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Journal of Applied Mathematics
Volume 2012, Article ID 521729, 19 pages
http://dx.doi.org/10.1155/2012/521729
Research Article

Extinction and Permanence of a General Predator-Prey System with Impulsive Perturbations

1Key Laboratory of Eco-Environments in Three Gorges Reservoir Region (Ministry of Education), School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
2Institute of Mathematics, Academy of Mathematics and System Sciences, Beijing 100080, China

Received 23 February 2012; Accepted 18 May 2012

Academic Editor: A. A. Soliman

Copyright © 2012 Xianning Liu and Lansun Chen. 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

A general predator-prey system is studied in a scheme where there is periodic impulsive perturbations. This scheme has the potential to protect the predator from extinction but under some conditions may also serve to lead to extinction of the prey. Conditions for extinction and permanence are obtained via the comparison methods involving monotone theory of impulsive systems and multiple Liapunov functions, which establish explicit bounds on solutions. The existence of a positive periodic solution is also studied by the bifurcation theory. Application is given to a Lotka-Volterra predator-prey system with periodic impulsive immigration of the predator. It is shown that the results are quite different from the corresponding system without impulsive immigration, where extinction of the prey can never be achieved. The prey will be extinct or permanent independent of whether the system without impulsive effect immigration is permanent or not. The model and its results suggest an approach of pest control which proves more effective than the classical one.

1. Introduction

Systems of differential equations with impulses are found in almost every domain of applied sciences. They generally describe phenomena which are subject to short-time perturbations or instantaneous changes. That is why in recent years these systems have been the object of many investigations [17], in which an abundance of basic theories has been developed. Systematic accounts of the subject can be found in [1, 3]. Some impulsive equations have been recently introduced in population dynamics in relation to impulsive birth [8], chemotherapeutic treatment [9], and pulse vaccination [10] of disease and the impulses could also be due to invasion or stocking and harvesting of species [11, 12].

The Lotka-Volterra system is a fundamental one to model the population dynamics. It can describe the basic interactions between species such as cooperation, competition, and predator-prey. It can be extended in many ways: Wang and Chen [13] considered stage-structure for the predator; Xiao and Chen [14] introduced diseases for the prey; while Xu and Chen [15] focused on the functional responses and diffusions of the predator. With regard to impulsive effects Lakmeche and Arino [9] studied a two-dimensional competing Lotka-Volterra system with impulses arising from chemotherapeutic treatment where the stability of a trivial periodic solution was studied and conditions for the existence of a positive periodic solution bifurcating from the trivial one were established. The Lotka-Volterra predator-prey system ̇𝑥1=𝑥1𝑟1𝑎11𝑥1𝑎12𝑥2,̇𝑥2=𝑥2𝑟2+𝑎21𝑥1(1.1) can be developed by introducing a constant periodic impulsive immigration for the predator. That is ̇𝑥1=𝑥1𝑟1𝑎11𝑥1𝑎12𝑥2,̇𝑥2=𝑥2𝑟2+𝑎21𝑥1𝑥,𝑡𝑛𝜏,1𝑡+=𝑥1(𝑡),𝑥2𝑡+=𝑥2(𝑥0𝑡)+𝑏,𝑡=𝑛𝜏,+=𝑥0=𝑥01,𝑥02,(1.2) where 𝑥1(𝑡), 𝑥2(𝑡) are the densities of the prey and predator at time 𝑡, respectively, 𝑟1 is the intrinsic growth rate of prey, 𝑟2 is the death rate of predator, 𝑎11 is the rate of intraspecific competition or density dependence, 𝑎12 is the per capita rate of predation of the predator, 𝑎21 denotes the product of the per capita rate of predation and the rate of conversing prey into predator, 𝜏 is the period of the impulsive immigration effect. This immigration could be artificially planting of predator in order to protect it from extinction. It could also be short-time invasion of predator as a disaster for the prey. For example, this has often been seen in recent years that a large amount of locusts may invade into some areas and cause damages to other species in the northwestern China of Xinjiang province and Inner Mongolia.

Usually biological pest control requires the introduction of a predator decreasing the pest population to an acceptable level as referred in [16, 17] and the references cited therein. It provides only short-term results as after some time this kind of predator-prey system will reach its coexisting equilibrium no matter how large the initial density of the predator is. In this case, system (1.1) can serve as a model of pest control, which will be called classical approach in this paper. However, the dynamics of system (1.1) are very simple. Either there is a positive equilibrium (𝑟1/𝑎11>𝑟2/𝑎21), in which case it is global asymptotically stable. Or there is no positive equilibrium (𝑟1/𝑎11𝑟2/𝑎21), in which case 𝑥2(𝑡) goes extinct and 𝑥1(𝑡) tends to 𝑟1/𝑎11, the capacity of the prey. In each case, the prey can never become extinct. That is why the classical approach of this kind in pest control is not so effective. System (1.2) serves as a different approach of biological pest control, in which predator is released impulsively. Besides this, Liu et al. [18] considered the following system which also includes impulsive chemical pest control of pesticide using into system (1.2): ̇𝑥1=𝑥1𝑟1𝑎11𝑥1𝑎12𝑥2,̇𝑥2=𝑥2𝑟2+𝑎21𝑥1𝑥,𝑡𝑛𝜏,1𝑡+=1𝑝1𝑥1(𝑡),𝑥2𝑡+=1𝑝2𝑥2(𝑥0𝑡)+𝑏,𝑡=𝑛𝜏,+=𝑥0=𝑥01,𝑥02,(1.3) where 0𝑝𝑖<1, 𝑖=1,2.

However, in order to check the effect of pest control, it is important to study the extinction and permanence of such kind of impulsive systems as (1.2) and (1.3). We will consider the following general impulsive predator-prey system, which includes (1.2) and (1.3) as special cases: ̇𝑥1=𝑥1𝑓1𝑥1,𝑥2,̇𝑥2=𝑥2𝑓2𝑥1,𝑥2𝑥,𝑡𝑛𝜏,1𝑡+=𝐼1𝑥1(𝑡),𝑥2𝑡+=𝐼2𝑥2(𝑥0𝑡),𝑡=𝑛𝜏,+=𝑥0=𝑥01,𝑥02.(1.4)

Ballinger and Liu [19] established some conditions to guarantee permanence of a general impulsive system by the method of Liapunov function and applied their results to the impulsive Lotka-Volterra system; however, their conditions include the existence of a positive equilibrium of the corresponding system without impulses. In Liu et al. [18], Liu and Chen [20], and Zhang et al. [21], extinction and permanence of impulsive prey-predator systems with different functional responses were established via comparison. But their methods and results depended on solving a prey eradicated periodic solution explicitly and obtaining its global asymptotical attractivity directly which is impossible for general systems. We will study the permanence of system (1.4) through some techniques of comparison methods involving monotone theory of impulsive systems and multiple Liapunov functions, which establish explicit bounds on solutions. The existence and global attractivity of the prey eradicated periodic solution are ensured by monotone theory of impulsive systems and extinction and permanence are obtained by generalizing the comparison skills to study the properties of solutions near boundary. Compared to [18, 20, 21], the model (1.4) and results in this paper have the following advantages. (i)Both the functional response and impulsive effect are in general functions which can be applied in many different settings. (ii)Extinction and permanence results do not depend on solving the boundary system and obtaining a trivial periodic solution and explicitly.

