Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 521729 | 19 pages | https://doi.org/10.1155/2012/521729

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

Academic Editor: A. A. Soliman
Received23 Feb 2012
Accepted18 May 2012
Published14 Aug 2012

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 Μ‡π‘₯1=π‘₯1ξ€·π‘Ÿ1βˆ’π‘Ž11π‘₯1βˆ’π‘Ž12π‘₯2ξ€Έ,Μ‡π‘₯2=π‘₯2ξ€·βˆ’π‘Ÿ2+π‘Ž21π‘₯1ξ€Έ(1.1) can be developed by introducing a constant periodic impulsive immigration for the predator. That is Μ‡π‘₯1=π‘₯1ξ€·π‘Ÿ1βˆ’π‘Ž11π‘₯1βˆ’π‘Ž12π‘₯2ξ€Έ,Μ‡π‘₯2=π‘₯2ξ€·βˆ’π‘Ÿ2+π‘Ž21π‘₯1ξ€Έπ‘₯,π‘‘β‰ π‘›πœ,1𝑑+ξ€Έ=π‘₯1(𝑑),π‘₯2𝑑+ξ€Έ=π‘₯2(π‘₯ξ€·0𝑑)+𝑏,𝑑=π‘›πœ,+ξ€Έ=π‘₯0=ξ€·π‘₯01,π‘₯02ξ€Έ,(1.2) where π‘₯1(𝑑), π‘₯2(𝑑) are the densities of the prey and predator at time 𝑑, respectively, π‘Ÿ1 is the intrinsic growth rate of prey, π‘Ÿ2 is the death rate of predator, π‘Ž11 is the rate of intraspecific competition or density dependence, π‘Ž12 is the per capita rate of predation of the predator, π‘Ž21 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 (π‘Ÿ1/π‘Ž11>π‘Ÿ2/π‘Ž21), in which case it is global asymptotically stable. Or there is no positive equilibrium (π‘Ÿ1/π‘Ž11β‰€π‘Ÿ2/π‘Ž21), in which case π‘₯2(𝑑) goes extinct and π‘₯1(𝑑) tends to π‘Ÿ1/π‘Ž11, 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): Μ‡π‘₯1=π‘₯1ξ€·π‘Ÿ1βˆ’π‘Ž11π‘₯1βˆ’π‘Ž12π‘₯2ξ€Έ,Μ‡π‘₯2=π‘₯2ξ€·βˆ’π‘Ÿ2+π‘Ž21π‘₯1ξ€Έπ‘₯,π‘‘β‰ π‘›πœ,1𝑑+ξ€Έ=ξ€·1βˆ’π‘1ξ€Έπ‘₯1(𝑑),π‘₯2𝑑+ξ€Έ=ξ€·1βˆ’π‘2ξ€Έπ‘₯2(π‘₯ξ€·0𝑑)+𝑏,𝑑=π‘›πœ,+ξ€Έ=π‘₯0=ξ€·π‘₯01,π‘₯02ξ€Έ,(1.3) where 0≀𝑝𝑖<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: Μ‡π‘₯1=π‘₯1𝑓1ξ€·π‘₯1,π‘₯2ξ€Έ,Μ‡π‘₯2=π‘₯2𝑓2ξ€·π‘₯1,π‘₯2ξ€Έπ‘₯,π‘‘β‰ π‘›πœ,1𝑑+ξ€Έ=𝐼1ξ€·π‘₯1(𝑑),π‘₯2𝑑+ξ€Έ=𝐼2ξ€·π‘₯2(ξ€Έπ‘₯ξ€·0𝑑),𝑑=π‘›πœ,+ξ€Έ=π‘₯0=ξ€·π‘₯01,π‘₯02ξ€Έ.(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+={π‘₯βˆˆπ‘…2∣π‘₯β‰₯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 𝑉0={π‘‰βˆΆπ‘…+×𝑅2+↦𝑅+βˆ£π‘‰iscontinuouson(π‘›πœ,(𝑛+1)𝜏]×𝑅2+andlim(𝑑,𝑦)β†’(π‘›πœ,π‘₯),𝑑>π‘›πœπ‘‰(𝑑,𝑦)=𝑉(π‘›πœ+,π‘₯)exists}.

Definition 2.1. π‘‰βˆˆπ‘‰0, then for (𝑑,π‘₯)∈(π‘›πœ,(𝑛+1)𝜏]×𝑅2+, the upper right derivative of 𝑉(𝑑,π‘₯) with respect to the impulsive differential system (1.4) is defined as 𝐷+𝑉(𝑑,π‘₯)=limsupβ„Žβ†’0+1β„Ž[].𝑉(𝑑+β„Ž,π‘₯+β„ŽπΉ(π‘₯))βˆ’π‘‰(𝑑,π‘₯)(2.1)
We will assume the following basic conditions for system (1.4) hold throughout this paper. (A1)π‘“π‘–βˆΆΓ—π‘…2+↦𝑅+ is differentiable and πœ•π‘“π‘–/πœ•π‘₯𝑖≀0, 𝑖=1,2, πœ•π‘“1/πœ•π‘₯2≀0, πœ•π‘“2/πœ•π‘₯1β‰₯0.(A2)πΌπ‘–βˆΆπ‘…+↦𝑅+ is continuous, 𝐼𝑖(0)β‰₯0, 𝐼𝑖(𝑒)>0 for 𝑒>0, and 𝐼𝑖 is nondecreasing for 𝑖=1,2.
The solution of system (1.4) is a piecewise continuous function π‘₯βˆΆπ‘…+↦𝑅2+, π‘₯(𝑑) is continuous on (π‘›πœ,(𝑛+1)𝜏], π‘›βˆˆπ‘, and π‘₯(π‘›πœ+)=limπ‘‘β†’π‘›πœ+π‘₯(𝑑) 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 π‘₯1 and π‘₯2 can be the densities of the prey and the predator at time 𝑑, 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 Μ‡π‘₯𝑖(𝑑)=0 whenever π‘₯𝑖(𝑑)=0, 𝑖=1,2, π‘‘β‰ π‘›πœ, π‘›βˆˆπ‘ and π‘₯1(π‘›πœ+)=𝐼1(π‘₯1(π‘›πœ)), π‘₯2(π‘›πœ+)=𝐼2(π‘₯2(π‘›πœ)), π‘›βˆˆπ‘, by (A2), we have the following lemma.

Lemma 2.2. Suppose π‘₯(𝑑) is a solution of (1.4) with π‘₯(0+)β‰₯0, then π‘₯(𝑑)β‰₯0 for all 𝑑β‰₯0. And further π‘₯(𝑑)>0, 𝑑β‰₯0 if π‘₯(0+)>0.

