International Journal of Differential Equations

Volume 2013 (2013), Article ID 890281, 13 pages

http://dx.doi.org/10.1155/2013/890281

Research Article

Existence of Positive Periodic Solutions for Periodic Neutral Lotka-Volterra System with Distributed Delays and Impulses

Zhenguo Luo1,2 and Liping Luo²

¹Department of Mathematics, National University of Defense Technology, Changsha 410073, China

²Department of Mathematics, Hengyang Normal University, Hengyang, Hunan 421008, China

Received 20 April 2013; Accepted 13 May 2013

Academic Editor: Norio Yoshida

Copyright © 2013 Zhenguo Luo and Liping Luo. 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.

By using a fixed-point theorem of strict-set-contraction, we investigate the existence of positive periodic solutions for a class of the following impulsive neutral Lotka-Volterra system with distributed delays: Some verifiable criteria are established easily.

1. Introduction

It is well known that natural environments are physically highly variable, and in response, birth rates, death rates, and other vital rates of populations vary greatly in time. Theoretical evidence suggests that many population and community patterns represent intricate interactions between biology and variation in the physical environment (see [1–4]). Thus, the focus in theoretical models of population and community dynamics must be not only on how populations depend on their own population densities or the population densities of other organisms, but also on how populations change in response to the physical environment. It is reasonable to study the models of population with periodic coefficients. In addition to the theoretical and practical significance, the Lotka-Volterra model is one of the famous models for dynamics of population; therefore it has been studied extensively [5–9]. In view of the above effects, by applying a fixed-point theorem of strict-set-contraction, Li [10] established criteria to guarantee the existence of positive periodic solutions of the following neutral Lotka-Volterra system with distributed delays:

where are -periodic functions and and satisfying , , . On the other side, birth of many species is an annual birth pulse or harvesting. To have a more accurate description of many mathematical ecology systems, we need to consider the use of impulsive differential equations [11–13]. Some qualitative properties such as oscillation, periodicity, asymptotic behavior, and stability properties have been investigated extensively by many authors over the past few years [14–18]. However, to our knowledge, there are few published papers discussing the existence of periodic solutions for neutral Lotka-Volterra system with distributed delays and impulses. In this paper, we are concerned with the following neutral Lotka-Volterra system with distributed delays and impulses:

where are -periodic functions and and satisfying , . Moreover,  (here represents the right limit of at the point ), ; that is, changes decreasingly suddenly at times ;   is a constant, , . We assume that there exists an integer such that , where .

The main purpose of this paper is by using a fixed-point theorem of strict-set-contraction [19, 20] to establish new criteria to guarantee the existence of positive periodic solutions of the system (2).

For convenience, we introduce the notation

where is a continuous -periodic function.

Throughout this paper, we assume that

; ; ; ; .

The paper is organized as follows. In Section 2, we give some definitions and lemmas to prove the main results of this paper. In Section 3, by using a fixed-point theorem of strict-set-contraction, we established some criteria to guarantee the existence of at least one positive periodic solution of system (2). Finally, in Section 4, we give an example to show the validity of our result.

2. Preliminaries

In order to obtain the existence of a periodic solution of system (2), we first introduce some definitions and lemmas.

Definition 1 (see [13]). A function is said to be a positive solution of (2), if the following conditions are satisfied:

(a) is absolutely continuous on each ;(b)for each , and exist and ;(c) satisfies the first equation of (2) for almost everywhere in and satisfies the second equation of (2) at impulsive point .

Definition 2 (see [21]). Let be a real Banach space and a closed, nonempty subset of . is a cone provided

(i) for all and all ,(ii) imply .

Definition 3 (see [21]). Let be a bounded subset in . Define

where denotes the diameter of the set , obviously,  . So is called the Kuratowski measure of noncompactness of  .

Definition 4 (see [21]). Let be two Banach spaces and , a continuous and bounded map , is called k-set contractive if, for any bounded set , we have

is called strict-set-contractive if it is -set-contractive for some .

Definition 5 (see [22]). The set is said to be quasi-equicontinuous in if for any there exists such that if ,  ,  ,  and , then .

