given in Section 6.

2. Preliminaries

In this section, some definitions and lemmas are introduced.

Let and  . Denote to be the map defined by the right hand of system (4). Let ; then is said to belong to class , if(i) is continuous in and for each , , and exists,(ii) is locally Lipschitzian in .

Definition 1. Let , and and , the upper right derivative of with respect to the impulsive differential system (4) is defined asThe solution of (4), denote by , is a piecewise continuous function , is continuous on andObviously the existence and uniqueness of solutions of (4) is guaranteed by the smoothness properties of , which denotes the map defined by the right hand of system (4).

The following lemmas are useful for the proof of the main results.

Lemma 2 (see [5, 7]). Consider the following differential equation:where , , , and are positive constants and for . We have the following:(i)if , then ;(ii)if , then .

Lemma 3 (see [14]). Consider the following system:Then, system (9) has a positive periodic solution with period , which is globally stable, whereif .

Lemma 4. There is a positive constant M such that ,  , and   for every solution of system (4) with all sufficiently large, where is a positive constant defined in system (4).

Proof. Define .
If , , we let ; thenIf , we haveand if , it is clear that .
Then, by Lemma 2.2 of [20], for all , we haveTherefore, there exists a positive constantsuch that , , and for large enough. This completes the proof.

3. Global Attractivity of a Predator-Extinction Periodic Solution

In this section, we will demonstrate the existence and global attractivity of a predator-extinction periodic solution, in which the predator individuals are entirely absent from the population permanently; that is, and   for all .

Firstly, by Lemma 3, we can easily obtain the existence of predator-extinction periodic solution for system (4).

Theorem 5. System (4) has a predator-extinction periodic solution which is globally stable; that is, for and any solution of system (4), we have as , whereand .
Next, we give the conditions on the global attractivity of the predator-extinction periodic solution of the system (4).

Theorem 6. The predator-extinction periodic solution of system (4) is globally attractive, if(),().

Proof. Let be any solution of system (4); from the first, the fourth, and the seventh equations of system (4), we haveConsider the auxiliary system of (16) as follows:By Lemma 3 and condition (A1), we havewhich is unique and positive periodic solution of system (17) and is globally attractive. Applying comparison theorem of impulsive differential equation [21], there exist and a sufficiently small constant such thatfor .
From inequality (19), we can obtainConsider the auxiliary system of (20):Applying Lemma 2, (), and comparison theorem, we have as . Then, for any small constant , there exists , such thatBy the second equation of system (4) and (22), we haveConsider the auxiliary system of (23) as follows:Integrating (24), one can easily get as . Then, for any small constant , there exists such thatAccording to the first, the fourth, and the seventh equations of (4), we havewhere . Again, consider the auxiliary system of (26):In view of Lemma 3, system (27) has a unique positive periodic solution as follows:Similarly, for any arbitrary small constant , there exists such that . Let ; then, , andIt follows from (29) and (19) that as . Let ; then, we have , as sufficiently large enough. The proof is complete.

4. Permanence of System (4)

In the real world, from the principle of ecosystem balance and saving resources, we only need to control the predator under the economic threshold level and not to eradicate the predator totally. Thus, we focus on the permanence of system (4).

First, we give the definition of permanence.

Definition 7. System (4) is said to be persistent if there exist positive constants and such that every positive solution of system (4) satisfies for sufficiently large enough.

Theorem 8. System (4) is permanent, if the following conditions hold:,,where , , and are defined in (14), (34), and (39), respectively.

Proof. It is obvious that the third equation of system (4) can be rewritten as follows:Define .
By computation, we haveApplying Lemma 4, we haveBy hypothesis , for the arbitrary small constant , we haveLet be determined by the following equation:Then, for any , it is impossible that for all . Suppose that the claim is invalid; then, there exists such that for all . It follows from the first equation of system (4) thatfor all . Consider the auxiliary system of (35) as follows:By hypothesis , we can obtain from Lemma 3 thatwhich is the unique positive periodic solution of system (36) and is globally asymptotically stable. By the comparison theorem of impulsive differential equation, we know there exists such that .
On the other hand, for all , we haveThus, By (33), we have Hence, by (32) and (40), we haveLet . We will show that for all . Otherwise, there exists a constant such that for , , and . However, by the third equation of system (4) and (40), we havewhich is a contradiction. Thus, we have for all .
By (40) and (41), therefore,which means that as . It is a contradiction with . Therefore, for any , the inequality cannot hold for all . So, there exist the following two cases: (i)if holds for all large enough, then our goal is obtained;(ii)if is oscillatory about , letWe prove that for all sufficiently large. Suppose that there exist positive constants such that and for all , and inequality (40) holds true for , where is sufficiently large enough. Since is continuous and bounded and is 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 . If , our aim is obtained. If , from the third equation of (4), we have that, for ,  . According to the assumptions and for , we have for . Then, we derive that . It is clear that for . If , then we have for . The same arguments can be continued. We can obtain for . Since the interval is arbitrarily chosen, we get that for sufficiently large. In view of our arguments above, the choice of is independent of the positive solution of (4) which satisfies that for large enough.
Next, from the second equation of system (4), we haveIntegrating (45) and using comparison theorem, we can easily getOn the other hand, by (29), we have . Let ; then, we have , , . By Lemma 4 and above discussion, system (4) is permanent. This completes the proof.

5. Examples and Numerical Simulations

In this section, we give some examples and numerical simulations to show the effectiveness of the main results.

In system (4), we let , , , , , , , , , , , , . By computation, one can obtain that the conditions of Theorem 6 are satisfied, so the predator-extinction periodic solution is globally attractive, which can be shown by Figure 1. If , , , , , , , , , , , , , then by computation, the conditions of Theorem 8 are also satisfied; hence, by Theorem 8, system (4) is permanent; see Figure 2.

Figure 1: Dynamical behavior of system (4) with , , , , , , , , , , , , . (a) Time series of the prey population; (b) time series of the immature predator population; (c) time series of the mature predator population.
Figure 2: Dynamical behavior of system (4) with , , , , , , , , , , , , . (a) Time series of the prey population; (b) time series of the immature predator population; (c) time series of the mature predator population.

6. Conclusion

In this paper, a stage-structured predator-prey model with Holling type III functional response, birth pulse, and impulsive harvesting at different moments is proposed. By using comparison theorem of impulsive differential equations and some analysis techniques, the sufficient conditions ensuring the predator-extinction periodic solution and the permanence of system (4) are obtained. Theorem 6 implies that reducing and appropriately and can be propitious to the global attractivity of the predator-extinction periodic solution . That is, if the mature predator is caught excessively or the immature predator is stocked too few, it will lose the merits of exploitative mature predator population. Similarly, we can see from Theorem 8 that reducing and appropriately and are in favor of the permanence of system (4). It implies that reasonable harvesting on mature predator and appropriate protecting on immature predator play an important role in the permanence of system (4). Moreover, by Theorems 6 and 8, we believe that there exists a sharp threshold, which is beneficial for people to make the best use of biological resources but will not break the biological balance. It will be interesting for us to continue to study the optimal harvesting policy of system (4) in the near future.

Conflict of Interests

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

Acknowledgments

This paper is supported by National Natural Science Foundation of China (11161015, 11361012) and Natural Science Foundation of Guangxi (2013GXNSFAA019003) and partially supported by the National High Technology Research and Development Program 863 under Grant no. 2013AA12A402.

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