Definition 2.3. System (1.4) is said to be permanent if there exist constants 𝑀β‰₯π‘š>0 such that π‘šβ‰€π‘₯𝑖(𝑑)≀𝑀, 𝑖=1,2 for all t sufficiently large, where π‘₯(𝑑) is any solution of (1.4) with π‘₯(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 π‘”βˆΆπ‘…+×𝑅+↦𝑅+ satisfies the following condition (H). (H)𝑔 is continuous in (π‘›πœ,(𝑛+1)𝜏]×𝑅+ and for π‘₯βˆˆπ‘…+, π‘›βˆˆπ‘, lim(𝑑,𝑦)β†’(π‘›πœ+,π‘₯)𝑔(𝑑,𝑦)=𝑔(π‘›πœ+,π‘₯) exists.

Lemma 2.4. Let π‘‰βˆˆπ‘‰0. Assume that 𝐷+𝑉𝑑𝑉(𝑑,π‘₯)≀𝑔(𝑑,𝑉(𝑑,π‘₯)),π‘‘β‰ π‘›πœ,𝑑,π‘₯+ξ€Έξ€Έβ‰€πœ“π‘›(𝑉(𝑑,π‘₯(𝑑))),𝑑=π‘›πœ,(2.2) where π‘”βˆΆπ‘…+×𝑅+↦𝑅+ satisfies (H) and πœ“π‘›βˆΆπ‘…+↦𝑅+ is nondecreasing. Let π‘Ÿ(𝑑) be the maximal solution of the scalar impulsive differential equation 𝑒𝑑̇𝑒=𝑔(𝑑,𝑒),π‘‘β‰ π‘›πœ,+ξ€Έ=πœ“π‘›π‘’ξ€·0(𝑒(𝑑)),𝑑=π‘›πœ,+ξ€Έ=𝑒0(2.3) existing on [0,∞). Then, 𝑉(0+,π‘₯0)≀𝑒0 implies that 𝑉(𝑑,π‘₯(𝑑))β‰€π‘Ÿ(𝑑),𝑑β‰₯0, where π‘₯(𝑑) 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 π‘Ÿ(𝑑) 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(𝑑)=𝑉(𝑑,π‘₯), π‘‰βˆˆπ‘‰0 such that the following conditions hold.
(i) 𝑉(𝑑,π‘₯)β‰₯𝑐1π‘₯1+𝑐2π‘₯2, for some 𝑐1,𝑐2>0;(ii)𝐷+𝑉𝑑𝑉(𝑑,π‘₯(𝑑))β‰€βˆ’πœ†π‘‰(𝑑)+𝐾,π‘‘β‰ π‘›πœ,𝑑,π‘₯+≀𝑉(𝑑,π‘₯(𝑑))+𝑏,𝑑=π‘›πœ,(3.1)where πœ†,𝐾,𝑏 are positive constants.
Then, system (1.4) is ultimately upper bounded.

Proof. Let 𝑉(0+)=𝑉(0,π‘₯(0+)). By (i), it suffices to prove that 𝑉(𝑑) 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): ̇𝑒=𝑒𝑓2𝑒𝑑(πœ€,𝑒),π‘‘β‰ π‘›πœ,+ξ€Έ=𝐼2𝑒0(𝑒(𝑑)),𝑑=π‘›πœ,+ξ€Έ=𝑒0>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 π‘Ž2, 𝑏2, and 𝛼2such that𝐼2(𝑒)β‰₯π‘Ž2𝑒+𝑏2 for 0≀𝑒≀𝛼2. Then, there exists 𝛿2>0 such that 𝑒(𝜏+)β‰₯𝑒0 for 𝑒0≀𝛿2, where 𝑒(𝑑) is the solution of (3.2).

Proof. Let 𝛿2=min{𝑏2,𝛼2,𝛼2/exp(πœπ‘“2(πœ€,0))} and 𝑒0≀𝛿2. Obviously, 𝑒(𝑑) is positive for 𝑑>0. By (A1), we have ̇𝑒≀𝑒𝑓2(πœ€,0),π‘‘βˆˆ(0,𝜏).(3.3) Hence, 𝑒(𝑑)≀𝑒0exp(𝑑𝑓2(πœ€,0))≀𝛿2exp(πœπ‘“2(πœ€,0))≀𝛼2, π‘‘βˆˆ(0,𝜏] if 𝑓2(πœ€,0)β‰₯0 and 𝑒(𝑑)≀𝑒0exp(𝑑𝑓2(πœ€,0))≀𝑒0≀𝛿2≀𝛼2, π‘‘βˆˆ(0,𝜏] if 𝑓2(πœ€,0)<0. By (3.2), we have 𝑒(𝜏)=𝑒0ξ‚΅ξ€œexp𝜏0𝑓2ξ‚Ά(πœ€,𝑒(𝑠))𝑑𝑠.(3.4) Hence, by (H2), π‘’ξ€·πœ+ξ€Έβ‰₯π‘Ž2𝑒0ξ‚΅ξ€œexp𝜏0𝑓2ξ‚Ά(πœ€,𝑒(𝑠))𝑑𝑠+𝑏2β‰₯𝑏2β‰₯𝛿2β‰₯𝑒0.(3.5) The proof is complete.

Lemma 3.3. Suppose that the following condition holds. (H3) There exist positive constants 𝐴2, 𝐡2, and 𝛽2 such that 𝐴2ξ€·expπœπ‘“2ξ€Έ(0,0)<1,𝑓2(0,𝑒)<0(3.6)and 𝐼2(𝑒)≀𝐴2𝑒+𝐡2 for 𝑒β‰₯𝛽2.
Then, there exist πœ€2>0,𝑀2>0 such that 𝑒(𝜏+)≀𝑒0 for 𝑒0β‰₯𝑀2, where 𝑒(𝑑) is the solution of (3.2) with 0β‰€πœ€β‰€πœ€2.

Proof. By (3.6), there exists πœ€2>0 such that 𝐴2ξ€·expπœπ‘“2ξ€Έ(πœ€,0)<1,𝑓2(πœ€,𝑒)<0(3.7) for 0β‰€πœ€β‰€πœ€2 and 𝑒β‰₯𝛽2. Let 𝑀=max{𝐼2(𝑒)∣0≀𝑒≀𝛽2} and 𝑀2=max{𝑀,𝛽2,𝐡2/(1βˆ’π΄2exp(πœπ‘“2(πœ€,0)))}>0. Let 𝑒0β‰₯𝑀2 and 𝑒(𝑑) be the solution of (3.2) with 0β‰€πœ€β‰€πœ€2. There are two cases for 𝑒(𝑑), π‘‘βˆˆ(0,𝜏].
Case  1. There exists 𝑑1∈(0,𝜏] such that 𝑒(𝑑1)<𝛽2.
Let π‘‘βˆ—=inf{π‘‘βˆˆ(0,𝜏]βˆ£π‘’(𝑑)<𝛽2}. Then, 𝑒(π‘‘βˆ—)=𝛽2. Since 𝑓2(πœ€,𝛽2)<0, we can conclude that 𝑒(𝑑)<𝛽2, π‘‘βˆˆ(π‘‘βˆ—,𝜏]. Hence, π‘’ξ€·πœ+ξ€Έ=𝐼2(𝑒(𝜏))≀𝑀≀𝑀2≀𝑒0.(3.8)
Case  2. 𝑒(𝑑)β‰₯𝛽2,π‘‘βˆˆ(0,𝜏].
By (A1), we have. ̇𝑒≀𝑒𝑓2(πœ€,0),π‘‘βˆˆ(0,𝜏).(3.9) Hence, 𝑒(𝜏)≀𝑒0ξ€·expπœπ‘“2ξ€Έ(πœ€,0).(3.10) Therefore, π‘’ξ€·πœ+≀𝐴2𝑒0ξ€·expπœπ‘“2ξ€Έ(πœ€,0)+𝐡2=𝑒0+𝐡2βˆ’ξ€·1βˆ’π΄2ξ€·expπœπ‘“2(π‘’πœ€,0)ξ€Έξ€Έ0≀𝑒0.(3.11) This completes the proof.