Lemma 6 (see [22]). The set is relatively compact if and only if

(1) is bounded; that is, ,  for each , and some ;(2) is quasi-equicontinuous in .

Lemma 7. is an-periodic solution of (2) is equivalent to is an-periodic solution of the following equation:

where

Proof. Assume that , is a periodic solution of (2). Then, we have

Integrating the previous equation with , we can have

where , , , . Therefore,

which can be transformed into

Thus, is a periodic solution of (6). If , is a periodic solution of (6), for any ,  from (6) we have

For any , , we have from (6) that

Hence is a positive -periodic solution of (2). Thus we complete the proof of Lemma 7.

Lemma 8 (see [20, 23]). Let be a cone of the real Banach space and with . Assume that is strict-set-contractive such that one of the following two conditions is satisfied:

(a),  for all   and ,  for all  ;(b),  for all   and ,  for all  .

Then has at least one fixed point in .

In order to apply Lemma 7 to system (1), one sets

Define

with the norm defined by , where , and

with the norm defined by , where . Then and are both Banach spaces. Define the cone in by

Let the map be defined by

where , ,

where

It is obvious to see that , and

In what follows, we will give some lemmas concerning and defined by (17) and (18), respectively.

Lemma 9. Assume that hold.

(i)If , then is well defined.(ii)If holds and , then is well defined.

Proof. For any , it is clear that . From (18), for , we have

That is, . So . In view of , for , we have

Therefore, for , we find

Now, we show that , , . From (18), we obtain

It follows from (23) and (25) that if , then

On the other hand, from (25) and , if , then

It follows from (27) and (28) that . So . By (25), we have . Hence, . This completes the proof of (i). In view of the proof of (ii), we only need to prove that , implies

From (23), (26),  , and , we have

The proof of (ii) is complete. Thus we complete the proof of Lemma 9.

Lemma 10. Assume that hold and .

(i)If , then is strict-set-contractive.(ii)If holds and , then is strict-set-contractive, where .

Proof. We only need to prove (i), since the proof of (ii) is similar. It is easy to see that is continuous and bounded. Now we prove that a for any bounded set . Let . Then, for any positive number , there is a finite family of subsets satisfying with . Therefore

As and are precompact in , it follows that there is a finite family of subsets of such that and

In addition, for any and , we have

Hence,

Applying the Arzela-Ascoli theorem, we know that is precompact in . Then, there is a finite family of subsets of such that and

From (23), (26), (31)–(34), and , for any , we have

where

From (36) and (37) we obtain

As is arbitrarily small, it follows that

Therefore, is strict-set-contractive. The proof of Lemma 10 is complete.

3. Main Results

Our main result of this paper is as follows.

Theorem 11. Assume that , hold.

(i)If , then system (2) has at least one positive -periodic solution.(ii)If holds and , then system (2) has at least one positive -periodic solution.

Proof. We only need to prove (i), since the proof of (ii) is similar. Let

Then it is easy to see that . From Lemmas 9 and 10, we know that is strict-set-contractive on . In view of (26), we see that if there exists such that , then is one positive -periodic solution of system (2). Now, we will prove that condition (ii) of Lemma 7 holds. First, we prove that , for all . Otherwise, there exists , such that . So and , which implies that

Moreover, for , we have

In view of (42) and (43), we obtain

which is a contradiction. Finally, we prove that ,  for all ,   also hold. For this case, suppose, for the sake of contradiction, that there exist such that . Furthermore, for any , we have

In addition, for any , we find

Thus, we have

From (45) and (47), we obtain

which is a contradiction. Therefore, condition (ii) of Lemma 7 holds. By Lemma 7, we see that has at least one nonzero fixed point in . Thus, the system (6) has at least one positive -periodic solution. Therefore, it follows from Lemma 6 that system (2) has a positive -periodic solution. The proof of Theorem 11 is complete.

Remark 12. If , we can easily derive the corresponding results in [10]. So we extend the corresponding results in [10].

4. Example

In this section, we give an example to show the effectiveness of our result.

Example 1. Consider the following nonimpulsive system:

where , and satisfying , . Obviously

Furthermore, we obtain

Therefore,

Hence, , hold and . According to Theorem 11, system (49) has at least one positive -periodic solution.