Our permanence results also known as practical persistence are stronger than permanence. Motivated by the approach of Wang and Ma [22], Cao and Gard [23] introduced the idea and methods, which were developed further in the context of reaction-diffusion models by Cantrell and Cosner [24]. And a discussion of how the methods are applied to various sorts of ecological models, including some discrete models, was given by Cosner [25]. Applying our results to (1.2), we can see that system (1.2) may be permanent or have at least one species reaching extinction, independent of whether system (1.1) is permanent or not.

The organization of this paper is as follows. In the next section, we introduce notations and definitions which will be used in this paper and give some basic assumptions on system (1.4). In Section 3, we present extinction and permanence results of system (1.4) and study the existence of a positive periodic solution by means of bifurcation theory. In Section 4, we apply our results to system (1.2) and interpret the biological meanings. And in the last section, we discuss our methods and results.

2. Notations and Definitions

In this section, we agree on some notations which will prove useful and give some definitions.

Let 𝑅+=[0,), 𝑅2+={𝑥𝑅2𝑥0}, and 𝑁 be the set of all nonnegative integers. Denote by 𝐹=(𝐹1,𝐹2) the map defined by the right hand of system (1.4). Let 𝑉0={𝑉𝑅+×𝑅2+𝑅+𝑉iscontinuouson(𝑛𝜏,(𝑛+1)𝜏]×𝑅2+andlim(𝑡,𝑦)(𝑛𝜏,𝑥),𝑡>𝑛𝜏𝑉(𝑡,𝑦)=𝑉(𝑛𝜏+,𝑥)exists}.

Definition 2.1. 𝑉𝑉0, then for (𝑡,𝑥)(𝑛𝜏,(𝑛+1)𝜏]×𝑅2+, the upper right derivative of 𝑉(𝑡,𝑥) with respect to the impulsive differential system (1.4) is defined as 𝐷+𝑉(𝑡,𝑥)=limsup0+1[].𝑉(𝑡+,𝑥+𝐹(𝑥))𝑉(𝑡,𝑥)(2.1)
We will assume the following basic conditions for system (1.4) hold throughout this paper. (A1)𝑓𝑖×𝑅2+𝑅+ is differentiable and 𝜕𝑓𝑖/𝜕𝑥𝑖0, 𝑖=1,2, 𝜕𝑓1/𝜕𝑥20, 𝜕𝑓2/𝜕𝑥10.(A2)𝐼𝑖𝑅+𝑅+ is continuous, 𝐼𝑖(0)0, 𝐼𝑖(𝑢)>0 for 𝑢>0, and 𝐼𝑖 is nondecreasing for 𝑖=1,2.
The solution of system (1.4) is a piecewise continuous function 𝑥𝑅+𝑅2+, 𝑥(𝑡) is continuous on (𝑛𝜏,(𝑛+1)𝜏], 𝑛𝑁, and 𝑥(𝑛𝜏+)=lim𝑡𝑛𝜏+𝑥(𝑡) exists. Obviously the smoothness properties of 𝑓𝑖 guarantee the global existence and uniqueness of solutions of system (1.4) (see [1, 3] for details on fundamental properties of impulsive systems). (A1) shows that 𝑥1 and 𝑥2 can be the densities of the prey and the predator at time 𝑡, respectively, and both the species are density dependent. With (A2), we can see that impulsive perturbations cannot make any species disappear instantly or in limited time interval. Since ̇𝑥𝑖(𝑡)=0 whenever 𝑥𝑖(𝑡)=0, 𝑖=1,2, 𝑡𝑛𝜏, 𝑛𝑁 and 𝑥1(𝑛𝜏+)=𝐼1(𝑥1(𝑛𝜏)), 𝑥2(𝑛𝜏+)=𝐼2(𝑥2(𝑛𝜏)), 𝑛𝑁, by (A2), we have the following lemma.

Lemma 2.2. Suppose 𝑥(𝑡) is a solution of (1.4) with 𝑥(0+)0, then 𝑥(𝑡)0 for all 𝑡0. And further 𝑥(𝑡)>0, 𝑡0 if 𝑥(0+)>0.

Definition 2.3. System (1.4) is said to be permanent if there exist constants 𝑀𝑚>0 such that 𝑚𝑥𝑖(𝑡)𝑀, 𝑖=1,2 for all t sufficiently large, where 𝑥(𝑡) is any solution of (1.4) with 𝑥(0+)>0.
We will use a basic comparison result from in [3, Theorem 3.1.1]. For convenience, we state it in our notations.
Suppose that 𝑔𝑅+×𝑅+𝑅+ satisfies the following condition (H). (H)𝑔 is continuous in (𝑛𝜏,(𝑛+1)𝜏]×𝑅+ and for 𝑥𝑅+, 𝑛𝑁, lim(𝑡,𝑦)(𝑛𝜏+,𝑥)𝑔(𝑡,𝑦)=𝑔(𝑛𝜏+,𝑥) exists.

Lemma 2.4. Let 𝑉𝑉0. Assume that 𝐷+𝑉𝑡𝑉(𝑡,𝑥)𝑔(𝑡,𝑉(𝑡,𝑥)),𝑡𝑛𝜏,𝑡,𝑥+𝜓𝑛(𝑉(𝑡,𝑥(𝑡))),𝑡=𝑛𝜏,(2.2) where 𝑔𝑅+×𝑅+𝑅+ satisfies (H) and 𝜓𝑛𝑅+𝑅+ is nondecreasing. Let 𝑟(𝑡) be the maximal solution of the scalar impulsive differential equation 𝑢𝑡̇𝑢=𝑔(𝑡,𝑢),𝑡𝑛𝜏,+=𝜓𝑛𝑢0(𝑢(𝑡)),𝑡=𝑛𝜏,+=𝑢0(2.3) existing on [0,). Then, 𝑉(0+,𝑥0)𝑢0 implies that 𝑉(𝑡,𝑥(𝑡))𝑟(𝑡),𝑡0, where 𝑥(𝑡) is any solution of (1.4).

Similar result can be obtained when all the directions of the inequalities in (2.2) are reversed. Note that if we have some smoothness conditions of 𝑔 to guarantee the existence and uniqueness of solutions for (2.3), then 𝑟(𝑡) is exactly the unique solution of (2.3).

3. Main Results

In this section, we will establish conditions for the extinction and permanence of system (1.4) and study the bifurcation of a positive periodic solution for system (1.4).

3.1. Uniformly Ultimate Upper Boundary

Firstly, we establish conditions for that all solutions of (1.4) are uniformly bounded above. This is usually valid from the biological interpreting of the system. Mathematically, it is easy to be achieved by using the method of Liapunov functions and the comparison results of Lemma 2.4. For example, we give one set of such conditions here.