Let 0<𝑒10≀𝛿2 and 𝑒20β‰₯𝛽2 for 𝛿2,𝛽2 in Lemmas 3.2 and 3.3, respectively. Consider the solution 𝑒(𝑑) of (3.2) with 𝑒0∈[𝑒10,𝑒20]. By Lemma 2.4, similar to [12, Theorem 3.1], we can define a map ξ€Ίπ‘’π‘ƒβˆΆ10,𝑒20ξ€»βŸΌξ€Ίπ‘’10,𝑒20ξ€»,𝑃𝑒0ξ€Έ=𝐼2(𝑒(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 πœ€2>0 such that system (3.2) has a positive 𝜏-periodic solution π‘’πœ€(𝑑) for each 0β‰€πœ€β‰€πœ€2.

Modify 𝑣(0),𝑣(𝑇) to 𝑣(0+),𝑣(𝑇+) and consider for the case with 𝑇=𝜏 and π‘ž=1 in the definition of lower and upper solutions of [26, Definition 3.1], the solution 𝑒(𝑑) of (3.2) with 0<𝑒0≀𝛿2 (𝑒0β‰₯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<𝑒0≀𝛿2 and 𝑒0β‰₯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 𝑒(𝑑) of (3.2) with initial value 𝑒0>0, we can always find a lower solution 𝑒1(𝑑) and an upper solution 𝑒2(𝑑) with initial values 0<𝑒1(0+)=𝑒10≀𝛼2 and 𝑒2(0+)=𝑒20β‰₯𝛽2, respectively, such that 𝑒10≀𝑒0≀𝑒20. Then since the solutions of system (3.2) are monotone with respect to initial values, 𝑒1(𝑑)≀𝑒(𝑑)≀𝑒2(𝑑). And since both 𝑒1(𝑑) and 𝑒2(𝑑) will tend to the unique positive 𝜏-periodic solution of (3.2), so is 𝑒(𝑑). 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 π‘ˆβˆΆπ‘…π‘›+↦𝑅𝑛+ is strongly concave on 𝑅𝑛+ if for any π‘₯βˆˆπ‘…π‘›+, π‘₯>0 and any number π‘ βˆˆ(0,1), there exists a positive number πœ‚ such that π‘ˆ(𝑠π‘₯)β‰₯(1+πœ‚)π‘ π‘ˆ(π‘₯).
Now let 𝑒(𝑑) be the solution of (3.2) with initial value 𝑒0>0. Let 𝑓2=𝑒𝑓2(πœ€,𝑒). Define 𝐹(𝑒(𝑑))=𝑓2(πœ€,𝑒(𝑑))βˆ’π·π‘“2(πœ€,𝑒(𝑑))𝑒(𝑑),(3.13) where 𝐷𝑓2 is the Jacobian matrix of 𝑓2 (here is just its derivative with respect to 𝑒 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(𝑒)>0 for any 𝑒>0 and 𝐼2 is strongly concave or is linear. Then, there exists aπœ€2>0 such that for each 0β‰€πœ€β‰€πœ€2, system (3.2) has a unique positive 𝜏-periodic solution π‘’πœ€(𝑑), which is a global attractor for any solution 𝑒(𝑑) of (3.2) with 𝑒0>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 (H4ξ…ž), which states the uniqueness directly. (H4ξ…ž)𝑃 has at most one positive fixed point.

3.3. Extinction of the Prey

Theorem 3.8. Let (π‘₯1(𝑑),π‘₯2(𝑑)) be the solution of (1.4). Suppose that (H2)–(H4) and the following (H5) hold.(H5) There exists 𝐴1>0 such that 𝐼1(π‘₯)≀𝐴1π‘₯ and 𝐴1ξ‚΅ξ€œexp𝜏0𝑓1ξ€·0,𝑒0ξ€Έξ‚Ά(𝑠)𝑑𝑠<1,(3.14)
where 𝑒0(𝑑) is the unique positive periodic solution of (3.2) for πœ€=0.
Then limπ‘‘β†’βˆžπ‘₯1(𝑑)=0, limπ‘‘β†’βˆž|π‘₯2(𝑑)βˆ’π‘’0(𝑑)|=0.

