Extinction and Permanence of a General Predator-Prey System with Impulsive Perturbations
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.
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 , chemotherapeutic treatment , and pulse vaccination  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  considered stage-structure for the predator; Xiao and Chen  introduced diseases for the prey; while Xu and Chen  focused on the functional responses and diffusions of the predator. With regard to impulsive effects Lakmeche and Arino  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 can be developed by introducing a constant periodic impulsive immigration for the predator. That is where , are the densities of the prey and predator at time , respectively, is the intrinsic growth rate of prey, is the death rate of predator, is the rate of intraspecific competition or density dependence, is the per capita rate of predation of the predator, denotes the product of the per capita rate of predation and the rate of conversing prey into predator, 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 (), in which case it is global asymptotically stable. Or there is no positive equilibrium (), in which case goes extinct and tends to , 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.  considered the following system which also includes impulsive chemical pest control of pesticide using into system (1.2): where , .
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:
Ballinger and Liu  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. , Liu and Chen , and Zhang et al. , 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 , Cao and Gard  introduced the idea and methods, which were developed further in the context of reaction-diffusion models by Cantrell and Cosner . And a discussion of how the methods are applied to various sorts of ecological models, including some discrete models, was given by Cosner . 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 , , and be the set of all nonnegative integers. Denote by the map defined by the right hand of system (1.4). Let .
Definition 2.1. , then for , the upper right derivative of with respect to the impulsive differential system (1.4) is defined as
We will assume the following basic conditions for system (1.4) hold throughout this paper. (A1) is differentiable and , , , .(A2) is continuous, , for , and is nondecreasing for .
The solution of system (1.4) is a piecewise continuous function , is continuous on , , and exists. Obviously the smoothness properties of 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 and can be the densities of the prey and the predator at time , 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 whenever , , , and , , , by (A2), we have the following lemma.
Lemma 2.2. Suppose is a solution of (1.4) with , then for all . And further , if .
Definition 2.3. System (1.4) is said to be permanent if there exist constants such that , for all t sufficiently large, where is any solution of (1.4) with .
We will use a basic comparison result from in [3, Theorem 3.1.1]. For convenience, we state it in our notations.
Suppose that satisfies the following condition (H). (H) is continuous in and for , , exists.
Lemma 2.4. Let . Assume that where satisfies (H) and is nondecreasing. Let be the maximal solution of the scalar impulsive differential equation existing on . Then, implies that , where 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 to guarantee the existence and uniqueness of solutions for (2.3), then is exactly the unique solution of (2.3).
3. Main Results
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 , such that the following conditions hold.
(i) , for some ;(ii)where are positive constants.
Then, system (1.4) is ultimately upper bounded.
Proof. Let . By (i), it suffices to prove that 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):
Since one-dimensional continuous differential system is naturally monotone system and 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 , , and such that for . Then, there exists such that for , where is the solution of (3.2).
Proof. Let and . Obviously, is positive for . By (A1), we have Hence, , if and , if . By (3.2), we have Hence, by (H2), The proof is complete.
Lemma 3.3. Suppose that the following condition holds. (H3) There exist positive constants , , and such that
and for .
Then, there exist such that for , where is the solution of (3.2) with .
Proof. By (3.6), there exists such that
for and . Let and . Let and be the solution of (3.2) with . There are two cases for , .
Case 1. There exists such that .
Let . Then, . Since , we can conclude that , . Hence,
Case 2. .
By (A1), we have. Hence, Therefore, This completes the proof.
Let and for in Lemmas 3.2 and 3.3, respectively. Consider the solution of (3.2) with . By Lemma 2.4, similar to [12, Theorem 3.1], we can define a map and show 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 such that system (3.2) has a positive -periodic solution for each .
Modify to and consider for the case with and in the definition of lower and upper solutions of [26, Definition 3.1], the solution of (3.2) with () 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 and will tend to a positive -periodic solution of (3.2). If the positive -periodic solution of (3.2) is unique, then its global attractivity will also be established. In fact, for any solution of (3.2) with initial value , we can always find a lower solution and an upper solution with initial values and , respectively, such that . Then since the solutions of system (3.2) are monotone with respect to initial values, . And since both and will tend to the unique positive -periodic solution of (3.2), so is . 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 defined above, we can simply use that result here. We first give the definition of strongly concave operator on .
Definition 3.5. An operator is strongly concave on if for any , and any number , there exists a positive number such that .
Now let be the solution of (3.2) with initial value . Let . Define where is the Jacobian matrix of (here is just its derivative with respect to 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) for any and is strongly concave or is linear. Then, there exists such that for each , system (3.2) has a unique positive -periodic solution , which is a global attractor for any solution of (3.2) with .