Theorem 3.1. Suppose that the following condition (H1) holds. (H1) There exists 𝑉(𝑡)=𝑉(𝑡,𝑥), 𝑉𝑉0 such that the following conditions hold.
(i) 𝑉(𝑡,𝑥)𝑐1𝑥1+𝑐2𝑥2, for some 𝑐1,𝑐2>0;(ii)𝐷+𝑉𝑡𝑉(𝑡,𝑥(𝑡))𝜆𝑉(𝑡)+𝐾,𝑡𝑛𝜏,𝑡,𝑥+𝑉(𝑡,𝑥(𝑡))+𝑏,𝑡=𝑛𝜏,(3.1)where 𝜆,𝐾,𝑏 are positive constants.
Then, system (1.4) is ultimately upper bounded.

Proof. Let 𝑉(0+)=𝑉(0,𝑥(0+)). By (i), it suffices to prove that 𝑉(𝑡) is ultimately upper bounded. In view of (ii), this is similar to the proof of [18, Lemma 3.2]. Thus we omit it here. The proof is complete.

3.2. Prey Eradicated Periodic Solution

To study the prey eradicated periodic solution, we consider the following scalar impulsive system, which will also serve as an comparison system for studying the permanence of system (1.4): ̇𝑢=𝑢𝑓2𝑢𝑡(𝜀,𝑢),𝑡𝑛𝜏,+=𝐼2𝑢0(𝑢(𝑡)),𝑡=𝑛𝜏,+=𝑢0>0.(3.2)

Since one-dimensional continuous differential system is naturally monotone system and 𝐼2 is nondecreasing, the solutions of system (3.2) are also monotone with respect to initial values [1, the proof of Theorem 12.5].

Lemma 3.2. Suppose that the following condition holds. (H2) There exist positive constants 𝑎2, 𝑏2, and 𝛼2such that𝐼2(𝑢)𝑎2𝑢+𝑏2 for 0𝑢𝛼2. Then, there exists 𝛿2>0 such that 𝑢(𝜏+)𝑢0 for 𝑢0𝛿2, where 𝑢(𝑡) is the solution of (3.2).

Proof. Let 𝛿2=min{𝑏2,𝛼2,𝛼2/exp(𝜏𝑓2(𝜀,0))} and 𝑢0𝛿2. Obviously, 𝑢(𝑡) is positive for 𝑡>0. By (A1), we have ̇𝑢𝑢𝑓2(𝜀,0),𝑡(0,𝜏).(3.3) Hence, 𝑢(𝑡)𝑢0exp(𝑡𝑓2(𝜀,0))𝛿2exp(𝜏𝑓2(𝜀,0))𝛼2, 𝑡(0,𝜏] if 𝑓2(𝜀,0)0 and 𝑢(𝑡)𝑢0exp(𝑡𝑓2(𝜀,0))𝑢0𝛿2𝛼2, 𝑡(0,𝜏] if 𝑓2(𝜀,0)<0. By (3.2), we have 𝑢(𝜏)=𝑢0exp𝜏0𝑓2(𝜀,𝑢(𝑠))𝑑𝑠.(3.4) Hence, by (H2), 𝑢𝜏+𝑎2𝑢0exp𝜏0𝑓2(𝜀,𝑢(𝑠))𝑑𝑠+𝑏2𝑏2𝛿2𝑢0.(3.5) The proof is complete.

Lemma 3.3. Suppose that the following condition holds. (H3) There exist positive constants 𝐴2, 𝐵2, and 𝛽2 such that 𝐴2exp𝜏𝑓2(0,0)<1,𝑓2(0,𝑢)<0(3.6)and 𝐼2(𝑢)𝐴2𝑢+𝐵2 for 𝑢𝛽2.
Then, there exist 𝜀2>0,𝑀2>0 such that 𝑢(𝜏+)𝑢0 for 𝑢0𝑀2, where 𝑢(𝑡) is the solution of (3.2) with 0𝜀𝜀2.

Proof. By (3.6), there exists 𝜀2>0 such that 𝐴2exp𝜏𝑓2(𝜀,0)<1,𝑓2(𝜀,𝑢)<0(3.7) for 0𝜀𝜀2 and 𝑢𝛽2. Let 𝑀=max{𝐼2(𝑢)0𝑢𝛽2} and 𝑀2=max{𝑀,𝛽2,𝐵2/(1𝐴2exp(𝜏𝑓2(𝜀,0)))}>0. Let 𝑢0𝑀2 and 𝑢(𝑡) be the solution of (3.2) with 0𝜀𝜀2. There are two cases for 𝑢(𝑡), 𝑡(0,𝜏].
Case  1. There exists 𝑡1(0,𝜏] such that 𝑢(𝑡1)<𝛽2.
Let 𝑡=inf{𝑡(0,𝜏]𝑢(𝑡)<𝛽2}. Then, 𝑢(𝑡)=𝛽2. Since 𝑓2(𝜀,𝛽2)<0, we can conclude that 𝑢(𝑡)<𝛽2, 𝑡(𝑡,𝜏]. Hence, 𝑢𝜏+=𝐼2(𝑢(𝜏))𝑀𝑀2𝑢0.(3.8)
Case  2. 𝑢(𝑡)𝛽2,𝑡(0,𝜏].
By (A1), we have. ̇𝑢𝑢𝑓2(𝜀,0),𝑡(0,𝜏).(3.9) Hence, 𝑢(𝜏)𝑢0exp𝜏𝑓2(𝜀,0).(3.10) Therefore, 𝑢𝜏+𝐴2𝑢0exp𝜏𝑓2(𝜀,0)+𝐵2=𝑢0+𝐵21𝐴2exp𝜏𝑓2(𝑢𝜀,0)0𝑢0.(3.11) This completes the proof.

Let 0<𝑢10𝛿2 and 𝑢20𝛽2 for 𝛿2,𝛽2 in Lemmas 3.2 and 3.3, respectively. Consider the solution 𝑢(𝑡) of (3.2) with 𝑢0[𝑢10,𝑢20]. By Lemma 2.4, similar to [12, Theorem 3.1], we can define a map 𝑢𝑃10,𝑢20𝑢10,𝑢20,𝑃𝑢0=𝐼2(𝑢(𝜏))(3.12) and show 𝑃 has a fixed point which corresponds to the initial value of a positive periodic solution of (3.2). Thus, we have the following theorem.

Theorem 3.4. Suppose that (H2) and (H3) hold. Then, there exists 𝜀2>0 such that system (3.2) has a positive 𝜏-periodic solution 𝑢𝜀(𝑡) for each 0𝜀𝜀2.

