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

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 [1–7], 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 𝑥̇₁=𝑥̇₂𝑥₁𝑟₁−𝑎₁₁𝑥₁−𝑎₁₂𝑥₂,=𝑥₂−𝑟₂+𝑎₂₁𝑥₁(1.1) can be developed by introducing a constant periodic impulsive immigration for the predator. That is 𝑥̇₁=𝑥₁𝑥𝑟₁−𝑎₁₁𝑥₁−𝑎₁₂𝑥₂,𝑥̇₂=𝑥₂−𝑟₂+𝑎₂₁𝑥₁,𝑡≠𝑛𝜏,₁𝑡⁺=𝑥𝑥₁(𝑡0),𝑥₂𝑡⁺=𝑥₂(𝑡)+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/𝑎₁₁≤𝑟₂/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): 𝑥𝑥̇₁=𝑥₁𝑟₁−𝑎₁₁𝑥₁−𝑎₁₂𝑥₂,𝑥̇₂=𝑥₂−𝑟₂+𝑎₂₁𝑥₁,𝑡≠𝑛𝜏,₁𝑡⁺=1−𝑝₁𝑥𝑥₁(𝑡0),𝑥₂𝑡⁺=1−𝑝₂𝑥₂(𝑡)+𝑏,𝑡=𝑛𝜏,₊=𝑥₀=𝑥₀₁,𝑥₀₂,(1.3) where 0≤𝑝_𝑖<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: 𝑥𝑥̇₁=𝑥₁𝑓₁𝑥₁,𝑥₂,𝑥̇₂=𝑥₂𝑓₂𝑥₁,𝑥₂,𝑡≠𝑛𝜏,₁𝑡⁺=𝐼₁𝑥𝑥₁(0𝑡),𝑥₂𝑡⁺=𝐼₂𝑥₂(𝑡),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⁺={𝑥∈𝑅²∣𝑥≥0}, 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 𝑉𝑉(₀𝑛𝜏=₊{𝑉∶𝑅₊×𝑅²⁺↦𝑅₊∣𝑉iscontinuouson(𝑛𝜏,(𝑛+1)𝜏]×𝑅²⁺andlim_{(𝑡,𝑦)→(𝑛𝜏,𝑥),𝑡>𝑛𝜏}𝑉(𝑡,𝑦)=,𝑥)exists}.

Definition 2.1. 𝑉∈𝑉₀, 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)=limsup_ℎ→0+1ℎ[𝑉(𝑡+ℎ,𝑥+ℎ𝐹(𝑥))−𝑉(𝑡,𝑥)].(2.1)
We will assume the following basic conditions for system (1.4) hold throughout this paper. (A1)𝑓_𝑖∶×𝑅²⁺↦𝑅₊ is differentiable and 𝜕𝑓_𝑖/𝜕𝑥_𝑖≤0, 𝑖=1,2, 𝜕𝑓₁/𝜕𝑥₂≤0, 𝜕𝑓₂/𝜕𝑥₁≥0.(A2)𝐼_𝑖∶𝑅₊↦𝑅₊ is continuous, 𝐼_𝑖(0)≥0, 𝐼_𝑖(𝑢)>0 for 𝑢>0, and 𝐼_𝑖 is nondecreasing for 𝑖=1,2.
The solution of system (1.4) is a piecewise continuous function 𝑥∶𝑅₊↦𝑅²⁺, 𝑥(𝑡) is continuous on (𝑛𝜏,(𝑛+1)𝜏], 𝑛∈𝑁, and 𝑥(𝑛𝜏⁺)=lim_{𝑡→𝑛𝜏+}𝑥(𝑡) exists. Obviously the smoothness properties of 𝑓_𝑖 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 𝑥̇_𝑖(𝑡)=0 whenever 𝑥_𝑖(𝑡)=0, 𝑖=1,2, 𝑡≠𝑛𝜏, 𝑛∈𝑁 and 𝑥₁(𝑛𝜏⁺)=𝐼₁(𝑥₁(𝑛𝜏)), 𝑥₂(𝑛𝜏⁺)=𝐼₂(𝑥₂(𝑛𝜏)), 𝑛∈𝑁, 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 𝑡≥0. And further 𝑥(𝑡)>0, 𝑡≥0 if 𝑥(0⁺)>0.

