Global Positive Periodic Solutions for Periodic Two-Species Competitive Systems with Multiple Delays and Impulses

Luo, Zhenguo; Luo, Liping; Yang, Liu; Zeng, Yunhui

doi:https://doi.org/10.1155/2014/785653

Abstract and Applied Analysis

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Analysis and Models in Interdisciplinary Mathematics

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Research Article | Open Access

Volume 2014 | Article ID 785653 | https://doi.org/10.1155/2014/785653

Global Positive Periodic Solutions for Periodic Two-Species Competitive Systems with Multiple Delays and Impulses

Zhenguo Luo,^1,2Liping Luo,¹Liu Yang,¹and Yunhui Zeng¹

Academic Editor: Francisco J. S. Lozano

Received19 Nov 2013

Accepted25 Feb 2014

Published03 Apr 2014

Abstract

A set of easily verifiable sufficient conditions are derived to guarantee the existence and the global stability of positive periodic solutions for two-species competitive systems with multiple delays and impulses, by applying some new analysis techniques. This improves and extends a series of the well-known sufficiency theorems in the literature about the problems mentioned previously.

1. Introduction

Throughout this paper, we make the following notation and assumptions:

let be a constant and , with the norm being defined by ; , with the norm being defined by ; , if exists, ; ; , with the norm being defined by ; , with the norm being defined by ;

then those spaces are all Banach spaces. We also denote that

In this paper, we investigate the existence, uniqueness, and global stability of the positive periodic solution for two corresponding periodic Lotka-Volterra competitive systems involving multiple delays and impulses: with initial conditions where , , , , , and are all in . Also , , , and are all in with , , , , , , , , , , . Furthermore, the intrinsic growth rates , are with , . For the ecological justification of (2) and (3) and similar types refer to [1–10].

In [1], Freedman and Wu proposed the following periodic single-species population growth models with periodic delay:

They had assumed that the net birth , the self-inhibition rate , and the delay are continuously differentiable -periodic functions, and , , , and for . The positive feedback term in the average growth rate of species has a positive time delay (the sign of the time delay term is positive), which is a delay due to gestation (see [1, 2]). They had established sufficient conditions which guarantee that system (5) has a positive periodic solution which is globally asymptotically stable.

In [3], Fan and Wang investigated the following periodic single-species population growth models with periodic delay:

They had assumed that the net birth , the self-inhibition rate , and the delay are continuously differentiable -periodic functions, and , , , and for . The negative feedback term in the average growth rate of species has a negative time delay (the sign of the time delay term is negative), which can be regarded as the deleterious effect of time delay on a species growth rate (see [4–6]). They had derived sufficient conditions for the existence and global attractivity of positive periodic solutions of system (6). But the discussion of global attractivity is only confined to the special case when the periodic delay is constant.

Alvarez and Lazer [7] and Ahmad [8] have studied the following two-species competitive system without delay:

They had derived sufficient conditions for the existence and global attractivity of positive periodic solutions of system (7) by using differential inequalities and topological degree, respectively. In fact, in many practical situations the time delay occurs so often. A more realistic model should include some of the past states of the system. Therefore, in [10], Liu et al. considered two corresponding periodic Lotka-Volterra competitive systems involving multiple delays: where , , , , , , , , , and are -periodic functions. Here, the intrinsic growth rates are -periodic functions with . They had derived the same criteria for the existence and globally asymptotic stability of positive periodic solutions of the above two competitive systems by using Gaines and Mawhin's continuation theorem of coincidence degree theory and by means of a suitable Lyapunov functional.

However, the ecological system is often deeply perturbed by human exploitation activities such as planting, harvesting, and so on, which makes it unsuitable to be considered continually. For having a more accurate description of such a system, we need to consider the impulsive differential equations. The theory of impulsive differential equations not only is richer than the corresponding theory of differential equations without impulses, but also represents a more natural framework for mathematical modeling of many real world phenomena [11–13]. Recently, some impulsive equations have been recently introduced in population dynamics in relation to population ecology [14–26] and chemotherapeutic treatment [27, 28]. However, to the best of the authors' knowledge, to this day, few scholars have done works on the existence, uniqueness, and global stability of positive periodic solution of (2) and (4). One could easily see that systems (5)–(9) are all special cases of systems (2) and (3). Therefore, we propose and study the systems (2) and (3) in this paper.