Modify 𝑣(0),𝑣(𝑇) to 𝑣(0+),𝑣(𝑇+) and consider for the case with 𝑇=𝜏 and 𝑞=1 in the definition of lower and upper solutions of [26, Definition 3.1], the solution 𝑢(𝑡) of (3.2) with 0<𝑢0𝛿2 (𝑢0𝛽2) here is factually also the lower solution (upper solution) of (3.2). Hence similar to [26, Theorem 3.6], the solutions of (3.2) with initial values 0<𝑢0𝛿2 and 𝑢0𝛽2 will tend to a positive 𝜏-periodic solution of (3.2). If the positive 𝜏-periodic solution of (3.2) is unique, then its global attractivity will also be established. In fact, for any solution 𝑢(𝑡) of (3.2) with initial value 𝑢0>0, we can always find a lower solution 𝑢1(𝑡) and an upper solution 𝑢2(𝑡) with initial values 0<𝑢1(0+)=𝑢10𝛼2 and 𝑢2(0+)=𝑢20𝛽2, respectively, such that 𝑢10𝑢0𝑢20. Then since the solutions of system (3.2) are monotone with respect to initial values, 𝑢1(𝑡)𝑢(𝑡)𝑢2(𝑡). And since both 𝑢1(𝑡) and 𝑢2(𝑡) will tend to the unique positive 𝜏-periodic solution of (3.2), so is 𝑢(𝑡). Using the theory of concave operators, [26, Theorem 3.8] established the uniqueness of positive periodic solution for a general n-dimensional monotone impulsive differential system. With the map 𝑃 defined above, we can simply use that result here. We first give the definition of strongly concave operator on 𝑅𝑛+.

Definition 3.5. An operator 𝑈𝑅𝑛+𝑅𝑛+ is strongly concave on 𝑅𝑛+ if for any 𝑥𝑅𝑛+, 𝑥>0 and any number 𝑠(0,1), there exists a positive number 𝜂 such that 𝑈(𝑠𝑥)(1+𝜂)𝑠𝑈(𝑥).
Now let 𝑢(𝑡) be the solution of (3.2) with initial value 𝑢0>0. Let 𝑓2=𝑢𝑓2(𝜀,𝑢). Define 𝐹(𝑢(𝑡))=𝑓2(𝜀,𝑢(𝑡))𝐷𝑓2(𝜀,𝑢(𝑡))𝑢(𝑡),(3.13) where 𝐷𝑓2 is the Jacobian matrix of 𝑓2 (here is just its derivative with respect to 𝑢 since (3.2) is one-dimensional). By [26, Theorems 3.8 and 3.9], we have the following result.

Theorem 3.6. Suppose that (H2), (H3), and the following (H4) hold.(H4)𝐹(𝑢)>0 for any 𝑢>0 and 𝐼2 is strongly concave or is linear. Then, there exists a𝜀2>0 such that for each 0𝜀𝜀2, system (3.2) has a unique positive 𝜏-periodic solution 𝑢𝜀(𝑡), which is a global attractor for any solution 𝑢(𝑡) of (3.2) with 𝑢0>0.

Remark 3.7. The unique positive 𝜏-periodic solution is corresponding to the unique positive fixed point of 𝑃. Since the map 𝑃 here is one-dimensional, it could be possible to establish its uniqueness of positive fixed point directly in some cases. Then (H4), can be replaced by the following (H4), which states the uniqueness directly. (H4)𝑃 has at most one positive fixed point.

3.3. Extinction of the Prey

Theorem 3.8. Let (𝑥1(𝑡),𝑥2(𝑡)) be the solution of (1.4). Suppose that (H2)–(H4) and the following (H5) hold.(H5) There exists 𝐴1>0 such that 𝐼1(𝑥)𝐴1𝑥 and 𝐴1exp𝜏0𝑓10,𝑢0(𝑠)𝑑𝑠<1,(3.14)
where 𝑢0(𝑡) is the unique positive periodic solution of (3.2) for 𝜀=0.
Then lim𝑡𝑥1(𝑡)=0, lim𝑡|𝑥2(𝑡)𝑢0(𝑡)|=0.

Proof. By (3.14), the continuity of 𝑓1, and the Lebesgue Theorem, we can choose an 𝜂>0 sufficiently small such that 𝐴1exp(𝜎0𝜂)<1, where 𝜎0𝜂=𝜏0𝑓1(0,𝑢0(𝑠)𝜂)𝑑𝑠. Note that ̇𝑥2𝑥2𝑓2(0,𝑥2), consider (3.2) with 𝜀=0 and 𝑢(0+)=𝑥2(0+). Let 𝑉(𝑡,𝑥)=𝑥2(𝑡). From Lemma 2.4 and Theorem 3.6, we have 𝑥2(𝑡)𝑢(𝑡) and |𝑢(𝑡)𝑢0(𝑡)|0 as 𝑡. Hence, there exists 𝑡1>0 such that 𝑥2(𝑡)𝑢(𝑡)>𝑢0(𝑡)𝜂(3.15) for all 𝑡𝑡1. For simplification and without loss of generality, we may assume (3.15) holds for all 𝑡0. By (A1), we get ̇𝑥1𝑥1𝑓10,𝑢0(𝑡)𝜂(3.16) which leads to 𝑥1((𝑛+1)𝜏)𝑥1𝑛𝜏+exp(𝑛+1)𝜏𝑛𝜏𝑓10,𝑢0(𝑠)𝜂𝑑𝑠=𝑥1𝑛𝜏+𝜎exp0𝜂𝑥1(𝑛𝜏)𝐴1𝜎exp0𝜂.(3.17) Hence 𝑥1(𝑛𝜏)𝑥1(𝜏)(𝐴1exp(𝜎0𝜂))𝑛1 and 𝑥1(𝑛𝜏)0 as 𝑛. Therefore, 𝑥1(𝑡)0 as 𝑡 since 0<𝑥1(𝑡)=𝑥1𝑛𝜏+exp𝑡𝑛𝜏𝑓10,𝑢0(𝑠)𝜂𝑑𝑠𝑥1(𝑛𝜏)𝐴1exp𝜏0||𝑓10,𝑢0||(𝑠)𝜂𝑑𝑠(3.18) for 𝑛𝜏<𝑡(𝑛+1)𝜏. Now, we prove that |𝑥2(𝑡)𝑢0(𝑡)|0 as 𝑡. For 0<𝜀𝜀2 in Theorem 3.6, there exists a 𝑡1>0 such that 0<𝑥1(𝑡)<𝜀, 𝑡𝑡1. Without loss of generality, we may assume that 0<𝑥1(𝑡)<𝜀 for all 𝑡0. Then, by (A1), we have 𝑥2𝑓20,𝑥2̇𝑥2𝑥2𝑓2𝜀,𝑥2.(3.19) From Lemma 2.4 and Theorem 3.6, we have 𝑢1(𝑡)𝑥2(𝑡)𝑢2(𝑡) and |𝑢1(𝑡)𝑢0(𝑡)|0, |𝑢2(𝑡)𝑢𝜀(𝑡)|0 as 𝑡, where 𝑢1(𝑡) is solution of (3.2) with 𝜀=0, 𝑢1(0+)=𝑥2(0+) and 𝑢2(𝑡) are solutions of (3.2) with 𝑢2(0+)=𝑥2(0+). Therefore, for any small enough 𝜂>0, we have 𝑢0(𝑡)𝜂<𝑥2(𝑡)<𝑢𝜀(𝑡)+𝜂(3.20) for large 𝑡. Let 𝜀0, we get 𝑢0(𝑡)𝜂<𝑥2(𝑡)𝑢0(𝑡)+𝜂(3.21) for large 𝑡, which implies lim𝑡|𝑥2(𝑡)𝑢0(𝑡)|=0. The proof is complete.