Definition 2.3. System (1.4) is said to be permanent if there exist constants 𝑀≥𝑚>0 such that 𝑚≤𝑥_𝑖(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 𝑔∶𝑅₊×𝑅₊↦𝑅₊ satisfies the following condition (H). (H)𝑔 is continuous in (𝑛𝜏,(𝑛+1)𝜏]×𝑅₊ and for 𝑥∈𝑅₊, 𝑛∈𝑁, lim_{(𝑡,𝑦)→(𝑛𝜏+,𝑥)}𝑔(𝑡,𝑦)=𝑔(𝑛𝜏⁺,𝑥) exists.

Lemma 2.4. Let 𝑉∈𝑉₀. Assume that 𝑉𝐷𝑡⁺,𝑥𝑉(𝑡𝑡₊,𝑥)≤𝑔(𝑡,𝑉(𝑡,𝑥)),𝑡≠𝑛𝜏,≤𝜓_𝑛(𝑉(𝑡,𝑥(𝑡))),𝑡=𝑛𝜏,(2.2) where 𝑔∶𝑅₊×𝑅₊↦𝑅₊ satisfies (H) and 𝜓_𝑛∶𝑅₊↦𝑅₊ is nondecreasing. Let 𝑟(𝑡) be the maximal solution of the scalar impulsive differential equation 𝑢𝑡𝑢̇=𝑔(𝑡,𝑢),𝑡≠𝑛𝜏,₊=𝑢𝜓_𝑛0(𝑢(𝑡)),𝑡=𝑛𝜏,₊=𝑢₀(2.3) existing on [0,∞). Then, 𝑉(0⁺,𝑥₀)≤𝑢₀ implies that 𝑉(𝑡,𝑥(𝑡))≤𝑟(𝑡),𝑡≥0, 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), 𝑉∈𝑉₀ such that the following conditions hold.
(i) 𝑉(𝑡,𝑥)≥𝑐₁𝑥₁+𝑐₂𝑥₂, for some 𝑐₁,𝑐₂>0;(ii)𝑉𝐷⁺𝑉𝑡,𝑥(𝑡,𝑡𝑥₊(𝑡))≤−𝜆𝑉(𝑡)+𝐾,𝑡≠𝑛𝜏,≤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𝑢𝑡̇=𝑢𝑓₂(𝜀,𝑢),𝑡≠𝑛𝜏,₊𝑢=0𝐼₂(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 0≤𝑢≤𝛼₂. Then, there exists 𝛿₂>0 such that 𝑢(𝜏⁺)≥𝑢₀ for 𝑢₀≤𝛿₂, where 𝑢(𝑡) is the solution of (3.2).

Proof. Let 𝛿₂=min{𝑏₂,𝛼₂,𝛼₂/exp(𝜏𝑓₂(𝜀,0))} and 𝑢₀≤𝛿₂. Obviously, 𝑢(𝑡) is positive for 𝑡>0. By (A1), we have 𝑢̇≤𝑢𝑓₂(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))≤𝑢₀≤𝛿₂≤𝛼₂, 𝑡∈(0,𝜏] if 𝑓₂(𝜀,0)<0. By (3.2), we have 𝑢(𝜏)=𝑢₀exp^𝜏0𝑓₂(𝜀,𝑢(𝑠))𝑑𝑠.(3.4) Hence, by (H2), 𝑢𝜏⁺≥𝑎₂𝑢₀exp^𝜏0𝑓₂(𝜀,𝑢(𝑠))𝑑𝑠+𝑏₂≥𝑏₂≥𝛿₂≥𝑢₀.(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 𝐴₂exp𝜏𝑓₂(0,0)<1,𝑓₂(0,𝑢)<0(3.6)and 𝐼₂(𝑢)≤𝐴₂𝑢+𝐵₂ for 𝑢≥𝛽₂.
Then, there exist 𝜀₂>0,𝑀₂>0 such that 𝑢(𝜏⁺)≤𝑢₀ for 𝑢₀≥𝑀₂, where 𝑢(𝑡) is the solution of (3.2) with 0≤𝜀≤𝜀₂.