For the sake of generality and convenience, we always make the following fundamental assumptions. , , , , , , , and are all in ; , , , and are all in with , , , , , , , , and . satisfies , , are constants, and there exists a positive integer such that . Without loss of generality, we can assume that and , and then . is a real sequence such that , , is an -periodic function.

Definition 1. A function , is said to be a positive solution of (2) and (3), 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) and (3) for almost everywhere (for short a.e.) in and satisfies for , .
Under the above hypotheses ()–(), we consider the following nonimpulsive delay differential equation: with the initial conditions where

The following lemmas will be used in the proofs of our results. The proof of Lemma 2 is similar to that of Theorem 1 in [25].

Lemma 2. Suppose that ()–() hold; then(1)if is a solution of (10)–(12) on , then is a solution of (2)–(4) on ;(2)if is a solution of (2)–(4) on , then is a solution of (10)–(12) on .

Proof. It is easy to see that is absolutely continuous on every interval , , , and On the other hand, for any , , thus which implies that is a solution of (2); similarly, we can prove that is also a solution of (3). Therefore, , are solutions of (2)–(4) on . Similarly, if is a solution of (10)–(12) on , we can prove that are solutions of (2)–(4) on .
Since is absolutely continuous on every interval , , , and in view of (15), it follows that for any , which implies that is continuous on . It is easy to prove that is absolutely continuous on . Similarly, we can prove that is absolutely continuous on . Similar to the proof of , we can check that are solutions of (10)–(12) on . If is a solution of (2)–(4) on by the same method, we can prove that are solutions of (10)–(12) on . The proof of Lemma 2 is completed.

From Lemma 2, if we want to discuss the existence and global asymptotic stability of positive periodic solutions of systems (2)–(4), we only discuss the existence of the existence and global asymptotic stability of positive periodic solutions of systems (10)–(12).

The organization of this paper is as follows. In Section 2, we introduce several useful definitions and lemmas. In Section 3, first, we study the existence of at least one periodic solution of systems (2)–(4) by using continuation theorem proposed by Gaines and Mawhin (see [9]). Second, we investigate the global asymptotic stability of positive periodic solutions of the above systems by using the method of Lyapunov functional. As applications in Section 4, we study some particular cases of systems (2)–(4) which have been investigated extensively in the references mentioned previously.

2. Preliminaries

In this section, we will introduce some concepts and some important lemmas which are useful for the next section.

Let , be two real Banach spaces, let be a linear mapping, and let be 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 . Let denote the space of -periodic functions which are continuous for , are continuous from the left for , and have discontinuities of the first kind at point . We also denote that .

Definition 3 (see [11]). The set is said to be quasiequicontinuous in if for any there exists such that if , , , , and , then .

Definition 4. Let be a strictly positive periodic solution of (2)–(4). One says that is globally attractive if any other solution of (2)–(4) has the property , .

Lemma 5. The region is the positive invariable region of the systems (2)–(4).

Proof. By the definition of we have . In view of having Then the solution of (2)–(4) is positive. The proof of Lemma 5 is completed.

Lemma 6 (see [19, 29]). Suppose that and , . Then the function has a unique inverse satisfying with , and if , , then .

Proof. Since , , and is continuous on , it follows that has a unique inverse function on . Hence, it suffices to show that , . For any , by the condition , one can find that exists as a unique solution and exists as a unique solution ; that is, and ; that is, and .
As it follows that . Since , we have and . We can easily obtain that if , , , then , , where is the unique inverse function of , which together with implies that . The proof of Lemma 6 is completed.

Lemma 7 (see [9]). Let and be two Banach spaces, and let be a Fredholm operator with index zero. is an open bounded set, and let be L-compact on . Suppose that (a) for each and ;(b) for each ;(c). Then, equation has at least one solution lying in .