Proof. By (3.14), the continuity of 𝑓1, and the Lebesgue Theorem, we can choose an πœ‚>0 sufficiently small such that 𝐴1exp(𝜎0πœ‚)<1, where 𝜎0πœ‚=∫𝜏0𝑓1(0,𝑒0(𝑠)βˆ’πœ‚)𝑑𝑠. Note that Μ‡π‘₯2β‰₯π‘₯2𝑓2(0,π‘₯2), consider (3.2) with πœ€=0 and 𝑒(0+)=π‘₯2(0+). Let 𝑉(𝑑,π‘₯)=π‘₯2(𝑑). From Lemma 2.4 and Theorem 3.6, we have π‘₯2(𝑑)β‰₯𝑒(𝑑) and |𝑒(𝑑)βˆ’π‘’0(𝑑)|β†’0 as π‘‘β†’βˆž. Hence, there exists 𝑑1>0 such that π‘₯2(𝑑)β‰₯𝑒(𝑑)>𝑒0(𝑑)βˆ’πœ‚(3.15) for all 𝑑β‰₯𝑑1. For simplification and without loss of generality, we may assume (3.15) holds for all 𝑑β‰₯0. By (A1), we get Μ‡π‘₯1≀π‘₯1𝑓1ξ€·0,𝑒0ξ€Έ(𝑑)βˆ’πœ‚(3.16) which leads to π‘₯1((𝑛+1)𝜏)≀π‘₯1ξ€·π‘›πœ+ξ€Έξ‚΅ξ€œexp(𝑛+1)πœπ‘›πœπ‘“1ξ€·0,𝑒0ξ€Έξ‚Ά(𝑠)βˆ’πœ‚π‘‘π‘ =π‘₯1ξ€·π‘›πœ+ξ€Έξ€·πœŽexp0πœ‚ξ€Έβ‰€π‘₯1(π‘›πœ)𝐴1ξ€·πœŽexp0πœ‚ξ€Έ.(3.17) Hence π‘₯1(π‘›πœ)≀π‘₯1(𝜏)(𝐴1exp(𝜎0πœ‚))π‘›βˆ’1 and π‘₯1(π‘›πœ)β†’0 as π‘›β†’βˆž. Therefore, π‘₯1(𝑑)β†’0 as π‘‘β†’βˆž since 0<π‘₯1(𝑑)=π‘₯1ξ€·π‘›πœ+ξ€Έξ‚΅ξ€œexpπ‘‘π‘›πœπ‘“1ξ€·0,𝑒0(𝑠)βˆ’πœ‚π‘‘π‘ β‰€π‘₯1(π‘›πœ)𝐴1ξ‚΅ξ€œexp𝜏0||𝑓1ξ€·0,𝑒0ξ€Έ||ξ‚Ά(𝑠)βˆ’πœ‚π‘‘π‘ (3.18) for π‘›πœ<𝑑≀(𝑛+1)𝜏. Now, we prove that |π‘₯2(𝑑)βˆ’π‘’0(𝑑)|β†’0 as π‘‘β†’βˆž. For 0<πœ€β‰€πœ€2 in Theorem 3.6, there exists a 𝑑1>0 such that 0<π‘₯1(𝑑)<πœ€, 𝑑β‰₯𝑑1. Without loss of generality, we may assume that 0<π‘₯1(𝑑)<πœ€ for all 𝑑β‰₯0. Then, by (A1), we have π‘₯2𝑓2ξ€·0,π‘₯2≀̇π‘₯2≀π‘₯2𝑓2ξ€·πœ€,π‘₯2ξ€Έ.(3.19) From Lemma 2.4 and Theorem 3.6, we have 𝑒1(𝑑)≀π‘₯2(𝑑)≀𝑒2(𝑑) and |𝑒1(𝑑)βˆ’π‘’0(𝑑)|β†’0, |𝑒2(𝑑)βˆ’π‘’πœ€(𝑑)|β†’0 as π‘‘β†’βˆž, where 𝑒1(𝑑) is solution of (3.2) with πœ€=0, 𝑒1(0+)=π‘₯2(0+) and 𝑒2(𝑑) are solutions of (3.2) with 𝑒2(0+)=π‘₯2(0+). Therefore, for any small enough πœ‚>0, we have 𝑒0(𝑑)βˆ’πœ‚<π‘₯2(𝑑)<π‘’πœ€(𝑑)+πœ‚(3.20) for large 𝑑. Let πœ€β†’0, we get 𝑒0(𝑑)βˆ’πœ‚<π‘₯2(𝑑)≀𝑒0(𝑑)+πœ‚(3.21) for large 𝑑, which implies limπ‘‘β†’βˆž|π‘₯2(𝑑)βˆ’π‘’0(𝑑)|=0. The proof is complete.

3.4. Permanence

Theorem 3.9. Suppose that (H1)–(H4) and the following (H6) hold.(H6) There exists π‘Ž1>0 such that 𝐼1(π‘₯)β‰₯π‘Ž1π‘₯ and π‘Ž1ξ‚΅ξ€œexp𝜏0𝑓1ξ€·0,𝑒0ξ€Έξ‚Ά(𝑠)𝑑𝑠>1,(3.22)
where 𝑒0(𝑠) is the unique positive periodic solution of (3.2) for πœ€=0.
Then, system (1.4) is permanent.

