International Journal of Differential EquationsVolume 2011 (2011), Article ID 259805, 28 pagesdoi:10.1155/2011/259805

Research Article

Existence and Global Attractivity of Positive Periodic Solutions for a Two-Species Competitive System with Stage Structure and Impulse

Changjin Xu¹ and Daxue Chen²

¹Guizhou Key Laboratory of Economics System Simulation, School of Mathematics and Statistics, Guizhou College of Finance and Economics, Guiyang 550004, China²Faculty of Science, Hunan Institute of Engineering, Xiangtan 411004, China

Received 10 August 2011; Accepted 17 September 2011

Academic Editor: Yuji Liu

Copyright © 2011 Changjin Xu and Daxue Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A class of nonautonomous two-species competitive system with stage structure and impulse is considered. By using the continuation theorem of coincidence degree theory, we derive a set of easily verifiable sufficient conditions that guarantee the existence of at least a positive periodic solution, and, by constructing a suitable Lyapunov functional, the uniqueness and global attractivity of the positive periodic solution are presented. Finally, an illustrative example is given to demonstrate the correctness of the obtained results.

1. Introduction

In recent years, with the increasing applications of theory of differential equations in mathematical ecology, various mathematical models have been proposed in the study of population [1–25]. But most of the previous results focused on the dynamical behaviors (including the stability, attractiveness, persistence, and periodicity of solution) of the systems which have fixed parameters and there is no impulse. Considering that harvest of many populations are not continuous and the periodic environmental factor, it is reasonable to investigate the systems with periodic coefficients and impulse. Impulsive differential systems display a combination of characteristics of both the continuous-time and discrete-time systems [26–30]. In 2006, Chen [1] studied the following non-autonomous almost periodic competitive two-species model with stage structure in one species: where and are immature and mature population densities of one species, respectively; represents the population density of another species; are all continuous, almost periodic functions. The competition is between and . Chen [1] obtained sufficient conditions for the existence of a unique, globally attractive, strictly positive almost periodic solution for system (1.1).

Considering that the harvest is an annual harvest pulse, to describe a system more accurately, we should consider the impulsive differential equation. Motivated by this point of view, we revised system (1.1) into the following form: where are the impulses at moments and is a strictly increasing sequence such that and are immature and mature population densities of one species, respectively, and represents the population density of another species. The competition is between and .

Throughout the paper, we always assume the following. are all continuous periodic; that is, for any . are all positive. for all , and there exists a positive integer such that .

The principle object of this paper is by using Mawhin’s continuation theorem of coincidence degree theory and by constructing the Lyapunov functions to investigate the stability and existence of periodic solutions of (1.2). To the best of my knowledge, it is the first time to deal with the existence and stability of periodic solutions of (1.2).

The organization of the paper is as follows. In Section 2, we introduce some notations and definitions and state some preliminary results needed in later sections. We then establish, in Section 3, some simple criteria for the existence of positive periodic solutions of system (1.2) by using the continuation theorem of coincidence degree theory proposed by Gaines and Mawhin [31]. The uniqueness and global attractivity of the positive periodic solution are presented in Section 4. In Section 5, an illustrative example is given to demonstrate the correctness of the obtained results.

2. Preliminaries

We will introduce some notations and definitions and state some preliminary results. Consider the impulsive system where is continuous and are continuous, and there exists a positive integer such that with . For . As we know, are called points of jump.

Let us recall some definitions. For the Canchy problem,

Definition 2.1. A map is said to be a solution of (2.2), if it satisfied the following conditions:(i) is a piecewise continuous map with first-class discontinuity points in , and at each discontinuity point it is continuous on the left;(ii) satisfies (2.2).

Definition 2.2. A map is said to be an periodic solution of (2.1), if (i) satisfies (i) and (ii) of Definition 2.1 in the interval and(ii) satisfies .

Obviously, if is a solution of (2.2) defined on , such that , then, by the periodicity of (2.2) in , the function defined by is a periodic solution of (2.1).