3.4. Permanence

Theorem 3.9. Suppose that (H1)–(H4) and the following (H6) hold.(H6) There exists 𝑎1>0 such that 𝐼1(𝑥)𝑎1𝑥 and 𝑎1exp𝜏0𝑓10,𝑢0(𝑠)𝑑𝑠>1,(3.22)
where 𝑢0(𝑠) is the unique positive periodic solution of (3.2) for 𝜀=0.
Then, system (1.4) is permanent.

Proof. Let 𝑥(𝑡) be any solution of (1.4) with 𝑥0int𝑅2+. From Theorem 3.1, we may assume 𝑥𝑖(𝑡)𝑀, 𝑡0, 𝑖=1,2 where 𝑀 is a positive constant independent of initial values. Since ̇𝑥2𝑥2𝑓20,𝑥2,(3.23) by Lemma 2.4 and Theorem 3.6, for sufficiently small 𝜂1>0, 𝑥2(𝑡)𝑢0(𝑡)𝜂1 for 𝑡 large enough. Hence obviously there exists 𝑚2>0 such that 𝑥2(𝑡)𝑚2 for all 𝑡 large enough. We shall next find an 𝑚1>0 such that 𝑥1(𝑡)𝑚1 for 𝑡 large enough. We will do it in the following two steps.
(1) Let 𝜀2 be the positive constant in the conclusion of Theorem 3.6. By (3.22), the continuity of 𝑓1, and the Lebesgue Theorem, we can choose 0<𝑚3𝜀2, 𝜂1>0 be small enough such that 𝜎=𝑎1exp(𝜏0𝑓2(𝑚3,𝑢𝑚3(𝑠)+𝜂1)𝑑𝑠)>1. We will prove 𝑥1(𝑡)𝑚3 cannot hold for all 𝑡0. Otherwise, ̇𝑥2𝑥2𝑓2𝑚3,𝑥2.(3.24)
By Lemma 2.4 and Theorem 3.6, we have 𝑥2(𝑡)𝑢(𝑡) and |𝑢(𝑡)𝑢𝑚3(𝑡)|0 as 𝑡, where 𝑢(𝑡) is the solution of (3.2) with 𝜀=𝑚3 and 𝑢(0+)=𝑥2(0+). Therefore, there exist, a 𝑇1>0 such that 𝑥2(𝑡)𝑢(𝑡)𝑢𝑚3(𝑡)+𝜂1,(3.25)̇𝑥1𝑥1𝑓1𝑥1,𝑢𝑚3(𝑡)+𝜂1(3.26) for 𝑡𝑇1.
Let 𝑁1𝑁 and 𝑁1𝜏𝑇1. Integrating (3.26) on (𝑛𝜏,(𝑛+1)𝜏] for 𝑛𝑁1, we have 𝑥1((𝑛+1)𝜏)𝑥1𝑛𝜏+exp(𝑛+1)𝜏𝑛𝜏𝑓1𝑚3,𝑢𝑚3(𝑠)+𝜂1𝑑𝑠=𝑥1𝑛𝜏+exp𝜏0𝑓1𝑚3,𝑢𝑚3(𝑠)+𝜂1𝑑𝑠𝑥1(𝑛𝜏)𝑎1exp𝜏0𝑓1𝑚3,𝑢𝑚3(𝑠)+𝜂1𝑑𝑠=𝑥1(𝑛𝜏)𝜎.(3.27) Then, 𝑥1((𝑁1+𝑘)𝜏)𝑥1(𝑁1𝜏)𝜎𝑘 as 𝑘, which is a contradiction. Hence, there exists a 𝑡1>0 such that 𝑥1(𝑡1)>𝑚3.
(2) If 𝑥1(𝑡)>𝑚3 for all 𝑡𝑡1, then our aim is obtained. Hence we need only to consider those solutions which leave the region 𝑅={𝑥𝑅2+𝑥1𝑚3} and reenter it again. Let 𝑡=inf𝑡𝑡1{𝑥1(𝑡)𝑚3}. Then, 𝑥1(𝑡)>𝑚3 for 𝑡[𝑡1,𝑡) and 𝑥1(𝑡)𝑚3. Suppose 𝑡(𝑛1𝜏,(𝑛1+1)𝜏],𝑛1𝑁. Let 𝑢𝑀(𝑡) be the solution of (3.2) with 𝜀=𝑚3 and 𝑢𝑀((𝑛1+1)𝜏+)=𝑀. Then there is 𝑡2>0 such that 𝑢𝑀(𝑡)<𝑢𝑚3(𝑡)+𝜂1 for 𝑡𝑡2+(𝑛1+1)𝜏. Note that 𝑡2 is independent of 𝑥(𝑡). Select 𝑛2,𝑛3𝑁 such that 𝑛2𝜏>𝑡2 and𝑎𝑛21||𝜎exp1||+𝑛2𝜎1𝜎𝑛3>1,(3.28) where 𝜎1=𝜏𝑓1(𝑚3,𝑀). We claim there must be a 𝑡3(𝑡,(𝑛1+1+𝑛2+𝑛3)𝜏] such that 𝑥1(𝑡3)>𝑚3. Otherwise for 𝑡(𝑡,(𝑛1+1+𝑛2+𝑛3)𝜏], 𝑥1(𝑡)𝑚3 and ̇𝑥2𝑓2𝑚3,𝑥2.(3.29) By Lemma 2.4, we have 𝑥2(𝑡)𝑢𝑀𝑛(𝑡),𝑡1𝑛+1𝜏,1+1+𝑛2+𝑛3𝜏.(3.30) Then, 𝑥2(𝑡)𝑢𝑀(𝑡)<𝑢𝑚3(𝑡)+𝜂1,(3.31) and (3.26) holds for 𝑡((𝑛1+1+𝑛2)𝜏,(𝑛1+1+𝑛2+𝑛3)𝜏]. As in step 1, we have 𝑥1𝑛1+1+𝑛2+𝑛3𝜏𝑥1𝑛1+1+𝑛2𝜏𝜎𝑛3.(3.32) Since ̇𝑥1𝑥1𝑓1𝑚3𝑡,𝑀,𝑡,𝑛1+1+𝑛2+𝑛3𝜏,(3.33) integrating it on [𝑡,(𝑛1+1+𝑛2)𝜏], we have 𝑥1𝑛1+1+𝑛2𝜏𝑚3𝑎𝑛21||𝜎exp1||+𝑛2𝜎1.(3.34) Thus, by (3.28), we have 𝑥1𝑛1+1+𝑛2+𝑛3𝜏𝑚3𝑎𝑛21||𝜎exp1||+𝑛2𝜎1𝜎𝑛3>𝑚3,(3.35) which is a contradiction. Let 𝑡=inf𝑡𝑡{𝑥1(𝑡)>𝑚3}. Then, 𝑡(𝑡,(𝑛1+1+𝑛2+𝑛3)𝜏]. Denote 𝑎=min{1,𝑎𝑛2+𝑛31}. For 𝑡(𝑡,𝑡], we have 𝑥1(𝑡)𝑥1𝑡𝑎exp1+𝑛2+𝑛3||𝜎1||𝑚3𝑎exp1+𝑛2+𝑛3||𝜎1||𝑚1.(3.36) For 𝑡>𝑡, the same arguments can be continued since 𝑥1(𝑡+)>𝑚3. Hence 𝑥1(𝑡)𝑚1 for all 𝑡𝑡1. The proof is complete.