Proof. Let π‘₯(𝑑) be any solution of (1.4) with π‘₯0∈int𝑅2+. From Theorem 3.1, we may assume π‘₯𝑖(𝑑)≀𝑀, 𝑑β‰₯0, 𝑖=1,2 where 𝑀 is a positive constant independent of initial values. Since Μ‡π‘₯2β‰₯π‘₯2𝑓2ξ€·0,π‘₯2ξ€Έ,(3.23) by Lemma 2.4 and Theorem 3.6, for sufficiently small πœ‚1>0, π‘₯2(𝑑)β‰₯𝑒0(𝑑)βˆ’πœ‚1 for 𝑑 large enough. Hence obviously there exists π‘š2>0 such that π‘₯2(𝑑)β‰₯π‘š2 for all 𝑑 large enough. We shall next find an π‘š1>0 such that π‘₯1(𝑑)β‰₯π‘š1 for 𝑑 large enough. We will do it in the following two steps.
(1) Let πœ€2 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<π‘š3β‰€πœ€2, πœ‚1>0 be small enough such that 𝜎=π‘Ž1∫exp(𝜏0𝑓2(π‘š3,π‘’π‘š3(𝑠)+πœ‚1)𝑑𝑠)>1. We will prove π‘₯1(𝑑)β‰€π‘š3 cannot hold for all 𝑑β‰₯0. Otherwise, Μ‡π‘₯2≀π‘₯2𝑓2ξ€·π‘š3,π‘₯2ξ€Έ.(3.24)
By Lemma 2.4 and Theorem 3.6, we have π‘₯2(𝑑)≀𝑒(𝑑) and |𝑒(𝑑)βˆ’π‘’π‘š3(𝑑)|β†’0 as π‘‘β†’βˆž, where 𝑒(𝑑) is the solution of (3.2) with πœ€=π‘š3 and 𝑒(0+)=π‘₯2(0+). Therefore, there exist, a 𝑇1>0 such that π‘₯2(𝑑)≀𝑒(𝑑)β‰€π‘’π‘š3(𝑑)+πœ‚1,(3.25)Μ‡π‘₯1β‰₯π‘₯1𝑓1ξ€·π‘₯1,π‘’π‘š3(𝑑)+πœ‚1ξ€Έ(3.26) for 𝑑β‰₯𝑇1.
Let 𝑁1βˆˆπ‘ and 𝑁1𝜏β‰₯𝑇1. Integrating (3.26) on (π‘›πœ,(𝑛+1)𝜏] for 𝑛β‰₯𝑁1, we have π‘₯1((𝑛+1)𝜏)β‰₯π‘₯1ξ€·π‘›πœ+ξ€Έξ‚΅ξ€œexp(𝑛+1)πœπ‘›πœπ‘“1ξ€·π‘š3,π‘’π‘š3(𝑠)+πœ‚1𝑑𝑠=π‘₯1ξ€·π‘›πœ+ξ€Έξ‚΅ξ€œexp𝜏0𝑓1ξ€·π‘š3,π‘’π‘š3(𝑠)+πœ‚1𝑑𝑠β‰₯π‘₯1(π‘›πœ)π‘Ž1ξ‚΅ξ€œexp𝜏0𝑓1ξ€·π‘š3,π‘’π‘š3(𝑠)+πœ‚1𝑑𝑠=π‘₯1(π‘›πœ)𝜎.(3.27) Then, π‘₯1((𝑁1+π‘˜)𝜏)β‰₯π‘₯1(𝑁1𝜏)πœŽπ‘˜β†’βˆž as π‘˜β†’βˆž, which is a contradiction. Hence, there exists a 𝑑1>0 such that π‘₯1(𝑑1)>π‘š3.
(2) If π‘₯1(𝑑)>π‘š3 for all 𝑑β‰₯𝑑1, then our aim is obtained. Hence we need only to consider those solutions which leave the region 𝑅={π‘₯βˆˆπ‘…2+∣π‘₯1β‰€π‘š3} and reenter it again. Let π‘‘βˆ—=inf𝑑β‰₯𝑑1{π‘₯1(𝑑)β‰€π‘š3}. Then, π‘₯1(𝑑)>π‘š3 for π‘‘βˆˆ[𝑑1,π‘‘βˆ—) and π‘₯1(π‘‘βˆ—)β‰₯π‘š3. Suppose π‘‘βˆ—βˆˆ(𝑛1𝜏,(𝑛1+1)𝜏],𝑛1βˆˆπ‘. Let 𝑒𝑀(𝑑) be the solution of (3.2) with πœ€=π‘š3 and 𝑒𝑀((𝑛1+1)𝜏+)=𝑀. Then there is 𝑑2>0 such that 𝑒𝑀(𝑑)<π‘’π‘š3(𝑑)+πœ‚1 for 𝑑β‰₯𝑑2+(𝑛1+1)𝜏. Note that 𝑑2 is independent of π‘₯(𝑑). Select 𝑛2,𝑛3βˆˆπ‘ such that 𝑛2𝜏>𝑑2 andπ‘Žπ‘›21ξ€·βˆ’||𝜎exp1||+𝑛2𝜎1ξ€ΈπœŽπ‘›3>1,(3.28) where 𝜎1=πœπ‘“1(π‘š3,𝑀). We claim there must be a 𝑑3∈(π‘‘βˆ—,(𝑛1+1+𝑛2+𝑛3)𝜏] such that π‘₯1(𝑑3)>π‘š3. Otherwise for π‘‘βˆˆ(π‘‘βˆ—,(𝑛1+1+𝑛2+𝑛3)𝜏], π‘₯1(𝑑)β‰€π‘š3 and Μ‡π‘₯2≀𝑓2ξ€·π‘š3,π‘₯2ξ€Έ.(3.29) By Lemma 2.4, we have π‘₯2(𝑑)≀𝑒𝑀𝑛(𝑑),π‘‘βˆˆξ€·ξ€·1𝑛+1𝜏,1+1+𝑛2+𝑛3ξ€Έπœξ€».(3.30) Then, π‘₯2(𝑑)≀𝑒𝑀(𝑑)<π‘’π‘š3(𝑑)+πœ‚1,(3.31) and (3.26) holds for π‘‘βˆˆ((𝑛1+1+𝑛2)𝜏,(𝑛1+1+𝑛2+𝑛3)𝜏]. As in step 1, we have π‘₯1𝑛1+1+𝑛2+𝑛3ξ€Έπœξ€Έβ‰₯π‘₯1𝑛1+1+𝑛2ξ€Έπœξ€ΈπœŽπ‘›3.(3.32) Since Μ‡π‘₯1β‰₯π‘₯1𝑓1ξ€·π‘š3𝑑,𝑀,π‘‘βˆˆβˆ—,𝑛1+1+𝑛2+𝑛3ξ€Έπœξ€»,(3.33) integrating it on [π‘‘βˆ—,(𝑛1+1+𝑛2)𝜏], we have π‘₯1𝑛1+1+𝑛2ξ€Έπœξ€Έβ‰₯π‘š3π‘Žπ‘›21ξ€·βˆ’||𝜎exp1||+𝑛2𝜎1ξ€Έ.