For system (1.2), seeking the periodic solutions is equivalent to seeking solutions of the following boundary value problem:

3. Existence of Positive Periodic Solutions

In this section, based on the Mawhin’s continuation theorem, we shall study the existence of at least one periodic solution of (1.1). To do so, we shall make some preparations.

Let be normed vector spaces; is a linear mapping; is a continuous mapping. The mapping will be called a Fredholm mapping of index zero if and is closed in . If is a Fredholm mapping of index zero and there exist continuous projectors and such that , it follows that is invertible. We denote the inverse of that map by . If is an open bounded subset of , the mapping will be called -compact on if is bounded and is compact. Since is isomorphic to , there exist isomorphisms .

Now we introduce Mawhin’s continuation theorem [31] as follows.

Lemma 3.1 (Continuation Theorem [31]). Let be a Fredholm mapping of index zero, and let be compact on . Suppose (a)for each , every solution of is such that .(b) for each , and .Then the equation has at least one solution lying in .

For convenience and simplicity in the following discussion, we always use the notations below throughout the paper: where is a continuous periodic function. For any nonnegative integer , let exist for , and let exist at , and with the norm , where is any norm of . It is easy to see that is a Banach space.

Now we are now in a position to state and prove the existence of periodic solutions of (2.4).

Theorem 3.2. In addition to , assume further that the following hold: where and are defined by (3.27), (3.32), and (3.61), respectively. Then the system (1.2) has at least a periodic solution.

Proof. According to the discussion above in Section 2, we need only to prove that the boundary value problem (2.4) has a solution. Since solutions of (2.4) remained positive for all , we let then system (2.4) can be translated to It is easy to see that if system (3.5) has one periodic solution , then is a positive solution of system (1.2). Therefore, to complete the proof, it suffices to show that system (3.5) has at least one periodic solution.In order to use the continuation theorem of coincidence degree theory, we take Then is a Banach space with norm , and is also a Banch space with norm .Let the following hold: Obviously, So, is closed in ; is a Fredholm mapping of index zero. Define two projectors It is easy to show that and are continuous and satisfy .Further, by direct computation, we can find that the inverse of , has the following form: Moreover, it is easy to check that where Obviously, and are continuous. Using the Ascoli-Arzela theorem, it is not difficult to show that is compact for any open bounded set . Moreover, is bounded. Thus, is -compact on with any open bounded set .Now we are at the point to search for an appropriate open, bounded subset for the application of the continuation theorem. Corresponding to the operator equation , we have Suppose that is an arbitrary solution of system (3.13) for a certain , integrating both sides of (3.13) over the interval with respect to , we obtain From (3.13) and (3.14), we obtain Let the following hold: From the second and the third equations of (3.14), we can obtain then Thus In the following, we will consider four cases.Case 1 (if ). From the first equation of (3.14), we have that is, Then Thus, from (3.22) and (3.25), we obtain By the first and the third equations of (3.14), we get then From (3.15), (3.17), (3.21), and (3.30), we have Thus, Case 2 (if ). By the first equation of (3.14), we have namely, Then From (3.22) and (3.37), we obtain By the first equation of (3.14), we also have Then that is, Thus, From (3.42), we have By the second equation of (3.14), we have Then that is, Therefore, we get Hence, we have Case 3 (if ). By the first equation of (3.14), we have Then namely, Therefore, So By the third equation of (3.14), we obtain that is, Thus, Case 4 (if ). By the second equation of (3.14), we have Then that is, Therefore, Thus, By the second equation of (3.14), we obtain that is, Thus, Then, from (3.16) and (3.20), we get Thus, By the first equation of (3.14), we have Then Thus, Hence, we have Based on the discussion above, we can easily obtain Obviously, are independent of . Similar to the proof of Theorem 2.1 of [17], we can easily find a sufficiently large so that we denote the set It is clear that satisfies the requirement (a) in Lemma 3.1.When and is a constant vector in with , then we have Letting and, by direct calculation, we get This proves that condition (b) in Lemma 3.1 is satisfied. By now, we have proved that verifies all requirements of Lemma 3.1, then it follows that has at least one solution in ; that is, to say, (3.5) has at least one periodic solution in . Then we know that is an periodic solution of system (2.4) with strictly positive components. This completes the proof.

