- Open Access
An impulsive prey-predator system with stage-structure and Holling II functional response
© Ju et al.; licensee Springer. 2014
Received: 7 May 2014
Accepted: 21 October 2014
Published: 31 October 2014
Taking into account that individual organisms usually go through immature and mature stages, in this paper, we investigate the dynamics of an impulsive prey-predator system with a Holling II functional response and stage-structure. Applying the comparison theorem and some analysis techniques, the sufficient conditions of the global attractivity of a mature predator periodic solution and the permanence are investigated. Examples and numerical simulations are shown to verify the validity of our results.
where , represent the densities of the prey and the predator, respectively. For the parameters , r is the intrinsic growth rate of the prey, is the Holling II function response, b denotes the death rate of the predator, , , , . One obtained the complex dynamics of the system.
However, in the real world, the development of an individual organism usually goes through two stages on the time: immaturity and maturity. Some stage-structured models for the prey-predator system consisting of immature and mature individuals were analyzed in [9–12]. For example, a stage-structured prey-predator model with impulsive stocking on prey and continuous harvesting on predator was considered in . Song and Chen  studied optimal harvesting and stability for a two species competitive system with stage structure. Shao and Dai  considered a predator-prey model with time delay and impulsive harvesting on prey and stocking on the immature predator. Actually, as the literature [13, 14] pointed out, stage-structured differential equations exhibit much more complicated behaviors than ordinary ones since time delays could cause a stable equilibrium to become unstable and cause the population to fluctuate. Therefore, it is important to consider the dynamics of a prey-predator system with stage-structure; see  and references cited therein.
On the other hand, with food safety gaining importance, green food is being paid more and more attention to. In order to plant green food, one can use a periodic harvesting or stocking prey or predator, instead of using high toxic or high residues pesticide.
where (), () denote the densities of immature (mature) prey and immature (mature) predator, respectively. The parameters r, k, λ, , , , , are all positive constants, r denotes the birth rate of the immature prey, k is the maximum number of the mature prey that can be eaten by a mature predator per unit of time, λ represents the rate of conversing prey into predator (i.e., the converse rate from mature prey to immature predator), (), are the mortality rates of the immature and mature prey, respectively, and (), are the mortality rates of the immature and mature predator, respectively, is the intra-specific competition rate of the mature prey, , represent a constant time to reach maturity of prey and predator, respectively, μ (≥0) denotes the stocking amount of the mature predator, p () is the catching rate of the immature prey at , , and , T is the period of the impulsive effect.
In this paper, we aim to investigate the global attractivity of a mature predator periodic solution and the permanence of system (1.1). In agreement with the biological point of view, we only consider (1.1) in the biological sense, region .
The organization of the paper is as follows. In Section 2, some preliminaries and lemmas are given. In Section 3, sufficient conditions for the global attractivity of a mature predator survival periodic solution are obtained. In Section 4, the permanence of system (1.1) is investigated. Some examples and numerical simulations are given to illustrate the main results in Section 5. Finally, in Section 6, a brief conclusion is presented.
2 Preliminaries and lemmas
In this section, some definitions and lemmas are introduced.
- (i)V is continuous in , for each , ,
V is locally Lipschitzian in x, then V is said to belong to class .
Next, we give some important lemmas which will be useful for our main results.
Lemma 2.1 
where , , , and are constants.
the sequence satisfies , with ;
and is left-continuous at , .
if , then ;
if , then .
Lemma 2.3 
with initial value , .
Hence, . The proof is completed. □
Lemma 2.5 There is a positive constant M such that , , , for every solution of system (1.1) with t sufficiently large, and λ is a positive constant defined in system (1.1).
Proof Define , , .
such that , , , with t large enough. This completes the proof. □
3 Global attractivity of mature predator periodic solution
In this section, we shall demonstrate the existence and global attractivity of the mature predator survival periodic solution of system (1.1).
Firstly, by Lemmas 2.2, 2.3, and 2.4, we can easily obtain the existence of a predator-extinction periodic solution for system (1.1).
Theorem 3.1 System (1.1) has a mature predator survival periodic solution . For , and each solution of system (1.1), we have as , where for , and .
Next, we give the conditions on the global attractivity of the predator-extinction periodic solution of the system (1.1).
By the comparison theorem, for sufficiently small constants , there exists such that , for all . Let , then and we have . On the other hand, we can conclude from (3.1), (3.2), and (3.3) that for t large enough, which implies as .
Since ε, , , are arbitrary small, we obtain , , , as t is large enough. The proof is completed. □
4 Permanence of system (1.1)
In the real world, from the principle of ecosystem balance and saving resources, we only need to control the prey under the economic threshold level, and not to eradicate the prey totally. Thus we focus on the permanence of system (1.1).
First, we give the definition of permanence.
Definition 4.1 System (1.1) is said to be permanent if there exist positive constants m and M such that each positive solution of system (1.1) satisfies , , , for t large enough.
where M, , , q are defined in (2.4), (4.7), (4.10), (4.13), respectively, then system (1.1) is permanent.
This is a contradiction. Thus, we have , .
This means that as . It is a contradiction with .
If holds for all t large enough, then our goal is obtained.
- (ii)If is oscillatory about . Setting(4.10)
we prove that for all t large enough. Suppose that there exist two positive constants γ, η such that and for all , where γ is large enough, and the inequality (4.8) holds true for . Since is continuous, 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 γ) such that for . If , our aim is obtained. If , from the second equation of (1.1), we obtain, for , . According to the assumption 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 obtain for . Since the interval is arbitrarily chosen, we get for t large enough. In view of our arguments above, the choice of is independent of the positive solution of (1.1), which satisfies for t large enough.
From (3.3), let , then .
Then taking , we have , . Considering Lemma 2.5 and the above discussion, we can find that system (1.1) is permanent. This completes the proof. □
5 Numerical simulations
In this paper, by using the comparison theorem of an impulsive differential equation and some analysis techniques, we obtain the sufficient conditions of the mature predator survival periodic solution and permanence of system (1.1). Theorem 3.2 implies that increasing T and μ is propitious to the global attractivity of the mature predator survival periodic solution . By Theorem 4.1, we may see that reducing T and μ plays an important role in the permanence of system (1.1). Combining the biological resource management, we believe that there exists a threshold value of economic benefits. Thus, it is unadvisable to make too much effort to destroy all the pest, and there must exist an optimal harvesting policy for system (1.1), that is, what we should do is to gain more, rather than wipe out all pest,so it is interesting for us to continue to study the optimal harvesting policy of system (1.1) in the near future.
This paper is supported by the National Natural Science Foundation of China (11161015, 11361012), and the 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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