3.5. Existence of a Positive Periodic Solution

Since system (1.4) may have a prey eradicated periodic solution, we can use the bifurcation theory in [9] to study the existence of positive periodic solution. System (1.4) can be rewritten as the following more general system: 𝑥𝑡̇𝑥=𝐹(𝑥),𝑡𝑛𝜏,+=Θ(𝑥(𝑡)),𝑡=𝑛𝜏,(3.37) where 𝐹=(𝑥1𝑓1(𝑥),𝑥2𝑓2(𝑥)), Θ=(𝐼1(𝑥),𝐼2(𝑥))𝑅2+𝑅2+, are suitable smooth. For convenience and using the same notations in [9], we have exchanged the subscripts of 𝑥1,𝑥2 and 𝑓1,𝑓2 in system (1.4). Suppose ̇𝑥1𝑥=𝑔1=𝐹1𝑥1𝑥,0,𝑡𝑛𝜏,1𝑡+𝑥=𝜃1(𝑡)=Θ1𝑥1(𝑡),0,𝑡=𝑛𝜏(3.38) has a periodic solution 𝑥𝑠(𝑡). Thus, 𝜁=(𝑥s,0)𝑇 is a trivial periodic solution of (3.37). By studying the local stability of 𝜁 and a standard computation of Floqet exponent, [9] establishes a bifurcation theory which gives the existence positive periodic solution of (3.37). The main idea of the process is to select the period 𝜏 as parameter and transform the problem of finding positive periodic solution into a fixed-point problem. Then, establish the conditions of the implicit function theorem. We will use their results to study the existence of positive periodic solution for (1.4). For simplification, we will only state some necessary notations and the bifurcation theorem of [9].

Let Φ be the flow associated to (3.37), we have 𝑥(𝑡)=Φ(𝑡,𝑥0), 0<𝑡𝜏, where 𝑥0=𝑥(0+). Now, we list following notations we will use from [9]: 𝑑0=1𝜕Θ2𝜕𝑥2𝜕Φ2𝜕𝑥2𝜏0,𝑥0,where𝜏0istherootof𝑑0𝑎=0,0=1𝜕Θ1𝜕𝑥1𝜕Φ1𝜕𝑥1𝜏0,𝑥0,𝑏0=𝜕Θ1𝜕𝑥1𝜕Φ1𝜕𝑥2+𝜕Θ1𝜕𝑥2𝜕Φ2𝜕𝑥2𝜏0,𝑥0,𝜕Φ1𝑡,𝑥0𝜕𝑥1=exp𝑡0𝜕𝐹1(𝜁(𝑟))𝜕𝑥1,𝑑𝑟𝜕Φ2𝑡,𝑥0𝜕𝑥2=exp𝑡0𝜕𝐹2(𝜁(𝑟))𝜕𝑥2,𝑑𝑟𝜕Φ1𝑡,𝑥0𝜕𝑥2=𝑡0exp𝑡𝑢𝜕𝐹1(𝜁(𝑟))𝜕𝑥1𝑑𝑟𝜕𝐹1(𝜁(𝑢))𝜕𝑥2exp𝑢0𝜕𝐹2(𝜁(𝑟))𝜕𝑥2𝜕𝑑𝑟𝑑𝑢,2Φ2𝑡,𝑥0𝜕𝑥1𝜕𝑥2=𝑡0exp𝑡𝑢𝜕𝐹2(𝜁(𝑟))𝜕𝑥2𝜕𝑑𝑟2𝐹2(𝜁(𝑢))𝜕𝑥1𝜕𝑥2exp𝑢0𝜕𝐹2(𝜁(𝑟))𝜕𝑥2𝜕𝑑𝑟𝑑𝑢,2Φ2𝑡,𝑥0𝜕𝑥22=𝑡0exp𝑡𝑢𝜕𝐹2(𝜁(𝑟))𝜕𝑥2𝜕𝑑𝑟2𝐹2(𝜁(𝑢))𝜕𝑥22exp𝑢0𝜕𝐹2(𝜁(𝑟))𝜕𝑥2+𝑑𝑟𝑑𝑢𝑡0exp𝑡𝑢𝜕𝐹2(𝜁(𝑟))𝜕𝑥2𝜕𝑑𝑟2𝐹2(𝜁(𝑢))𝜕𝑥2𝜕𝑥1×𝑢0exp𝑢𝑝𝜕𝐹1(𝜁(𝑟))𝜕𝑥1𝑑𝑟𝜕𝐹1(𝜁(𝑢))𝜕𝑥2exp𝑝0𝜕𝐹2(𝜁(𝑟))𝜕𝑥2𝜕𝑑𝑟𝑑𝑝𝑑𝑢,2Φ2𝑡,𝑥0𝜕𝜏𝜕𝑥2=𝜕𝐹2(𝜁(𝑡))𝜕𝑥2exp𝑡0𝜕𝐹2(𝜁(𝑟))𝜕𝑥2,𝑑𝑟𝜕Φ1𝜏0,𝑥0𝜕𝜏=̇𝑥𝑠𝜏0𝜕𝐵=2Θ2𝜕𝑥1𝜕𝑥2𝜕Φ1𝜏0,𝑥0𝜕𝜏+𝜕Φ1𝜏0,𝑥0𝜕𝑥11𝑎0𝜕Θ1𝜕𝑥1𝜕Φ1𝜏0,𝑥0𝜕𝑡𝑎𝑢𝜕Φ2𝜏0,𝑥0𝜕𝑥2𝜕Θ2𝜕𝑥2𝜕2Φ2𝜏0,𝑥0𝜕𝜏𝜕𝑥2+𝜕2Φ2𝜏0,𝑥0𝜕𝑥1𝜕𝑥21𝑎0𝜕Θ1𝜕𝑥1𝜕Φ1𝜏0,𝑥0𝜕𝜏,𝜕𝐶=22Θ2𝜕𝑥1𝜕𝑥2𝑏0𝑎0𝜕Φ1𝜏0,𝑥0𝜕𝑥1+𝜕Φ1𝜏0,𝑥0𝜕𝑥2×𝜕Φ2𝜏0,𝑥0𝜕𝑥2𝜕2Θ2𝜕𝑥22𝜕Φ2𝜏0,𝑥0𝜕𝑥22+2𝜕Θ2𝜕𝑥2𝑏0𝑎0𝜕2Φ2𝜏0,𝑥0𝜕𝑥2𝜕𝑥1𝜕Θ2𝜕𝑥2𝜕2Φ2𝜏0,𝑥0𝜕𝑥22.(3.39)

Theorem 3.10 (see [9]). If |1𝑎0|<1 and 𝑑0=0, then one has the following: (a)If 𝐵𝐶0, then there is a bifurcation of nontrivial periodic solution. Moreover, there is a bifurcation of supercritical case if 𝐵𝐶<0 and a subcritical case if 𝐵𝐶>0. (b)If 𝐵𝐶=0, then there is an undetermined case.