Proof. By (3.6), there exists 𝜀₂>0 such that 𝐴₂exp𝜏𝑓₂(𝜀,0)<1,𝑓₂(𝜀,𝑢)<0(3.7) for 0≤𝜀≤𝜀₂ and 𝑢≥𝛽₂. Let 𝑀=max{𝐼₂(𝑢)∣0≤𝑢≤𝛽₂} and 𝑀₂=max{𝑀,𝛽₂,𝐵₂/(1−𝐴₂exp(𝜏𝑓₂(𝜀,0)))}>0. Let 𝑢₀≥𝑀₂ and 𝑢(𝑡) be the solution of (3.2) with 0≤𝜀≤𝜀₂. 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, 𝑢𝜏⁺=𝐼₂(𝑢(𝜏))≤𝑀≤𝑀₂≤𝑢₀.(3.8)
Case  2. 𝑢(𝑡)≥𝛽₂,𝑡∈(0,𝜏].
By (A1), we have. 𝑢̇≤𝑢𝑓₂(𝜀,0),𝑡∈(0,𝜏).(3.9) Hence, 𝑢(𝜏)≤𝑢₀exp𝜏𝑓₂(𝜀,0).(3.10) Therefore, 𝑢𝜏⁺≤𝐴₂𝑢₀exp𝜏𝑓₂(𝜀,0)+𝐵₂=𝑢₀+𝐵₂−1−𝐴₂exp𝜏𝑓₂(𝜀,0)𝑢₀≤𝑢₀.(3.11) This completes the proof.

Let 0<𝑢¹⁰≤𝛿₂ and 𝑢²⁰≥𝛽₂ 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 𝑃∶𝑃𝑢¹⁰𝑢,₀𝑢²⁰⟼𝑢¹⁰,𝑢²⁰,=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 𝑢^𝜀(𝑡) for each 0≤𝜀≤𝜀₂.

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<𝑢₀≤𝛿₂ (𝑢₀≥𝛽₂) 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<𝑢₀≤𝛿₂ and 𝑢₀≥𝛽₂ 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⁺)=𝑢¹⁰≤𝛼₂ and 𝑢₂(0⁺)=𝑢²⁰≥𝛽₂, respectively, such that 𝑢¹⁰≤𝑢₀≤𝑢²⁰. 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 𝑅^𝑛+.

Definition 3.5. An operator 𝑈∶𝑅^𝑛+↦𝑅^𝑛+ is strongly concave on 𝑅^𝑛+ if for any 𝑥∈𝑅^𝑛+, 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 0≤𝜀≤𝜀₂, system (3.2) has a unique positive 𝜏-periodic solution 𝑢^𝜀(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 𝐴₁exp^𝜏0𝑓₁0,𝑢⁰(𝑠)𝑑𝑠<1,(3.14)
where 𝑢⁰(𝑡) is the unique positive periodic solution of (3.2) for 𝜀=0.
Then lim_𝑡→∞𝑥₁(𝑡)=0, lim_𝑡→∞|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(𝜎^0𝜂)<1, where 𝜎^0𝜂=∫^𝜏0𝑓₁(0,𝑢⁰(𝑠)−𝜂)𝑑𝑠. Note that 𝑥̇₂≥𝑥₂𝑓₂(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 𝑡≥𝑡₁. For simplification and without loss of generality, we may assume (3.15) holds for all 𝑡≥0. By (A1), we get 𝑥̇₁≤𝑥₁𝑓₁0,𝑢⁰(𝑡)−𝜂(3.16) which leads to 𝑥₁((𝑛+1)𝜏)≤=𝑥𝑥₁₁𝑛𝜏⁺exp^{(𝑛+1)𝜏𝑛𝜏}𝑓₁0,𝑢⁰(𝑠)−𝜂𝑑𝑠≤𝑥₁𝑛𝜏⁺exp𝜎^0𝜂(𝑛𝜏)𝐴₁exp𝜎^0𝜂.