4. Uniqueness and Global Attractivity of Periodic Solutions

Under the hypotheses , we consider the following ordinary differential equation without impulsive: with the initial conditions .

The following lemmas will be helpful in the proofs of our results. The proof of the following Lemma 4.1 is similar to that of Theorem 1 in [18], and it will be omitted.

Lemma 4.1. Assume that hold, then one has the following. (i)If is a solution of (4.1) on , then is a solution of (2.4) on .(ii)If is a solution of (2.4) on , then is a solution of (4.1) on .

Lemma 4.2. Let denote any positive solution of system (4.1) with initial conditions . Assume that the following condition holds, Then there exists a such that where

Proof. From the second equation of (4.1), we can obtain By (4.5), we can derive the following.If , then .If , let . Then there exists such that , then , and also we have From what has been discussed above, we can easily conclude that, if , then is strictly monotone decreasing with speed at least . Therefore, there exists a such that , then .From the third equation of (4.1), we can obtain By (4.7), we can derive the followingIf , then .If , let . Then there exists such that , then , and also we have From what has been discussed above, we can easily conclude that, if , then is strictly monotone decreasing with speed at least . Therefore, there exists a such that , then .From the first equation of (4.1), we can obtain Then we have Set , then we have The proof is complete.

Lemma 4.3. Let hold. Assume that the following condition holds. Then there exists positive constants and such that, for all , in which

Proof. By the second equation of (4.1), It is easy to obtain that, for , where is defined in Lemma 4.1. If , then . If and let then there exists such that , then , and also we have Then we know that if , will strictly monotonically increase with speed . Thus, there exists such that if , then .By the third equation of (4.1), It is easy to obtain that for , where is defined in Lemma 4.2. If , then .If , and let then there exists such that , then , and also we have Then we know that if , will strictly monotonically increase with speed . Thus, there exists such that, if , then .Finally, by the third equation of (4.1), we obtain Thus, we have for . Set , then we have In the sequel, we formulate the uniqueness and global attractivity of the periodic solution in Theorem 4.4. It is immediate that if is global attractivity, then is in fact unique.

Theorem 4.4. In addition to –, assume further , where Then system (2.4) has a unique positive periodic solution which is global attractivity.

Proof. According to the conclusion of Theorem 3.2, we only need to show that the positive periodic solution of (2.4) is global asymptotical stable. Let be a positive periodic solution of system (2.4) let be any positive solution of system (2.4). Then is the positive periodic solution of (4.1), and is the positive solution of (4.1). It follows from Lemma 4.2 and 4.3 that there exists positive constants and (defined by Lemmas 4.2 and 4.3, resp.) such that, for all , Define Calculating the upper-right derivative of along the solution of (4.1), it follows for that where In the sequel, we will estimate under the following two cases.(i)If , then(ii)If , then Combining the conclusions of (4.29) and (4.30), we obtain It follows from (4.31) that where and are defined in Theorem 4.4. By hypothesis , there exist constants and such that Integrating both sides of (4.32) on interval yields It follows from (4.33) and (4.34) that Since and () are bounded for , so are uniformly continuous on . By Barbalat’s Lemma [32], we have Thus, By (4.37) and the first equation of (2.4), one can easily obtain that By Theorems 7.4 and 8.2 in [33], we know that the positive periodic solution of (2.4) is uniformly asymptotically stable. The proof of Theorem 4.4 is complete.

5. An Example

As an application of our main results, we consider the following system: in which , and . By direct computation, we can obtain Then . It is easy to check that (5.1) satisfies all the conditions of Theorems 3.2 and 4.4; hence, (5.1) has a positive periodic solution which is global attractivity.

Acknowledgments

The authors would like to thank the referees for their helpful comments and valuable suggestions regarding this paper. This work is supported by the National Natural Science Foundation of China (no. 10771215), the Scientific Research Fund of Hunan Provincial Education Department (no. 10C0560), the Doctoral Foundation of Guizhou College of Finance and Economics (2010) and the Science and technology Program of Hunan Province (no. 2010FJ6021).

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