Lemma 8 (see [11]). The set is relatively compact if only if(1) is bounded, that is, , for each , and some ;(2) is quasiequicontinuous in .

Lemma 9 (see [30]). Assume that , are continuous nonnegative functions defined on the interval ; then there exists such that .

Lemma 10 (see [20, 31]). Suppose that is a differently continuous -periodic function on with (); then, for any , the following inequality holds:

Lemma 11 (see Barbalat's Lemma [32]). Let be a nonnegative function defined on such that is integrable and uniformly continuous on ; then .
In the following section, we only discuss the existence and global asymptotic stability of positive periodic solutions of systems (10)–(12).

3. Existence and Global Asymptotic Stability

Since , , , , , by Lemma 6, we see that all have their inverse functions. Throughout the following part, we set , , , and to represent the inverse function of , , , and , respectively. Obviously, , , , . We also denote that

Theorem 12. In addition to ()–(), assume that one of the following conditions hold: , ; , .
Then systems (3) and (4) have at least one positive -periodic solution, where , , , and are defined in (22).

Proof. Since the solutions of systems (11) and (12) remain positive for , we carry out the change of variable ; then (11) can be transformed to It is easy to see that if system (23) has one -periodic solution , then is a positive -periodic solution of system (10); that is to say, is a positive -periodic solution of system (2). Therefore, it suffices to prove that system (23) has a -periodic solution. In order to use Lemma 7 to (23), we take and define Then and are Banach spaces when they are endowed with the norm . Let withand define It is not difficult to show that and . So, is closed in , and is a Fredholm mapping of index zero. It is trivial to show that , are continuous projectors such that Furthermore, the generalized inverse (to L) exists and is given by Thus, for
Clearly, and are continuous. By applying Ascoli-Arzela theorem, one can easily show that , are relatively compact for any open bounded set . Moreover, is obviously bounded. Thus, is -compact on for any open bounded set . Now, we reach the position to search for an appropriate open bounded set for the application of Lemma 7. Corresponding to the operating equation , , we have Since is a -periodic function, we need only to prove the result in the interval . Integrating (33) over the interval leads to the following: Hence, we have Note that , and then there exists , such that Since , we can let , that is, ; then According to Lemma 7, we know that . Thus, By (37) and (38), we have Similarly, we obtain It follows from (35), (39), and (40) that we get Thus, from (41) we get where , , , and are defined by (22). On the other hand, by Lemma 7, we can see that , so we can derive Thus, from (43) we get On one hand, by (42), we have which implies that On the other hand, by (42), the integral mean value theorem that there are , , , and such that By , we have , which, together with (47), lead to Again, by , one can deduce that the following inequalities: It follows from (46), (48), and (49) that which together with (36) yield which implies that From the first equation of (32), we get where , , are defined by (22). By (46) and (53), we obtain Similarly, by the second equation of (32), we get where , , are defined by (22). From (52), (54), and (55) and Lemma 10, it follows that for that Let It follows from (56)–(57) that Clearly, , , are independent of , respectively. Note that , . From (44), we have which deduces that which implies that Hence From (61) and (62), it is easy to show that the system of algebraic equations has a unique solution . In view of (58), we can take sufficiently large R such that , and define , and it is clear that satisfies condition (a) of Lemma 7. Letting , then is a constant vector in with . Then that is, condition (b) of Lemma 7 holds. In order to verify condition (c) in the Lemma 7, by (62) and the formula for Brouwer degree, a straightforward calculation shows that By now we have proved that all the requirements in Lemma 7 hold. Hence system (32) has at least one -periodic solution, say . Setting , , then has at least one positive -periodic solution of systems (11) and (12). Furthermore, setting , , then has at least one positive -periodic solution of systems (3) and (4). If holds, similarly, we can prove that systems (2) and (4) have at least one positive -periodic solution. The proof of Theorem 12 is complete.

We now proceed to the discussion on the uniqueness and global stability of the -periodic solution in Theorem 12. It is immediate that if is globally asymptotically stable, then is unique in fact.

