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Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 864367, 10 pages
Research Article

Analysis of an Impulsive One-Predator and Two-Prey System with Stage-Structure and Generalized Functional Response

1School of Science, Guilin University of Technology, Guilin, Guangxi 541004, China
2Department of Mathematics, Hengyang Normal University, Hengyang, Hunan 421008, China

Received 8 June 2015; Revised 23 September 2015; Accepted 28 September 2015

Academic Editor: Luca Gori

Copyright © 2015 Xiangmin Ma 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.


An impulsive one-predator and two-prey system with stage-structure and generalized functional response is proposed and analyzed. By reasonable assumption and theoretical analysis, we obtain conditions for the existence and global attractivity of the predator-extinction periodic solution. Sufficient conditions for the permanence of this system are established via impulsive differential comparison theorem. Furthermore, abundant results of numerical simulations are given by choosing two different and concrete functional responses, which indicate that impulsive effects, stage-structure, and functional responses are vital to the dynamical properties of this system. Finally, the biological meanings of the main results and some control strategies are given.

1. Introduction and Model Formulation

In real world, the properties of one-predator and one-prey system have been studied widely and many valuable results have been obtained. If examining the cases that there are two preys for a predator, then the above system cannot reflect the real behaviors of individuals accurately, so scholars proposed three-species predator-prey system. The relationship between species in three-species system may take many forms, such as one prey and two predators [1], a food chain [2, 3], or two preys and one predator [4, 5]. On the other hand, for predator-prey model, in description of the relationship between predator and prey, a crucial element is the classic definition of a predator’s functional response. Recently, the dynamics of predator-prey systems with different kinds of functional responses have been studied in relevant literature, such as Holling type [6], Crowley-Martin type [79], Beddington-DeAngelis type [10, 11], Watt type [12, 13], and Ivlev type [14]. For example, Gakkhar and Naji [15] investigated the dynamical behaviors of the following three-species system with nonlinear functional response:where and represent the two preys densities, respectively, and represents the density of predators depending on the two preys.

However, as Pei et al. [16] pointed out that system (1) could not provide an effective approach because there was no impulsive spraying pesticides or harvesting pest at different fixed moment. We know that pests may bring disastrous effects to their existing system when their amount reaches a certain level. For preventing large economic loss, chemical pesticides are often used in the process of pest management. As a matter of fact, the control on pests often makes pests reduce instantaneously in a short time. In the modeling process, these perturbations are often assumed to be in the form of impulses. Based on traditional models, impulsive differential equations are proposed and extensively used in some applied fields, especially in population dynamics; see [1719]. The theory of impulsive differential equation is now being recognized richer than the corresponding differential equation without impulses, which plays a key role in the development of biomathematics; see monographs [20, 21] and references cited therein.

On the other hand, the stage-structure for predator was also not considered in system (1). In real world, many species go through two or more life stages when they proceed from birth to death. For many animals, their babies are raised by their parents or are dependent on the nutrition from the eggs they stay in. The babies are too weak to produce babies or capture their prey; hence their competition with other individuals of the community can be ignored. Therefore, it is reasonable to introduce stage-structure into competitive or predator-prey models. Many researchers have incorporated it into biological models, where stage-structure is modeled by using a time delay [2224]. Authors [5] pointed out that when the system contained time delay, it had more interesting behaviors. Their results showed that time delay could cause a stable equilibrium to become unstable and Hopf bifurcation could occur as the time delay crossed some critical values. These obtained results have shown that stage-structure plays a vital role in predator-prey models and stage-structured systems exhibit complicated properties. Moreover, Xu [25] showed that an important factor in modeling of predator-prey is the choice of functional response. Model with generalized functional response exhibited many universal properties, which could be applied to many fields because of its flexibility. Shao and Li [26] considered a predator-prey system with generalized functional response. Their results indicated that generalized functional response caused dynamical behaviors of the system to be very complex.

Based on these backgrounds, in this paper, developing system (1) with stage-structure, generalized function response, and impulsive spraying pesticides, we will consider the following one-predator and two-prey system:where and represent the densities of two different preys, respectively, and we assume that there is no competition between the two preys. and denote the densities of immature predator and mature predator, respectively. is the natural growth rate of     (). and are coefficients of internal competition of prey () and mature predator , respectively. is capture rate of mature predator for (). represents the conversion rate of two preys into reproduction of mature predator. and are death rates of immature predator and mature predator, respectively. is the mean length of juvenile period of predator. The term denotes the mature rate of immature predator. Function () is adult predator’s functional response. (,  ) is partial impulsive harvesting of prey by catching or pesticides at moment   ().