(3.34) Thus, by (3.28), we have π‘₯1𝑛1+1+𝑛2+𝑛3ξ€Έπœξ€Έβ‰₯π‘š3π‘Žπ‘›21ξ€·βˆ’||𝜎exp1||+𝑛2𝜎1ξ€ΈπœŽπ‘›3>π‘š3,(3.35) which is a contradiction. Let 𝑑=inf𝑑β‰₯π‘‘βˆ—{π‘₯1(𝑑)>π‘š3}. Then, π‘‘βˆˆ(π‘‘βˆ—,(𝑛1+1+𝑛2+𝑛3)𝜏]. Denote π‘Ž=min{1,π‘Žπ‘›2+𝑛31}. For π‘‘βˆˆ(π‘‘βˆ—,𝑑], we have π‘₯1(𝑑)β‰₯π‘₯1ξ€·π‘‘βˆ—ξ€Έξ€·βˆ’ξ€·π‘Žexp1+𝑛2+𝑛3ξ€Έ||𝜎1||ξ€Έβ‰₯π‘š3ξ€·βˆ’ξ€·π‘Žexp1+𝑛2+𝑛3ξ€Έ||𝜎1||ξ€Έβ‰œπ‘š1.(3.36) For 𝑑>𝑑, the same arguments can be continued since π‘₯1(𝑑+)>π‘š3. Hence π‘₯1(𝑑)β‰₯π‘š1 for all 𝑑β‰₯𝑑1. 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: π‘₯𝑑̇π‘₯=𝐹(π‘₯),π‘‘β‰ π‘›πœ,+ξ€Έ=Θ(π‘₯(𝑑)),𝑑=π‘›πœ,(3.37) where 𝐹=(π‘₯1𝑓1(π‘₯),π‘₯2𝑓2(π‘₯)), Θ=(𝐼1(π‘₯),𝐼2(π‘₯))βˆΆπ‘…2+↦𝑅2+, are suitable smooth. For convenience and using the same notations in [9], we have exchanged the subscripts of π‘₯1,π‘₯2 and 𝑓1,𝑓2 in system (1.4). Suppose Μ‡π‘₯1ξ€·π‘₯=𝑔1ξ€Έ=𝐹1ξ€·π‘₯1ξ€Έπ‘₯,0,π‘‘β‰ π‘›πœ,1𝑑+ξ€Έξ€·π‘₯=πœƒ1(𝑑)=Θ1ξ€·π‘₯1(𝑑),0,𝑑=π‘›πœ(3.38) has a periodic solution π‘₯𝑠(𝑑). Thus, 𝜁=(π‘₯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 Ξ¦ be the flow associated to (3.37), we have π‘₯(𝑑)=Ξ¦(𝑑,π‘₯0), 0<π‘‘β‰€πœ, where π‘₯0=π‘₯(0+). Now, we list following notations we will use from [9]: π‘‘ξ…ž0=1βˆ’πœ•Ξ˜2πœ•π‘₯2πœ•Ξ¦2πœ•π‘₯2ξ€·πœ0,π‘₯0ξ€Έ,where𝜏0istherootofπ‘‘ξ…ž0π‘Ž=0,ξ…ž0ξ‚΅=1βˆ’πœ•Ξ˜1πœ•π‘₯1πœ•Ξ¦1πœ•π‘₯1ξ‚Άξ€·πœ0,π‘₯0ξ€Έ,π‘ξ…ž0ξ‚΅=βˆ’πœ•Ξ˜1πœ•π‘₯1πœ•Ξ¦1πœ•π‘₯2+πœ•Ξ˜1πœ•π‘₯2πœ•Ξ¦2πœ•π‘₯2ξ‚Άξ€·πœ0,π‘₯0ξ€Έ,πœ•Ξ¦1𝑑,π‘₯0ξ€Έπœ•π‘₯1ξ‚΅ξ€œ=exp𝑑0πœ•πΉ1(𝜁(π‘Ÿ))πœ•π‘₯1ξ‚Ά,π‘‘π‘Ÿπœ•Ξ¦2𝑑,π‘₯0ξ€Έπœ•π‘₯2ξ‚΅ξ€œ=exp𝑑0πœ•πΉ2(𝜁(π‘Ÿ))πœ•π‘₯2ξ‚Ά,π‘‘π‘Ÿπœ•Ξ¦1𝑑,π‘₯0ξ€Έπœ•π‘₯2=ξ€œπ‘‘0ξ‚΅ξ€œexpπ‘‘π‘’πœ•πΉ1(𝜁(π‘Ÿ))πœ•π‘₯1ξ‚Άπ‘‘π‘Ÿπœ•πΉ1(𝜁(𝑒))πœ•π‘₯2ξ‚΅ξ€œexp𝑒0πœ•πΉ2(𝜁(π‘Ÿ))πœ•π‘₯2ξ‚Άπœ•π‘‘π‘Ÿπ‘‘π‘’,2Ξ¦2𝑑,π‘₯0ξ€Έπœ•π‘₯1πœ•π‘₯2=ξ€œπ‘‘0ξ‚΅ξ€œexpπ‘‘π‘’πœ•πΉ2(𝜁(π‘Ÿ))πœ•π‘₯2ξ‚Άπœ•π‘‘π‘Ÿ2𝐹2(𝜁(𝑒))πœ•π‘₯1πœ•π‘₯2ξ‚΅ξ€œexp𝑒0πœ•πΉ2(𝜁(π‘Ÿ))πœ•π‘₯2ξ‚Άπœ•π‘‘π‘Ÿπ‘‘π‘’,2Ξ¦2𝑑,π‘₯0ξ€Έπœ•π‘₯22=ξ€œπ‘‘0ξ‚΅ξ€œexpπ‘‘π‘’πœ•πΉ2(𝜁(π‘Ÿ))πœ•π‘₯2ξ‚Άπœ•π‘‘π‘Ÿ2𝐹2(𝜁(𝑒))πœ•π‘₯22ξ‚΅ξ€œexp𝑒0πœ•πΉ2(𝜁(π‘Ÿ))πœ•π‘₯2ξ‚Ά+ξ€œπ‘‘π‘Ÿπ‘‘π‘’π‘‘0ξ‚»ξ‚΅ξ€œexpπ‘‘π‘’πœ•πΉ2(𝜁(π‘Ÿ))πœ•π‘₯2ξ‚Άπœ•π‘‘π‘Ÿ2𝐹2(𝜁(𝑒))πœ•π‘₯2πœ•π‘₯1ξ‚ΌΓ—ξ‚»ξ€œπ‘’0ξ‚΅ξ€œexpπ‘’π‘πœ•πΉ1(𝜁(π‘Ÿ))πœ•π‘₯1ξ‚Άπ‘‘π‘Ÿπœ•πΉ1(𝜁(𝑒))πœ•π‘₯2ξ‚΅ξ€œexp𝑝0πœ•πΉ2(𝜁(π‘Ÿ))πœ•π‘₯2ξ‚Άξ‚Όπœ•π‘‘π‘Ÿπ‘‘π‘π‘‘π‘’,2Ξ¦2𝑑,π‘₯0ξ€Έπœ•πœπœ•π‘₯2=πœ•πΉ2(𝜁(𝑑))πœ•π‘₯2ξ‚΅ξ€œexp𝑑0πœ•πΉ2(𝜁(π‘Ÿ))πœ•π‘₯2ξ‚Ά,π‘‘π‘Ÿπœ•Ξ¦1ξ€·πœ0,π‘₯0ξ€Έπœ•πœ=Μ‡π‘₯π‘ ξ€·πœ0ξ€Έπœ•π΅=2Θ2πœ•π‘₯1πœ•π‘₯2ξƒ©πœ•Ξ¦1ξ€·πœ0,π‘₯0ξ€Έπœ•πœ+πœ•Ξ¦1ξ€·πœ0,π‘₯0ξ€Έπœ•π‘₯11π‘Žξ…ž0πœ•Ξ˜1πœ•π‘₯1πœ•Ξ¦1ξ€·πœ0,π‘₯0ξ€Έπœ•π‘‘ξƒͺπ‘Žπ‘’πœ•Ξ¦2ξ€·πœ0,π‘₯0ξ€Έπœ•π‘₯2βˆ’πœ•Ξ˜2πœ•π‘₯2ξƒ©πœ•2Ξ¦2ξ€·πœ0,π‘₯0ξ€Έπœ•πœπœ•π‘₯2+πœ•2Ξ¦2ξ€·πœ0,π‘₯0ξ€Έπœ•π‘₯1πœ•π‘₯21π‘Žξ…ž0πœ•Ξ˜1πœ•π‘₯1πœ•Ξ¦1ξ€·πœ0,π‘₯0ξ€Έπœ•πœξƒͺ,πœ•πΆ=22Θ2πœ•π‘₯1πœ•π‘₯2ξƒ©βˆ’π‘ξ…ž0π‘Žξ…ž0πœ•Ξ¦1ξ€·πœ0,π‘₯0ξ€Έπœ•π‘₯1+πœ•Ξ¦1ξ€·πœ0,π‘₯0ξ€Έπœ•π‘₯2ξƒͺΓ—πœ•Ξ¦2ξ€·πœ0,π‘₯0ξ€Έπœ•π‘₯2βˆ’πœ•2Θ2πœ•π‘₯22ξƒ©πœ•Ξ¦2ξ€·πœ0,π‘₯0ξ€Έπœ•π‘₯2ξƒͺ2+2πœ•Ξ˜2πœ•π‘₯2π‘ξ…ž0π‘Žξ…ž0πœ•2Ξ¦2ξ€·πœ0,π‘₯0ξ€Έπœ•π‘₯2πœ•π‘₯1βˆ’πœ•Ξ˜2πœ•π‘₯2πœ•2Ξ¦2ξ€·πœ0,π‘₯0ξ€Έπœ•π‘₯22.(3.39)