4. Application

System (1.2) developed the Lotka-Volterra predator-prey system with periodic constant impulsive immigration effect on the predator which is quite natural. For example, we can use the impulsive effects for the purpose of protecting the predator or eliminating the prey. Similar results could be achieved by impulsive invasion of predator. Applying the results in Section 3, we first establish conditions both for the system to be permanent and driving the prey to extinction.

Theorem 4.1. There exists a constant 𝑀>0 such that 𝑥𝑖(𝑡)𝑀, 𝑖=1,2 for each solution 𝑥(𝑡) of (1.2) with all 𝑡 large enough.

Proof. Suppose 𝑥(𝑡) is any solution of (1.2). Let 𝑉(𝑡)=𝑉(𝑡,𝑥(𝑡))=𝑎21𝑥1(𝑡)+𝑎12𝑥2(𝑡). Let 0<𝜆𝑟2. Then, when 𝑡𝑛𝜏, 𝐷+𝑉(𝑡)+𝜆𝑉(𝑡)=𝑎21𝑥1𝑟1+𝜆𝑎11𝑥1𝑎12𝑟2𝑥𝜆2𝑎21𝑥1𝑟1+𝜆𝑎11𝑥1𝐾,(4.1) for some positive constant 𝐾. When 𝑡=𝑛𝜏, 𝑉(𝑛𝜏+)=𝑉(𝑛𝜏)+𝑎12𝑏. Thus, (H1) is satisfied, the conclusion comes from Theorem 3.1. The proof is complete.

Theorem 4.2. Let 𝑥(𝑡) be any solution of (1.2). Then, lim𝑡𝑥1(𝑡)=0, lim𝑡|𝑥2(𝑡)𝑥2(𝑡)|=0 if 𝑏>𝑟1𝑟2𝜏/𝑎12, where 𝑥2(𝑡)=𝑏exp(𝑟2(𝑡𝑛𝜏))/(1exp(𝑟2𝜏)), 𝑡(𝑛𝜏,(𝑛+1)𝜏], 𝑛𝑁, 𝑥2(0+)=𝑏/(1exp(𝑟2𝜏)).

Proof. Consider system (3.2) with the 𝑓2 and 𝐼2 taking the forms in (1.2). Since 𝐼2(𝑢)=𝑢+𝑏, 𝑓2(0,0)=𝑓2(0,𝑢)=𝑟2, obviously, (H2) and (H3) are satisfied. Clearly 𝑥2(𝑡) is a positive 𝜏 periodic solution when 𝜀=0. Note that 𝑃(𝑢)=𝑢exp(𝑟2𝜏)+𝑏, it has a unique positive fixed point since 𝑃(0)>0,lim𝑢(𝑃(𝑢)𝑢)= and 𝑃(𝑢)𝑢 is strictly decreasing. Since 𝐼1(𝑥1)=𝑥1 and exp𝜏0𝑓10,𝑥2(𝑠)𝑑𝑠=exp𝜏0𝑟1𝑎12𝑥2𝑟(𝑠)𝑑𝑠=exp1𝑎𝜏12𝑏𝑟2<1(4.2) when 𝑏>𝑟1𝑟2𝜏/𝑎12, (H5) is satisfied. Thus, the results follow from Theorem 3.8 and Remark 3.7. The proof is complete.

Theorem 4.3. System (1.2) is permanent if 𝑏<𝑟1𝑟2𝜏/𝑎12.

Proof. Let 𝑥(𝑡) be any solution of (1.2) with 𝑥0int𝑅2+. Obviously, (H1)–(H3) and (H4) are satisfied. Since 𝐼1(𝑥1)=𝑥1 and exp𝜏0𝑓10,𝑥2(𝑠)𝑑𝑠=exp𝜏0𝑟1𝑎12𝑥2𝑟(𝑠)𝑑𝑠=exp1𝑎𝜏12𝑏𝑟2>1(4.3) when 𝑏<𝑟1𝑟2𝜏/𝑎12, (H6) is also satisfied. Hence the result follows from Theorem 3.9 and Remark 3.7. The proof is complete.

Next, we show that system (1.2) has positive periodic solution when it is permanent. Note (1.2) has a trivial periodic solution (0,𝑥2(𝑡))𝑇. We also exchange the subscripts of 𝑥1 and 𝑥2 as in Theorem 3.10. Thus, 𝐹1𝑥1,𝑥2=𝑥1𝑟2+𝑎21𝑥2,𝐹2𝑥1,𝑥2=𝑥2𝑟1𝑎11𝑥2𝑎12𝑥1,Θ1𝑥1,𝑥2=𝑥1Θ+𝑏,2𝑥1,𝑥2=𝑥2,𝑥𝜁(𝑡)=𝑠(𝑡),0𝑇=𝑥2.(𝑡),0(4.4) Then, we can compute that 𝑎0=1exp𝑟2𝜏0𝑏>0,0𝑎=21𝑏exp𝑟2𝜏01exp𝑟2𝜏0𝜏00𝑟exp1𝑎𝑢12𝑏1exp𝑟2𝑢𝑟21exp𝑟2𝜏0𝜕𝑑𝑢<0,2Φ2𝜏0,𝑥0𝜕𝜏𝜕𝑥2=𝑟1𝑎12𝑏exp𝑟2𝜏01exp𝑟2𝜏0,𝜕2Φ2𝜏0,𝑥0𝜕𝑥1𝜕𝑥2=𝑎12𝜏0𝜕<0,2Φ2𝜏0,𝑥0𝜕𝑥22=𝑎11𝜏0𝜏00𝑎12𝑎21𝑏exp𝑟2𝑢exp()1exp𝑟2𝜏0×𝑢0𝑟exp1𝑎𝑝12𝑏1exp𝑟2𝑝𝑟21exp𝑟2𝜏0𝑑𝑝𝑑𝑢<0,𝜕Φ1𝜏0,𝑥0𝜕𝜏=̇𝑥𝑠𝜏0𝑟=2𝑏exp𝑟2𝜏01exp𝑟2𝜏0<0,(4.5) where denotes 𝑟1(𝜏0𝑢)𝑎12𝑏(exp(𝑟2𝑢)exp(𝑟2𝜏0))/(𝑟2(1exp(𝑟2𝜏0))).