(3.17) Hence 𝑥₁(𝑛𝜏)≤𝑥₁(𝜏)(𝐴₁exp(𝜎^0𝜂))^𝑛−1 and 𝑥₁(𝑛𝜏)→0 as 𝑛→∞. Therefore, 𝑥₁(𝑡)→0 as 𝑡→∞ since 0<𝑥₁(𝑡)≤=𝑥𝑥₁₁𝑛𝜏⁺exp^𝑡𝑛𝜏𝑓₁0,𝑢⁰(𝑠)−𝜂𝑑𝑠(𝑛𝜏)𝐴₁exp^𝜏0||𝑓₁0,𝑢⁰(𝑠)−𝜂||𝑑𝑠(3.18) for 𝑛𝜏<𝑡≤(𝑛+1)𝜏. Now, we prove that |𝑥₂(𝑡)−𝑢⁰(𝑡)|→0 as 𝑡→∞. For 0<𝜀≤𝜀₂ in Theorem 3.6, there exists a 𝑡₁>0 such that 0<𝑥₁(𝑡)<𝜀, 𝑡≥𝑡₁. Without loss of generality, we may assume that 0<𝑥₁(𝑡)<𝜀 for all 𝑡≥0. Then, by (A1), we have 𝑥₂𝑓₂0,𝑥₂≤𝑥̇₂≤𝑥₂𝑓₂𝜀,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)−𝑢^𝜀(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)<𝑢^𝜀(𝑡)+𝜂(3.20) for large 𝑡. Let 𝜀→0, we get 𝑢⁰(𝑡)−𝜂<𝑥₂(𝑡)≤𝑢⁰(𝑡)+𝜂(3.21) for large 𝑡, which implies lim_𝑡→∞|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 𝑎₁exp^𝜏0𝑓₁0,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 𝑥₀∈int𝑅²⁺. From Theorem 3.1, we may assume 𝑥_𝑖(𝑡)≤𝑀, 𝑡≥0, 𝑖=1,2 where 𝑀 is a positive constant independent of initial values. Since 𝑥̇₂≥𝑥₂𝑓₂0,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<𝑚₃≤𝜀₂, 𝜂₁>0 be small enough such that 𝜎=𝑎₁exp(∫^𝜏0𝑓₂(𝑚₃,𝑢^𝑚3(𝑠)+𝜂₁)𝑑𝑠)>1. We will prove 𝑥₁(𝑡)≤𝑚₃ cannot hold for all 𝑡≥0. Otherwise, 𝑥̇₂≤𝑥₂𝑓₂𝑚₃,𝑥₂.(3.24)
By Lemma 2.4 and Theorem 3.6, we have 𝑥₂(𝑡)≤𝑢(𝑡) and |𝑢(𝑡)−𝑢^𝑚3(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 𝑥𝑥̇₂₁(𝑡)≤𝑢(𝑡)≤𝑢^𝑚3(𝑡)+𝜂₁,(3.25)≥𝑥₁𝑓₁𝑥₁,𝑢^𝑚3(𝑡)+𝜂₁(3.26) for 𝑡≥𝑇₁.
Let 𝑁₁∈𝑁 and 𝑁₁𝜏≥𝑇₁. Integrating (3.26) on (𝑛𝜏,(𝑛+1)𝜏] for 𝑛≥𝑁₁, we have 𝑥₁((𝑛+1)𝜏)≥=𝑥𝑥₁₁𝑛𝜏⁺exp^{(𝑛+1)𝜏𝑛𝜏}𝑓₁𝑚₃,𝑢^𝑚3(𝑠)+𝜂₁𝑑𝑠≥𝑥₁𝑛𝜏⁺exp^𝜏0𝑓₁𝑚₃,𝑢^𝑚3(𝑠)+𝜂₁𝑑𝑠=𝑥₁(𝑛𝜏)𝑎₁exp^𝜏0𝑓₁𝑚₃,𝑢^𝑚3(𝑠)+𝜂₁𝑑𝑠(𝑛𝜏)𝜎.(3.27) Then, 𝑥₁((𝑁₁+𝑘)𝜏)≥𝑥₁(𝑁₁𝜏)𝜎^𝑘→∞ as 𝑘→∞, which is a contradiction. Hence, there exists a 𝑡₁>0 such that 𝑥₁(𝑡₁)>𝑚₃.