Theorem 3.10 (see [9]). If |1βˆ’π‘Žξ…ž0|<1 and π‘‘ξ…ž0=0, then one has the following: (a)If 𝐡𝐢≠0, then there is a bifurcation of nontrivial periodic solution. Moreover, there is a bifurcation of supercritical case if 𝐡𝐢<0 and a subcritical case if 𝐡𝐢>0. (b)If 𝐡𝐢=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 π‘₯𝑖(𝑑)≀𝑀, 𝑖=1,2 for each solution π‘₯(𝑑) of (1.2) with all 𝑑 large enough.

Proof. Suppose π‘₯(𝑑) is any solution of (1.2). Let 𝑉(𝑑)=𝑉(𝑑,π‘₯(𝑑))=π‘Ž21π‘₯1(𝑑)+π‘Ž12π‘₯2(𝑑). Let 0<πœ†β‰€π‘Ÿ2. Then, when π‘‘β‰ π‘›πœ, 𝐷+𝑉(𝑑)+πœ†π‘‰(𝑑)=π‘Ž21π‘₯1ξ€·π‘Ÿ1+πœ†βˆ’π‘Ž11π‘₯1ξ€Έβˆ’π‘Ž12ξ€·π‘Ÿ2ξ€Έπ‘₯βˆ’πœ†2β‰€π‘Ž21π‘₯1ξ€·π‘Ÿ1+πœ†βˆ’π‘Ž11π‘₯1≀𝐾,(4.1) for some positive constant 𝐾. When 𝑑=π‘›πœ, 𝑉(π‘›πœ+)=𝑉(π‘›πœ)+π‘Ž12𝑏. Thus, (H1) is satisfied, the conclusion comes from Theorem 3.1. The proof is complete.

Theorem 4.2. Let π‘₯(𝑑) be any solution of (1.2). Then, limπ‘‘β†’βˆžπ‘₯1(𝑑)=0, limπ‘‘β†’βˆž|π‘₯2(𝑑)βˆ’π‘₯βˆ—2(𝑑)|=0 if 𝑏>π‘Ÿ1π‘Ÿ2𝜏/π‘Ž12, where π‘₯βˆ—2(𝑑)=𝑏exp(βˆ’π‘Ÿ2(π‘‘βˆ’π‘›πœ))/(1βˆ’exp(βˆ’π‘Ÿ2𝜏)), π‘‘βˆˆ(π‘›πœ,(𝑛+1)𝜏], π‘›βˆˆπ‘, π‘₯βˆ—2(0+)=𝑏/(1βˆ’exp(βˆ’π‘Ÿ2𝜏)).

Proof. Consider system (3.2) with the 𝑓2 and 𝐼2 taking the forms in (1.2). Since 𝐼2(𝑒)=𝑒+𝑏, 𝑓2(0,0)=𝑓2(0,𝑒)=βˆ’π‘Ÿ2, obviously, (H2) and (H3) are satisfied. Clearly π‘₯βˆ—2(𝑑) is a positive 𝜏 periodic solution when πœ€=0. Note that 𝑃(𝑒)=𝑒exp(βˆ’π‘Ÿ2𝜏)+𝑏, it has a unique positive fixed point since 𝑃(0)>0,limπ‘’β†’βˆž(𝑃(𝑒)βˆ’π‘’)=βˆ’βˆž and 𝑃(𝑒)βˆ’π‘’ is strictly decreasing. Since 𝐼1(π‘₯1)=π‘₯1 and ξ‚΅ξ€œexp𝜏0𝑓1ξ€·0,π‘₯βˆ—2ξ€Έξ‚Άξ‚΅ξ€œ(𝑠)𝑑𝑠=exp𝜏0ξ€·π‘Ÿ1βˆ’π‘Ž12π‘₯βˆ—2ξ€Έξ‚Άξ‚΅π‘Ÿ(𝑠)𝑑𝑠=exp1π‘Žπœβˆ’12π‘π‘Ÿ2ξ‚Ά<1(4.2) when 𝑏>π‘Ÿ1π‘Ÿ2𝜏/π‘Ž12, (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<π‘Ÿ1π‘Ÿ2𝜏/π‘Ž12.

Proof. Let π‘₯(𝑑) be any solution of (1.2) with π‘₯0∈int𝑅2+. Obviously, (H1)–(H3) and (H4ξ…ž) are satisfied. Since 𝐼1(π‘₯1)=π‘₯1 and ξ‚΅ξ€œexp𝜏0𝑓1ξ€·0,π‘₯βˆ—2ξ€Έξ‚Άξ‚΅ξ€œ(𝑠)𝑑𝑠=exp𝜏0ξ€·π‘Ÿ1βˆ’π‘Ž12π‘₯βˆ—2ξ€Έξ‚Άξ‚΅π‘Ÿ(𝑠)𝑑𝑠=exp1π‘Žπœβˆ’12π‘π‘Ÿ2ξ‚Ά>1(4.3) when 𝑏<π‘Ÿ1π‘Ÿ2𝜏/π‘Ž12, (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,π‘₯βˆ—2(𝑑))𝑇. We also exchange the subscripts of π‘₯1 and π‘₯2 as in Theorem 3.10. Thus, 𝐹1ξ€·π‘₯1,π‘₯2ξ€Έ=π‘₯1ξ€·βˆ’π‘Ÿ2+π‘Ž21π‘₯2ξ€Έ,𝐹2ξ€·π‘₯1,π‘₯2ξ€Έ=π‘₯2ξ€·π‘Ÿ1βˆ’π‘Ž11π‘₯2βˆ’π‘Ž12π‘₯1ξ€Έ,Θ1ξ€·π‘₯1,π‘₯2ξ€Έ=π‘₯1Θ+𝑏,2ξ€·π‘₯1,π‘₯2ξ€Έ=π‘₯2,ξ€·π‘₯𝜁(𝑑)=𝑠(𝑑),0𝑇=ξ€·π‘₯βˆ—2ξ€Έ.(𝑑),0(4.4) Then, we can compute that π‘Žξ…ž0ξ€·=1βˆ’expβˆ’π‘Ÿ2𝜏0𝑏>0,ξ…ž0π‘Ž=βˆ’21𝑏expβˆ’π‘Ÿ2𝜏0ξ€Έξ€·1βˆ’expβˆ’π‘Ÿ2𝜏0ξ€Έξ€œπœ00ξƒ©π‘Ÿexp1π‘Žπ‘’βˆ’12𝑏1βˆ’expβˆ’π‘Ÿ2π‘’ξ€Έξ€Έπ‘Ÿ2ξ€·ξ€·1βˆ’expβˆ’π‘Ÿ2𝜏0ξƒͺπœ•ξ€Έξ€Έπ‘‘π‘’<0,2Ξ¦2ξ€·πœ0,π‘₯0ξ€Έπœ•πœπœ•π‘₯2=π‘Ÿ1βˆ’π‘Ž12𝑏expβˆ’π‘Ÿ2𝜏0ξ€Έξ€·1βˆ’expβˆ’π‘Ÿ2𝜏0ξ€Έ,πœ•2Ξ¦2ξ€·πœ0,π‘₯0ξ€Έπœ•π‘₯1πœ•π‘₯2=βˆ’π‘Ž12𝜏0πœ•<0,2Ξ¦2ξ€·πœ0,π‘₯0ξ€Έπœ•π‘₯22=βˆ’π‘Ž11𝜏0βˆ’ξ€œπœ00ξƒ―π‘Ž12π‘Ž21𝑏expβˆ’π‘Ÿ2𝑒exp(ℝ)ξ€·1βˆ’expβˆ’π‘Ÿ2𝜏0ξ€Έξƒ°Γ—ξƒ―ξ€œπ‘’0ξƒ©π‘Ÿexp1π‘Žπ‘βˆ’12𝑏1βˆ’expβˆ’π‘Ÿ2π‘ξ€Έξ€Έπ‘Ÿ2ξ€·ξ€·1βˆ’expβˆ’π‘Ÿ2𝜏0ξƒͺ𝑑𝑝𝑑𝑒<0,πœ•Ξ¦1ξ€·πœ0,π‘₯0ξ€Έπœ•πœ=Μ‡π‘₯π‘ ξ€·πœ0ξ€Έπ‘Ÿ=βˆ’2𝑏expβˆ’π‘Ÿ2𝜏0ξ€Έξ€·1βˆ’expβˆ’π‘Ÿ2𝜏0ξ€Έ<0,(4.5) where ℝ denotes π‘Ÿ1(𝜏0βˆ’π‘’)βˆ’π‘Ž12𝑏(exp(βˆ’π‘Ÿ2𝑒)βˆ’exp(βˆ’π‘Ÿ2𝜏0))/(π‘Ÿ2(1βˆ’exp(βˆ’π‘Ÿ2𝜏0))).