Theorem 13. In addition to ()–(), assume further that .
Then systems (3) and (4) have a unique positive -periodic solution which is globally asymptotically stable.

Proof. Letting be a positive -periodic solution of (3) and (4), then , is the positive -periodic solution of system (11) and (12), and let be any positive solution of system (11) with the initial conditions (12). It follows from Theorem 12 that there exist positive constants , , , such that, for all , By the assumptions of Theorem 12, we can obtain , and then there exist constants , ; we can choose a positive constant such that In the following, we always assume that and satisfy (67). We define Calculating the upper right derivative of along solutions of (11), it follows that We also define Calculating the upper right derivative of along solutions of (11), it follows that We define a Lyapunov functional as follows: Calculating the upper right derivative of along solutions of (11), it follows that So by (73), we have where which implies that By (76), it is obvious that is bounded.
On the other hand, we know that which implies that which, together with (66), yield From (66) and (79), it follows that is bounded for . Hence, , , and their derivatives remain bounded on . So , are uniformly continuous on . By Lemma 11, we have Therefore By Theorems 7.4 and 8.2 in [30], we know that the periodic positive solution is uniformly asymptotically stable. The proof of Theorem 13 is completed.

Theorem 14. In addition to ()–(), assume that one of the following conditions holds: , ; , .
Then systems (2) and (4) have at least one positive -periodic solution, where , , , and are defined in (22).

Proof. Since the solutions of systems (10) and (12) remain positive for , we carry out the change of variable , and then (10) can be transformed to It is easy to see that if system (82) has one -periodic solution , then is a positive -periodic solution of systems (10) and (12); that is to say, is a positive -periodic solution of systems (2) and (4). Therefore, it suffices to prove that system (82) has a -periodic solution. In order to use Lemma 6 for (81), we take and define Then and are Banach spaces when they are endowed with the norm . Let withand define It is not difficult to show that and . So, is closed in , and is a Fredholm mapping of index zero. It is trivial to show that , are continuous projectors such that . Furthermore, the generalized inverse (to L) exists and is given by Thus, for Clearly, and are continuous. By applying Ascoli-Arzela theorem, one can easily show that , are relatively compact for any open bounded set . Moreover, is obviously bounded. Thus, is -compact on for any open bounded set . Now, we reach the position to search for an appropriate open bounded set for the application of Lemma 6. Corresponding to the operating equation , , we have Since is a -periodic function, we need only to prove the result in the interval . Integrating (90) over the interval leads to the following: Hence, we have Noting that , then there exists , such that Since , we can let , that is, , , and then According to Lemma 6, we know that . Thus, By (37) and (38), we have Similarly, we obtain It follows from (92), (96), and (97) that we get Thus from (98) we get where , , , and are defined by (22). On the other hand, by Lemma 6, we can see that , so we can derive Thus, from (99) and (100), we get By (99), on one hand, we have which implies that On the other hand, by (99) the integral mean value theorem that there is , , , and such that By , we have , which together with (104), lead to the following: Again, by , one can deduce that the following inequalities: It follows from (103), (105), and (106) that which, together with (92) yield which implies that From the first equation of (90), we get where , , are defined by (22). By (103) and (110), we obtain Similarly, by the second equation of (90), we get where , , are defined by (22). From (109), (111), and (112) and Lemma 10, it follows that for Let It follows from (113)–(115) that Clearly, , , are independent of , respectively. Note that , , . From (44), we have which deduces that which implies that Hence From (119) and (120), it is easy to show that the system of algebraic equations has a unique solution . In view of (116), we can take sufficiently large such that , and define , and it is clear that satisfies condition (a) of Lemma 7. Letting , then is a constant vector in with . Then That is, condition (b) of Lemma 7 holds. In order to verify condition (c) in the Lemma 7, by (120) and the formula for Brouwer degree, a straightforward calculation shows that By now we have proved that all requirements in Lemma 7 hold. Hence system (82) has at least one -periodic solution, say . Setting , , then has at least one positive -periodic solution of systems (10) and (12). Furthermore, setting , , then has at least one positive -periodic solution of systems (2) and (4). If holds, similarly we can prove that systems (2) and (4) have at least one positive -periodic solution. The proof of Theorem 14 is complete.