(2) If 𝑥₁(𝑡)>𝑚₃ for all 𝑡≥𝑡₁, then our aim is obtained. Hence we need only to consider those solutions which leave the region 𝑅={𝑥∈𝑅²⁺∣𝑥₁≤𝑚₃} and reenter it again. Let 𝑡^∗=inf_𝑡≥𝑡1{𝑥₁(𝑡)≤𝑚₃}. Then, 𝑥₁(𝑡)>𝑚₃ for 𝑡∈[𝑡₁,𝑡^∗) and 𝑥₁(𝑡^∗)≥𝑚₃. Suppose 𝑡^∗∈(𝑛₁𝜏,(𝑛₁+1)𝜏],𝑛₁∈𝑁. Let 𝑢_𝑀(𝑡) be the solution of (3.2) with 𝜀=𝑚₃ and 𝑢_𝑀((𝑛₁+1)𝜏⁺)=𝑀. Then there is 𝑡₂>0 such that 𝑢_𝑀(𝑡)<𝑢^𝑚3(𝑡)+𝜂₁ for 𝑡≥𝑡₂+(𝑛₁+1)𝜏. Note that 𝑡₂ is independent of 𝑥(𝑡). Select 𝑛₂,𝑛₃∈𝑁 such that 𝑛₂𝜏>𝑡₂ and𝑎^𝑛2₁exp−||𝜎₁||+𝑛₂𝜎₁𝜎^𝑛3>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 𝑥̇₂≤𝑓₂𝑚₃,𝑥₂.(3.29) By Lemma 2.4, we have 𝑥₂(𝑡)≤𝑢_𝑀(𝑡),𝑡∈𝑛₁+1𝜏,𝑛₁+1+𝑛₂+𝑛₃𝜏.(3.30) Then, 𝑥₂(𝑡)≤𝑢_𝑀(𝑡)<𝑢^𝑚3(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 𝑥₁𝑛₁+1+𝑛₂+𝑛₃𝜏≥𝑥₁𝑛₁+1+𝑛₂𝜏𝜎^𝑛3.(3.32) Since 𝑥̇₁≥𝑥₁𝑓₁𝑚₃,𝑀,𝑡∈𝑡^∗,𝑛₁+1+𝑛₂+𝑛₃𝜏,(3.33) integrating it on [𝑡^∗,(𝑛₁+1+𝑛₂)𝜏], we have 𝑥₁𝑛₁+1+𝑛₂𝜏≥𝑚₃𝑎^𝑛2₁exp−||𝜎₁||+𝑛₂𝜎₁.(3.34) Thus, by (3.28), we have 𝑥₁𝑛₁+1+𝑛₂+𝑛₃𝜏≥𝑚₃𝑎^𝑛2₁exp−||𝜎₁||+𝑛₂𝜎₁𝜎^𝑛3>𝑚₃,(3.35) which is a contradiction. Let 𝑡=inf_{𝑡≥𝑡∗}{𝑥₁(𝑡)>𝑚₃}. Then, 𝑡∈(𝑡^∗,(𝑛₁+1+𝑛₂+𝑛₃)𝜏]. Denote 𝑎=min{1,𝑎^𝑛2+𝑛3₁}. For 𝑡∈(𝑡^∗,𝑡], we have 𝑥₁(𝑡)≥𝑥₁𝑡^∗𝑎exp−1+𝑛₂+𝑛₃||𝜎₁||≥𝑚₃𝑎exp−1+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 𝑡≥𝑡₁. 