By use of impulsive differential equation theory and some analysis techniques, we aim to investigate the existence and global attractivity of predator-extinction periodic solution and the permanence of (2). Further, by numerical analysis, we try to find out the effects of impulsive and stage-structure on this system.

Since does not appear in the first, the second, and the fourth equation of system (2), we can simplify (2) and restrict our attention to the following system:with initial conditions:

From biological point of view, without loss of generality, in this paper, we assumed that   () is strictly increasing, differential with , satisfying (a constant) for all . Further, we only consider (3) in the following biological meaning region:

The rest of this paper is organized as follows. In Section 2, we give some notations, definitions, and lemmas. By using lemmas and impulsive comparison theorem, we discuss the existence of predator-extinction solution and permanence of system (3) in Sections 3 and 4, respectively. In Section 5, numerical simulations are given to show the complicated dynamical behaviors of (3). Finally, we end this paper by a brief discussion in Section 6.

2. Preliminaries

In this section, some definitions and lemmas are introduced which are useful for our main results. Solution of (3), denoted by , is piecewise and continuous function: ,   is continuous on ,  , and exists. Obviously, the global existence and uniqueness of solution of (3) are guaranteed by the smoothness of , where denotes the mapping defined by right side of system (2). For more details refer to [20, 21].

Lemma 1 (see [27]). Consider the following differential equation:where    is a positive constant and ,  ; then we have the following:(i)if , then ;(ii)if , then .

Lemma 2 (see [10]). Consider the following impulsive system:where , , and . If , then system (7) has a positive periodic solution and for any solution of system (7), we have , as , whereBy Lemma 2, we can easily know that if the following hypotheses (H1) and (H2) hold,(H1):,(H2):.Then (3) has a mature predator-extinction periodic solution for , and for any solution of system (3), we have , , and , , wherewith

3. Global Attractivity of the Predator-Extinction Periodic Solution

In this section, we investigate the global attractivity of predator-extinction periodic solution of system (3).

Theorem 3. Predator-extinction periodic solution of system (3) is globally attractive if (H1), (H2), and(H3):hold, where and are defined in (15) and (16), respectively.

Proof. Since (H3) holds and , are differential for all , we can choose two positive constants and to be sufficiently small such thatFrom the first equation of system (3), we have .
Consider the following impulsive comparison system:In view of Lemma 2, we obtain thatwithwhich is unique and globally asymptotically stable positive periodic solution of (12). By use of comparison theorem of impulsive differential equation, there exists such that, for the sufficiently small constant and all (), we haveSimilarly, there exists such that, for the sufficiently small constant and all (), we haveThrough observation of the third equation of (3), we haveConsider the following differential comparison system:According to (11) and Lemma 1, we have . Since , , is the solution of (18) with initial conditions , ; by comparison theorem, we have . In view of the positivity of , we have . It implies that for arbitrarily small positive constant and large enough, we haveFurther, from the first and the fourth equation of (3), we haveConsidering the following comparison system of (20),by Lemma 2, we get the positive periodic solution of system (21) as follows:withBy comparison theorem, for given constant and large enough, we have . Let , then , so we have . It follows from (15) that for sufficiently large, which implies that as . Similarly, we can obtain as . This is the end of the proof.

4. Permanence of System (3)

Now we investigate the permanence of system (3). Before stating the theorem, we give the definition of permanence for system (3).

Definition 4. System (3) is said to be permanent, if there exist two positive constants and , such that, for any solution of (3), , , holds for sufficiently large.

Theorem 5. Suppose that conditions of (H1) and (H2) hold; moreover if the following conditions: (H4):, ,(H5):,are satisfied, where , , and are defined in (27), (40), and (42), respectively, then system (3) is permanent.

Proof. Firstly, in view of (15) and (16), noticing that positive constants and are arbitrarily chosen and can be sufficiently small, we haveSecondly, from the third equation of system (3), we have the following inequality:Considering the following comparison equation,by (H4) and Lemma 1, we have . According to comparison theorem of differential equation, we get