We now proceed to the discussion on the uniqueness and global stability of the -periodic solution in Theorem 14. It is immediate that if is globally asymptotically stable, then is unique in fact.

Theorem 15. In addition to ()–(), assume further that .
Then systems (2) and (4) have a unique positive -periodic solution which is globally asymptotically stable.

Proof. Letting be a positive -periodic solution of (2) and (4), then , is the positive -periodic solution of systems (10) and (12), and let be any positive solution of system (10) with the initial conditions (12). It follows from Theorem 14 that there exist positive constants , such that for all By the assumptions of Theorem 14, we can obtain ; then there exist constants , ; we can choose a positive constant such that In the following, we always assume that and satisfy (67). We define Calculating the upper right derivative of along solutions of (10), it follows that We also define Calculating the upper right derivative of along solutions of (10), it follows that We define a Lyapunov functional as follows: Calculating the upper right derivative of along solutions of (10), it follows that So by (131), we have where which implies that By (134), it is obvious that is bounded.
On the other hand, we know that which implies that which, together with (123), yield From (124) and (137), it follows that are bounded for . Hence, , , and their derivatives remain bounded on . So , are uniformly continuous on . By Lemma 11, we have Therefore By Theorems 7.4 and 8.2 in [30], we know that the periodic positive solution is uniformly asymptotically stable. The proof of Theorem 15 is completed.

4. Applications

In this section, for some applications of our main results, we will consider some special cases of systems (2) and (3), which have been investigated extensively in [10].

Application 1. consider the following equations: which are special cases of systems (2) and (3) without impulse, respectively. By applying Theorems 12–15 to systems (140) and (141), respectively, we obtain the following theorems.

Theorem 16. In addition to , assume that the following conditions hold: , .
Then system (140) has a unique positive -periodic solution which is globally asymptotically stable, where , , , and are defined in (22).

Proof. It is similar to the proof of Theorems 12 and 13, so we omit the details here.

Theorem 17. In addition to , assume further that , .
Then system (140) has a unique positive -periodic solution which is globally asymptotically stable, where , , , and are defined in (22).

Proof. It is similar to the proof of Theorems 14 and 15, so we omit the details here.
We consider the following systems: which are special cases of systems (140) and (141), respectively. From Theorems 17 and 18, we have the following corollary.

Corollary 18. In addition to , assume that the following condition holds: .
Then systems (142) and (143) have a unique positive -periodic solution which is globally asymptotically stable, where are the inverses of functions .

Proof. It is similar to the proof of Theorems 12 and 13, so we omit the details here.
Application 2. Let us consider two delayed two-species competitive systems: which are special cases of systems (2) and (3) without impulse and , respectively. By applying Theorems 12–15 to systems (144) and (145), respectively, we obtain the following theorems.

Theorem 19. In addition to , assume that the following conditions hold: , .
Then system (144) has a unique positive -periodic solution which is globally asymptotically stable, where , , , are are defined as follows: And , , , and represent the inverse function of , , , and , respectively.

Proof. It is similar to the proof of Theorems 12 and 13, so we omit the details here.

Theorem 20. In addition to , assume further that , .
Then system (145) has a unique positive -periodic solution which is globally asymptotically stable, where , , , and are defined as follow: And , , , and represent the inverse function of , , , and , respectively.

Proof. It is similar to the proof of Theorems 14 and 15, so we omit the details here.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by NSF of China (nos. 11161015, 11371367, and 11361012), PSF of China (nos. 2012M512162 and 2013T60934), NSF of Hunan Province (nos. 11JJ900, 12JJ9001, and 13JJ4098), the Education Foundation of Hunan Province (nos. 12C0541, 12C0361, and 13C084), and the Construct Program of the Key Discipline in Hunan Province.

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Copyright © 2014 Zhenguo Luo et al. 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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