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: 𝑥𝑡𝑥̇=𝐹(𝑥),𝑡≠𝑛𝜏,₊=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 𝑥𝑥̇₁=𝑔𝑥₁=𝐹₁𝑥₁,0,𝑡≠𝑛𝜏,₁𝑡⁺=𝜃𝑥₁(𝑡)=Θ₁𝑥₁(𝑡),0,𝑡=𝑛𝜏(3.38) has a periodic solution 𝑥_𝑠(𝑡). Thus, 𝜁=(𝑥_s,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<𝑡≤𝜏, where 𝑥₀=𝑥(0⁺). Now, we list following notations we will use from [9]: 𝑑^0=1−𝜕Θ²𝜕𝑥₂𝜕Φ₂𝜕𝑥₂𝜏₀𝑎,𝑥₀,where𝜏₀istherootof𝑑^0=0,_0=1−𝜕Θ¹𝜕𝑥₁𝜕Φ₁𝜕𝑥₁𝜏⁰𝑏_0,𝑥₀,=−𝜕Θ¹𝜕𝑥₁𝜕Φ₁𝜕𝑥₂+𝜕Θ¹𝜕𝑥₂𝜕Φ₂𝜕𝑥₂𝜏⁰𝜕Φ₁,𝑥₀,𝑡,𝑥₀𝜕𝑥₁=exp^𝑡0𝜕𝐹₁(𝜁(𝑟))𝜕𝑥₁𝜕Φ₂𝑑𝑟,𝑡,𝑥₀𝜕𝑥₂=exp^𝑡0𝜕𝐹₂(𝜁(𝑟))𝜕𝑥₂𝜕Φ₁𝑑𝑟,𝑡,𝑥₀𝜕𝑥₂=^𝑡0exp^𝑡𝑢𝜕𝐹₁(𝜁(𝑟))𝜕𝑥₁𝑑𝑟𝜕𝐹¹(𝜁(𝑢))𝜕𝑥₂exp^𝑢0𝜕𝐹₂(𝜁(𝑟))𝜕𝑥₂𝜕𝑑𝑟𝑑𝑢,₂Φ₂𝑡,𝑥₀𝜕𝑥₁𝜕𝑥₂=^𝑡0exp^𝑡𝑢𝜕𝐹₂(𝜁(𝑟))𝜕𝑥₂𝑑𝑟𝜕²𝐹₂(𝜁(𝑢))𝜕𝑥₁𝜕𝑥₂exp^𝑢0𝜕𝐹₂(𝜁(𝑟))𝜕𝑥₂𝜕𝑑𝑟𝑑𝑢,₂Φ₂𝑡,𝑥₀𝜕𝑥²²=^𝑡0exp^𝑡𝑢𝜕𝐹₂(𝜁(𝑟))𝜕𝑥₂𝑑𝑟𝜕²𝐹₂(𝜁(𝑢))𝜕𝑥²²exp^𝑢0𝜕𝐹₂(𝜁(𝑟))𝜕𝑥₂+𝑑𝑟𝑑𝑢_𝑡0exp^𝑡𝑢𝜕𝐹₂(𝜁(𝑟))𝜕𝑥₂𝑑𝑟𝜕²𝐹₂(𝜁(𝑢))×_𝑢0𝜕𝑥₂𝜕𝑥₁exp^𝑢𝑝𝜕𝐹₁(𝜁(𝑟))𝜕𝑥₁𝑑𝑟𝜕𝐹¹(𝜁(𝑢))𝜕𝑥₂exp^𝑝0𝜕𝐹₂(𝜁(𝑟))𝜕𝑥₂𝜕𝑑𝑟𝑑𝑝𝑑𝑢,₂Φ₂𝑡,𝑥₀𝜕𝜏𝜕𝑥₂=𝜕𝐹²(𝜁(𝑡))𝜕𝑥₂exp^𝑡0𝜕𝐹₂(𝜁(𝑟))𝜕𝑥₂𝜕Φ₁𝑑𝑟,𝜏₀,𝑥₀𝜕𝜏𝐵==𝑥̇𝜕_𝑠2𝜏₀Θ₂𝜕𝑥₁𝜕𝑥₂𝜕Φ¹𝜏₀,𝑥₀𝜕𝜏+𝜕Φ¹𝜏₀,𝑥₀𝜕𝑥₁1𝑎^0𝜕Θ₁𝜕𝑥₁𝜕Φ₁𝜏₀,𝑥₀𝜕𝑡𝑎𝑢𝜕Φ²𝜏₀,𝑥₀𝜕𝑥₂−𝜕Θ²𝜕𝑥₂𝜕²Φ₂𝜏₀,𝑥₀𝜕𝜏𝜕𝑥₂+𝜕²Φ₂𝜏₀,𝑥₀𝜕𝑥₁𝜕𝑥₂1𝑎^0𝜕Θ₁𝜕𝑥₁𝜕Φ₁𝜏₀,𝑥₀𝐶=2𝜕₂𝜕𝜏,Θ₂𝜕𝑥₁𝜕𝑥₂−𝑎𝑏^00𝜕Φ₁𝜏₀,𝑥₀𝜕𝑥₁+𝜕Φ¹𝜏₀,𝑥₀𝜕𝑥₂×𝜕Φ²𝜏₀,𝑥₀𝜕𝑥₂−𝜕²Θ₂𝜕𝑥²²𝜕Φ²𝜏₀,𝑥₀𝜕𝑥₂²+2𝜕Θ²𝜕𝑥₂𝑏^0𝑎^0𝜕²Φ₂𝜏₀,𝑥₀𝜕𝑥₂𝜕𝑥₁−𝜕Θ²𝜕𝑥₂𝜕²Φ₂𝜏₀,𝑥₀𝜕𝑥²².(3.39)

Theorem 3.10 (see [9]). If |1−𝑎^0|<1 and 𝑑^0=0, then one has the following: (a)If 𝐵𝐶≠0, 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 𝑥_𝑖(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<𝜆≤𝑟₂. Then, when 𝑡≠𝑛𝜏, 