Remark 3.7. The unique positive -periodic solution is corresponding to the unique positive fixed point of . Since the map 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 (), which states the uniqueness directly. () has at most one positive fixed point.
3.3. Extinction of the Prey
Proof. By (3.14), the continuity of , and the Lebesgue Theorem, we can choose an sufficiently small such that , where . Note that , consider (3.2) with and . Let . From Lemma 2.4 and Theorem 3.6, we have and as . Hence, there exists such that for all . For simplification and without loss of generality, we may assume (3.15) holds for all . By (A1), we get which leads to Hence and as . Therefore, as since for . Now, we prove that as . For in Theorem 3.6, there exists a such that , . Without loss of generality, we may assume that for all . Then, by (A1), we have From Lemma 2.4 and Theorem 3.6, we have and , as , where is solution of (3.2) with , and are solutions of (3.2) with . Therefore, for any small enough , we have for large . Let , we get for large , which implies . The proof is complete.
Proof. Let be any solution of (1.4) with . From Theorem 3.1, we may assume , , where is a positive constant independent of initial values. Since
by Lemma 2.4 and Theorem 3.6, for sufficiently small , for large enough. Hence obviously there exists such that for all large enough. We shall next find an such that for large enough. We will do it in the following two steps.
(1) Let be the positive constant in the conclusion of Theorem 3.6. By (3.22), the continuity of , and the Lebesgue Theorem, we can choose , be small enough such that . We will prove cannot hold for all . Otherwise,
By Lemma 2.4 and Theorem 3.6, we have and as , where is the solution of (3.2) with and . Therefore, there exist, a such that for .
Let and . Integrating (3.26) on for , we have Then, as , which is a contradiction. Hence, there exists a such that .
(2) If for all , then our aim is obtained. Hence we need only to consider those solutions which leave the region and reenter it again. Let . Then, for and . Suppose . Let be the solution of (3.2) with and . Then there is such that for . Note that is independent of . Select such that and where . We claim there must be a such that . Otherwise for , and By Lemma 2.4, we have Then, and (3.26) holds for . As in step 1, we have Since integrating it on , we have Thus, by (3.28), we have which is a contradiction. Let . Then, . Denote . For , we have For , the same arguments can be continued since . Hence for all . 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  to study the existence of positive periodic solution. System (1.4) can be rewritten as the following more general system: where , , are suitable smooth. For convenience and using the same notations in , we have exchanged the subscripts of and in system (1.4). Suppose has a periodic solution . Thus, is a trivial periodic solution of (3.37). By studying the local stability of and a standard computation of Floqet exponent,  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 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 .
Theorem 3.10 (see ). If and , then one has the following: (a)If , then there is a bifurcation of nontrivial periodic solution. Moreover, there is a bifurcation of supercritical case if and a subcritical case if . (b)If , then there is an undetermined case.
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 such that , for each solution of (1.2) with all large enough.
Theorem 4.2. Let be any solution of (1.2). Then, , if , where , , , .
Proof. Consider system (3.2) with the and taking the forms in (1.2). Since , , obviously, (H2) and (H3) are satisfied. Clearly is a positive periodic solution when . Note that , it has a unique positive fixed point since and is strictly decreasing. Since and when , (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 .
Proof. Let be any solution of (1.2) with . Obviously, (H1)–(H3) and () are satisfied. Since and when , (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 . We also exchange the subscripts of and as in Theorem 3.10. Thus, Then, we can compute that where denotes .
Since , , , , and , it is easy to verify that and To determine the sign of , let . We have . Thus, we conclude that since and is strictly increasing. Therefore, we have from (4.6) and the following result.
Theorem 4.4. System (1.2) has a supercritical bifurcation of positive periodic solution at the point , that is, system (1.2) has a positive periodic solution if and is close to , where is the root of .
Remark 4.5. From Theorems 4.2 and 4.3, we know that the trivial periodic solution is a global attractor if and is unstable if . 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 with . Then, and .
Proof. Since is a positive periodic solution of system (1.2), we have which gives 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 . It can be rewritten as or . If there is no prey or its density is very small, the density of the predator is . It is clear that is the amount of prey that the predator can eat in period of time and means the increasing amount of prey in such a period of time. Thus, can be interpreted as the amount of immigrating predator which the prey could supply for with its increment during period of time. Then, it is easy to understand that the prey will go extinct when 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, , 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 or . 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.
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 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 where all the parameters are positive. Let be the right-hand side of the second equation. As defined in Section 3, we can compute that
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.
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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