Since 𝜕Θ𝑖/𝜕𝑥𝑗=0, 𝑖𝑗, 𝜕Θ𝑖/𝜕𝑥𝑖=1, 𝜕2Θ𝑖/𝜕𝑥1𝜕𝑥2=0, 𝑖=1,2 and 𝜕2Θ2/𝜕𝑥22=0, it is easy to verify that 𝐶>0 and 𝑟𝐵=1𝑎12𝑏exp𝑟2𝜏01exp𝑟2𝜏0+𝑎12𝜏0𝑟2𝑏exp𝑟2𝜏01exp𝑟2𝜏02.(4.6) To determine the sign of 𝐵, let 𝜙(𝑡)=𝑟1𝑎12𝑏exp(𝑟2𝑡)/(1exp(𝑟2𝜏0)). We have 𝑑𝜙/𝑑𝑡=𝑟2𝑎12𝑏exp(𝑟2𝑡)/(1exp(𝑟2𝜏0))>0. Thus, we conclude that 𝜙(𝜏0)>0 since 𝜏00𝜙(𝑡)𝑑𝑡=𝑟1𝜏0𝑎12𝑏/𝑟2=0 and 𝜙(𝑡) is strictly increasing. Therefore, we have 𝐵<0 from (4.6) and the following result.

Theorem 4.4. System (1.2) has a supercritical bifurcation of positive periodic solution at the point 𝜏0=𝑎12𝑏/(𝑟1𝑟2), that is, system (1.2) has a positive periodic solution if 𝜏>𝜏0 and is close to 𝜏0, where 𝜏0 is the root of 𝑑0=0.

Remark 4.5. From Theorems 4.2 and 4.3, we know that the trivial periodic solution 𝜁 is a global attractor if 𝜏<𝜏0=𝑎12𝑏(𝑟1𝑟2) and is unstable if 𝜏>𝜏0=𝑎12𝑏(𝑟1𝑟2). Thus, the bifurcation, if it exists, should be a supercritical one and the positive periodic solution is stable.
To find the biological implications of the results for system (1.2), we now evaluate the average density of the species when system (1.2) has a positive periodic solution.

Theorem 4.6. Suppose system (1.2) has a positive periodic solution ̂𝑥(𝑡) with ̂𝑥(0+)=̂𝑥0=(̂𝑥01,̂𝑥02)𝑇. Then, (1/𝜏)𝜏0̂𝑥1(𝑡)𝑑𝑡<min{𝑟1/𝑎11,𝑟2/𝑎21} and (1/𝜏)𝜏0̂𝑥2(𝑡)𝑑𝑡<𝑟1/𝑎12.

Proof. Since ̂𝑥(𝑡) is a positive periodic solution of system (1.2), we have ̂𝑥(0+)=̂𝑥(𝜏+) which gives ̂𝑥01=̂𝑥01𝑟exp1𝜏𝑎11𝜏0̂𝑥1(𝑡)𝑑𝑡𝑎12𝜏0̂𝑥2,(𝑡)𝑑𝑡̂𝑥02=̂𝑥02exp𝑟2𝜏+𝑎21𝜏0̂𝑥1(𝑡)𝑑𝑡+𝑏.(4.7) Hence, the conclusion is quite clear and the proof is complete.

To end this section, we explain the biological implications for the results of (1.2). Let us recall the condition 𝑏=𝑟1𝑟2𝜏/𝑎12. It can be rewritten as 𝑟1𝜏=𝑎12𝑏/𝑟2 or (𝑛+1)𝜏𝑛𝜏𝑟1𝑑𝑡=(𝑛+1)𝜏𝑛𝜏𝑎12𝑥2(𝑡)𝑑𝑡. If there is no prey or its density is very small, the density of the predator is 𝑥2(𝑡). It is clear that 𝑎12𝑏/𝑟2 is the amount of prey that the predator can eat in 𝜏 period of time and 𝑟1𝜏 means the increasing amount of prey in such a period of time. Thus, 𝑟1𝑟2𝜏/𝑎12 can be interpreted as the amount of immigrating predator which the prey could supply for with its increment during 𝜏 period of time. Then, it is easy to understand that the prey will go extinct when 𝑏>𝑟1𝑟2𝜏/𝑎12 and should be permanent otherwise.

It is the impulsive immigration of the predator that makes the dynamics of system (1.2) quite different from that of system (1.1). We note that the conditions in Theorems 4.2 and 4.3 have no relations with that for permanence or extinction of the corresponding system (1.1), which means that system (1.2) can be permanent or extinct no matter whether (1.1) is permanent or not. Therefore, our results suggest a biological approach in pest control by adding some amount of predator impulsively after a fixed period of time. If the amount is large enough, it can drive the pest to extinction which the classical approach can never achieve. When the magnitude of the impulse, 𝑏, is not too large, system (1.2) is permanent and has a positive periodic solution. Our further numeric results show that the periodic solution is a global attractor. In this case, if we use the classical way, the density of prey will tend to either 𝑟1/𝑎11 or 𝑟2/𝑎21. As Theorem 4.6 shows, the average density of the prey is smaller than each of them which means that this approach is still better than the classical one.

5. Discussion

In this paper, we established some conditions of extinction and permanence for a general impulsive predator-prey system. These two concepts are important for a biological system and are useful in protecting the diversities of species. As a simple application, we applied the results to the Lotka-Volterra predator-prey system with periodic constant impulsive immigration effect on the predator. Similarly, our results can also be applied to the models in [18, 20, 21] and we obtain all results therein directly. The analysis process of are the same as that of Section 4. The methods and results in [18, 20, 21] are based on explicitly solving the prey eradicated periodic solution. Different from this, the existence and global attractivity of the prey-eradication periodic solution are ensured by monotone theory and all the conditions in this paper are given for the parameters or the functions in the right-hand side of system (1.4). This makes our results may be easily applied to some more general predator-prey systems with different functional responses and nonlinear impulsive perturbations. Since both system (1.2) and (1.3) do not include density dependent of predator, we can check that the map 𝑃 has a unique positive fixed point directly, which ensures the uniqueness of prey-eradication periodic solution. If we add the density dependent to the (1.1), that is, consider ̇𝑥1=𝑥1𝑟1𝑎11𝑥1𝑎12𝑥2,̇𝑥2=𝑥2𝑟2+𝑎21𝑥1𝑎22𝑥2,(5.1) where all the parameters are positive. Let 𝑓2 be the right-hand side of the second equation. As defined in Section 3, we can compute that 𝐹(𝑢(𝑡))=𝑓2(𝜀,𝑢(𝑡))𝐷𝑓2(𝜀,𝑢(𝑡))𝑢(𝑡)=𝑎22𝑢(𝑡).(5.2)

Thus, (H4) may be easily satisfied and it is also easy to apply our results to the above Lotka-Volterra system when introducing some more general practical impulsive effects.

Acknowledgments

The authors are grateful to the reviewers for their valuable comments and suggestions that greatly improved the presentation of this paper. This work was supported by Chonqqing Natural Science Foundation (CSTC, 2008BB0009) and Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China.

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