𝐷⁺𝑉(𝑡)+𝜆𝑉(𝑡)=𝑎₂₁𝑥₁𝑟₁+𝜆−𝑎₁₁𝑥₁−𝑎₁₂𝑟₂−𝜆𝑥₂≤𝑎₂₁𝑥₁𝑟₁+𝜆−𝑎₁₁𝑥₁≤𝐾,(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, lim_𝑡→∞𝑥₁(𝑡)=0, lim_𝑡→∞|𝑥₂(𝑡)−𝑥^∗2(𝑡)|=0 if 𝑏>𝑟₁𝑟₂𝜏/𝑎₁₂, where 𝑥^∗2(𝑡)=𝑏exp(−𝑟₂(𝑡−𝑛𝜏))/(1−exp(−𝑟₂𝜏)), 𝑡∈(𝑛𝜏,(𝑛+1)𝜏], 𝑛∈𝑁, 𝑥^∗2(0⁺)=𝑏/(1−exp(−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 𝑥^∗2(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,lim_𝑢→∞(𝑃(𝑢)−𝑢)=−∞ and 𝑃(𝑢)−𝑢 is strictly decreasing. Since 𝐼₁(𝑥₁)=𝑥₁ and exp^𝜏0𝑓₁0,𝑥^∗2(𝑠)𝑑𝑠=exp^𝜏0𝑟₁−𝑎₁₂𝑥^∗2(𝑠)𝑑𝑠=exp𝑟₁𝜏−𝑎¹²𝑏𝑟₂<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 𝑥₀∈int𝑅²⁺. Obviously, (H1)–(H3) and (H4^) are satisfied. Since 𝐼₁(𝑥₁)=𝑥₁ and exp^𝜏0𝑓₁0,𝑥^∗2(𝑠)𝑑𝑠=exp^𝜏0𝑟₁−𝑎₁₂𝑥^∗2(𝑠)𝑑𝑠=exp𝑟₁𝜏−𝑎¹²𝑏𝑟₂>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,𝑥^∗2(𝑡))^𝑇. We also exchange the subscripts of 𝑥₁ and 𝑥₂ as in Theorem 3.10. Thus, 𝐹₂𝐹₁𝑥₁,𝑥₂=𝑥₁−𝑟₂+𝑎₂₁𝑥₂,𝑥₁,𝑥Θ₂₁=𝑥₂𝑟₁−𝑎₁₁𝑥₂−𝑎₁₂𝑥₁,Θ𝑥₁,𝑥₂=𝑥₁+𝑏,₂𝜁(𝑡)=𝑥_𝑠𝑥₁,𝑥₂=𝑥₂,(𝑡),0^𝑇=𝑥^∗2(𝑡),0.(4.4) Then, we can compute that 𝑎𝑏^0=1−exp−𝑟₂𝜏₀>0,_0=−𝑎²¹𝑏exp−𝑟₂𝜏₀1−exp−𝑟₂𝜏₀^𝜏00exp𝑟₁𝑢−𝑎¹²𝑏1−exp−𝑟₂𝑢𝜕𝑟₂1−exp−𝑟₂𝜏₀𝑑𝑢<0,₂Φ₂𝜏₀,𝑥₀𝜕𝜏𝜕𝑥₂=𝑟₁−𝑎₁₂𝑏exp−𝑟₂𝜏₀𝜕₂1−exp−𝑟₂𝜏₀,Φ₂𝜏₀,𝑥₀𝜕𝑥₁𝜕𝑥₂𝜕=−𝑎₁₂𝜏₀<0,₂Φ₂𝜏₀,𝑥₀𝜕𝑥²²=−𝑎₁₁𝜏₀−^𝜏0₀𝑎¹²𝑎₂₁𝑏exp−𝑟₂𝑢exp(ℝ)×_𝑢01−exp−𝑟₂𝜏₀exp𝑟₁𝑝−𝑎¹²𝑏1−exp−𝑟₂𝑝𝜕Φ₁𝑟₂1−exp−𝑟₂𝜏₀𝑑𝑝𝑑𝑢<0,𝜏₀,𝑥₀𝜕𝜏=𝑥̇^𝑠𝜏₀=−𝑟²𝑏exp−𝑟₂𝜏₀1−exp−𝑟₂𝜏₀<0,(4.5) where ℝ denotes 𝑟₁(𝜏₀−𝑢)−𝑎₁₂𝑏(exp(−𝑟₂𝑢)−exp(−𝑟₂𝜏₀))/(𝑟₂(1−exp(−𝑟₂𝜏₀))).