Since πœ•Ξ˜π‘–/πœ•π‘₯𝑗=0, 𝑖≠𝑗, πœ•Ξ˜π‘–/πœ•π‘₯𝑖=1, πœ•2Ξ˜π‘–/πœ•π‘₯1πœ•π‘₯2=0, 𝑖=1,2 and πœ•2Θ2/πœ•π‘₯22=0, it is easy to verify that 𝐢>0 and ξƒ©π‘Ÿπ΅=βˆ’1βˆ’π‘Ž12𝑏expβˆ’π‘Ÿ2𝜏0ξ€Έξ€·1βˆ’expβˆ’π‘Ÿ2𝜏0ξ€Έ+π‘Ž12𝜏0π‘Ÿ2𝑏expβˆ’π‘Ÿ2𝜏0ξ€Έξ€·ξ€·1βˆ’expβˆ’π‘Ÿ2𝜏0ξ€Έξ€Έ2ξƒͺ.(4.6) To determine the sign of 𝐡, let πœ™(𝑑)=π‘Ÿ1βˆ’π‘Ž12𝑏exp(βˆ’π‘Ÿ2𝑑)/(1βˆ’exp(βˆ’π‘Ÿ2𝜏0)). We have π‘‘πœ™/𝑑𝑑=π‘Ÿ2π‘Ž12𝑏exp(βˆ’π‘Ÿ2𝑑)/(1βˆ’exp(βˆ’π‘Ÿ2𝜏0))>0. Thus, we conclude that πœ™(𝜏0)>0 since ∫𝜏00πœ™(𝑑)𝑑𝑑=π‘Ÿ1𝜏0βˆ’π‘Ž12𝑏/π‘Ÿ2=0 and πœ™(𝑑) is strictly increasing. Therefore, we have 𝐡<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=π‘Ž12𝑏/(π‘Ÿ1π‘Ÿ2), that is, system (1.2) has a positive periodic solution if 𝜏>𝜏0 and is close to 𝜏0, where 𝜏0 is the root of π‘‘ξ…ž0=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=π‘Ž12𝑏(π‘Ÿ1π‘Ÿ2) and is unstable if 𝜏>𝜏0=π‘Ž12𝑏(π‘Ÿ1π‘Ÿ2). 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 Μ‚π‘₯(𝑑) with Μ‚π‘₯(0+)=Μ‚π‘₯0=(Μ‚π‘₯01,Μ‚π‘₯02)𝑇. Then, ∫(1/𝜏)𝜏0Μ‚π‘₯1(𝑑)𝑑𝑑<min{π‘Ÿ1/π‘Ž11,π‘Ÿ2/π‘Ž21} and ∫(1/𝜏)𝜏0Μ‚π‘₯2(𝑑)𝑑𝑑<π‘Ÿ1/π‘Ž12.

Proof. Since Μ‚π‘₯(𝑑) is a positive periodic solution of system (1.2), we have Μ‚π‘₯(0+)=Μ‚π‘₯(𝜏+) which gives Μ‚π‘₯01=Μ‚π‘₯01ξ‚΅π‘Ÿexp1πœβˆ’π‘Ž11ξ€œπœ0Μ‚π‘₯1(𝑑)π‘‘π‘‘βˆ’π‘Ž12ξ€œπœ0Μ‚π‘₯2ξ‚Ά,(𝑑)𝑑𝑑̂π‘₯02=Μ‚π‘₯02ξ‚΅expβˆ’π‘Ÿ2𝜏+π‘Ž21ξ€œπœ0Μ‚π‘₯1ξ‚Ά(𝑑)𝑑𝑑+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=π‘Ÿ1π‘Ÿ2𝜏/π‘Ž12. It can be rewritten as π‘Ÿ1𝜏=π‘Ž12𝑏/π‘Ÿ2 or ∫(𝑛+1)πœπ‘›πœπ‘Ÿ1βˆ«π‘‘π‘‘=(𝑛+1)πœπ‘›πœπ‘Ž12π‘₯βˆ—2(𝑑)𝑑𝑑. If there is no prey or its density is very small, the density of the predator is π‘₯βˆ—2(𝑑). It is clear that π‘Ž12𝑏/π‘Ÿ2 is the amount of prey that the predator can eat in 𝜏 period of time and π‘Ÿ1𝜏 means the increasing amount of prey in such a period of time. Thus, π‘Ÿ1π‘Ÿ2𝜏/π‘Ž12 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>π‘Ÿ1π‘Ÿ2𝜏/π‘Ž12 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 π‘Ÿ1/π‘Ž11 or π‘Ÿ2/π‘Ž21. 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 Μ‡π‘₯1=π‘₯1ξ€·π‘Ÿ1βˆ’π‘Ž11π‘₯1βˆ’π‘Ž12π‘₯2ξ€Έ,Μ‡π‘₯2=π‘₯2ξ€·βˆ’π‘Ÿ2+π‘Ž21π‘₯1βˆ’π‘Ž22π‘₯2ξ€Έ,(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(𝑒(𝑑))=𝑓2(πœ€,𝑒(𝑑))βˆ’π·π‘“2(πœ€,𝑒(𝑑))𝑒(𝑑)=π‘Ž22𝑒(𝑑).(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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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.


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