Acknowledgments

The authors are thankful to the referees and editor for the improvement of the paper. This work was supported by the Construct Program of the Key Discipline in Hunan Province. Research was supported by the National Natural Science Foundation of China (10971229, 11161015), the China Postdoctoral Science Foundation (2012M512162), and Hunan Provincial Natural Science-Hengyang United Foundation of China (11JJ9002).

References

P. Chesson, “Understanding the role of environmental variation in population and community dynamics,” Theoretical Population Biology, vol. 64, no. 3, pp. 253–254, 2003.

J. Zhen and Z. E. Ma, “Periodic solutions for delay differential equations model of plankton allelopathy,” Computers & Mathematics with Applications, vol. 44, no. 3-4, pp. 491–500, 2002.

X. Liu and G. Ballinger, “Boundedness for impulsive delay differential equations and applications to population growth models,” Nonlinear Analysis: Theory, Methods & Applications, vol. 53, no. 7-8, pp. 1041–1062, 2003.

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, NY, USA, 1993.

Y. K. Li and Y. Kuang, “Periodic solutions of periodic delay Lotka-Volterra equations and systems,” Journal of Mathematical Analysis and Applications, vol. 255, no. 1, pp. 260–280, 2001.

Y. K. Li and L. F. Zhu, “Existence of periodic solutions of discrete Lotka-Volterra systems with delays,” Bulletin of the Institute of Mathematics, Academia Sinica, vol. 33, no. 4, pp. 369–380, 2005.

H. Zhao and N. Ding, “Existence and global attractivity of positive periodic solution for competition-predator system with variable delays,” Chaos, Solitons & Fractals, vol. 29, no. 1, pp. 162–170, 2006.

Y. Song, M. Han, and Y. Peng, “Stability and Hopf bifurcations in a competitive Lotka-Volterra system with two delays,” Chaos, Solitons & Fractals, vol. 22, no. 5, pp. 1139–1148, 2004.

Z. Yang and J. Cao, “Positive periodic solutions of neutral Lotka-Volterra system with periodic delays,” Applied Mathematics and Computation, vol. 149, no. 3, pp. 661–687, 2004.

Y. K. Li, “Positive periodic solutions of periodic neutral Lotka-Volterra system with distributed delays,” Chaos, Solitons & Fractals, vol. 37, no. 1, pp. 288–298, 2008.

A. M. Samoikleno and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.

S. T. Zavalishchin and A. N. Sesekin, Dynamic Impulse Systems, Theory and Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.

V. Lakshmikantham, D. D. Bainov, and P. S. Simenov, Theory of Impulsive Differential Equations, World Science, Singapore, 1989.

J. R. Yan, A. P. Zhao, and J. J. Nieto, “Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems,” Mathematical and Computer Modelling, vol. 40, no. 5-6, pp. 509–518, 2004.

X. Liu and G. Ballinger, “Boundedness for impulsive delay differential equations and applications to population growth models,” Nonlinear Analysis: Theory, Methods & Applications, vol. 53, no. 7-8, pp. 1041–1062, 2003.

W. Zhang and M. Fan, “Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays,” Mathematical and Computer Modelling, vol. 39, no. 4-5, pp. 479–493, 2004.

S. L. Sun and L. S. Chen, “Existence of positive periodic solution of an impulsive delay logistic model,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 617–623, 2007.

X. Li, X. Lin, D. Jiang, and X. Zhang, “Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects,” Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 4, pp. 683–701, 2005.

R. E. Gaines and J. L. Mawhin, Coincidence Degree Theory and Nonlinear Differential Equations, Springer, Berlin, Germany, 1977.

N. P. Các and J. A. Gatica, “Fixed point theorems for mappings in ordered Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 71, no. 2, pp. 547–557, 1979.

D. J. Guo, Nonlinear Functional Analysis, in Chinese, ShanDong Science and Technology Press, 2001.

D. Baĭnov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66 of Pitman Monographs and Surveys in Pure and Applied Mathematics, 1993.

D. J. Guo, “Positive solutions to nonlinear operator equations and their applications to nonlinear integral equations,” Advances in Mathematics, vol. 13, no. 4, pp. 294–310, 1984 (Chinese).