Since 𝜕Θ_𝑖/𝜕𝑥_𝑗=0, 𝑖≠𝑗, 𝜕Θ_𝑖/𝜕𝑥_𝑖=1, 𝜕²Θ_𝑖/𝜕𝑥₁𝜕𝑥₂=0, 𝑖=1,2 and 𝜕²Θ₂/𝜕𝑥²²=0, it is easy to verify that 𝐶>0 and 𝐵=−𝑟₁−𝑎₁₂𝑏exp−𝑟₂𝜏₀1−exp−𝑟₂𝜏₀+𝑎¹²𝜏₀𝑟₂𝑏exp−𝑟₂𝜏₀1−exp−𝑟₂𝜏₀².(4.6) To determine the sign of 𝐵, let 𝜙(𝑡)=𝑟₁−𝑎₁₂𝑏exp(−𝑟₂𝑡)/(1−exp(−𝑟₂𝜏₀)). We have 𝑑𝜙/𝑑𝑡=𝑟₂𝑎₁₂𝑏exp(−𝑟₂𝑡)/(1−exp(−𝑟₂𝜏₀))>0. Thus, we conclude that 𝜙(𝜏₀)>0 since ∫^𝜏0₀𝜙(𝑡)𝑑𝑡=𝑟₁𝜏₀−𝑎₁₂𝑏/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 𝑑^0=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 𝑥̂(𝑡) with 𝑥̂(0⁺)=𝑥̂₀=(𝑥̂₀₁,𝑥̂₀₂)^𝑇. Then, (1/𝜏)∫^𝜏0𝑥̂₁(𝑡)𝑑𝑡<min{𝑟₁/𝑎₁₁,𝑟₂/𝑎₂₁} and (1/𝜏)∫^𝜏0𝑥̂₂(𝑡)𝑑𝑡<𝑟₁/𝑎₁₂.

Proof. Since 𝑥̂(𝑡) is a positive periodic solution of system (1.2), we have 𝑥̂(0⁺)=𝑥̂(𝜏⁺) which gives 𝑥̂₀₁=𝑥̂₀₁exp𝑟₁𝜏−𝑎₁₁^𝜏0𝑥̂₁(𝑡)𝑑𝑡−𝑎₁₂^𝜏0𝑥̂₀₂𝑥̂₂(𝑡)𝑑𝑡,=𝑥̂₀₂exp−𝑟₂𝜏+𝑎₂₁^𝜏0𝑥̂₁(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)𝜏𝑛𝜏}𝑟₁𝑑𝑡=∫^{(𝑛+1)𝜏𝑛𝜏}𝑎₁₂𝑥^∗2(𝑡)𝑑𝑡. If there is no prey or its density is very small, the density of the predator is 𝑥^∗2(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 𝑥𝑥̇̇₁₂=𝑥₁𝑟₁−𝑎₁₁𝑥₁−𝑎₁₂𝑥₂,=𝑥₂−𝑟₂+𝑎₂₁𝑥₁−𝑎₂₂𝑥₂,(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.