International Journal of Differential Equations

Volume 2013 (2013), Article ID 617824, 13 pages

http://dx.doi.org/10.1155/2013/617824

Research Article

Global Positive Periodic Solutions of Generalized -Species Gilpin-Ayala Delayed Competition Systems with Impulses

Zhenguo Luo,1,2 Liping Luo,¹ Jianhua Huang,² and Binxiang Dai³

¹Department of Mathematics, Hengyang Normal University, Hengyang 421008, China

²Department of Mathematics, National University of Defense Technology, Changsha 410073, China

³School of Mathematical Sciences and Statistics, Central South University, Changsha 410075, China

Received 9 September 2013; Accepted 4 October 2013

Academic Editor: Tongxing Li

Copyright © 2013 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.

Abstract.

We consider the following generalized -species Lotka-Volterra type and Gilpin-Ayala type competition systems with multiple delays and impulses: , , ; , , By applying the Krasnoselskii fixed-point theorem in a cone of Banach space, we derive some verifiable necessary and sufficient conditions for the existence of positive periodic solutions of the previously mentioned. As applications, some special cases of the previous system are examined and some earlier results are extended and improved.

1. Introduction

In the recent decades, the traditional Lotka-Volterra competition systems have been studied extensively. One of the models is the following competition system:

Many results concerned with the permanence, global asymptotic stability, and the existence of positive periodic solutions of system (1) are obtained; we refer to [1–10] and the reference therein. However, the Lotka-Volterra type models have often been severely criticized. One of the criticisms is that, in such a model, the per capita rate of change of the density of each species is a linear function of densities of the interacting species. In 1973, Ayala et al. [11] conducted experiments on fruit fly dynamics to test the validity of ten models of competitions. One of the models accounting best for the experimental results is given by

In order to fit data in their experiments and to yield significantly more accurate results, Gilpin and Ayala [12] claimed that a slightly more complicated model was needed and proposed the following competition model:

whereis the population density of the ith species,is the intrinsic exponential growth rate of the ith species,is the environmental carrying capacity of speciesin the absence of competition,provides a nonlinear measure of interspecific interference, andprovides a measure of interspecific interference. [13–15] obtained sufficient conditions which guarantee the global asymptotic stability of system (3). Chen [16] investigated the following-species nonautonomous Gilpin-Ayala competitive model:

For each, they established a series of criteria under whichof the species of system (4) were permanent while the remainingspecies were driven to extinction. In [17], Fan and Wang further studied the following delay Gilpin-Ayala type competition model:

They obtained a set of easily verifiable sufficient conditions for the existence of at least one positive periodic solution of the system (5) by applying the coincidence degree theory. Recently, in [18], Chen investigated the following-species Gilpin-Ayala type competition systems:

He established a series of criteria under whichof the species in the system (6) were permanent while the remainingspecies were driven to extinction. In [19], Xia et al. considered the following almost periodic nonlinear-species competitive Lotka-Volterra model:

By using comparison theorem and constructing suitable Lyapunov functional, they derived a set of sufficient conditions for the existence and global attractivity of a unique positive almost periodic solution of the previously mentioned model. Motivated by the previous ideas, in [20], Yan considered the following generalized periodic-species Gilpin-Ayala type competition models in periodic environments with deviating arguments of the form:

By using a fixed point theorem in cone and the proof by contradiction, he obtained a necessary and sufficient condition for the existence of positive periodic solutions (with strictly positive components) of the system (8).

However, the ecological system is often deeply perturbed by human exploitation activities such as planting and harvesting, 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 is not only 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 (see [21–23]). In recent years, some impulsive equations have been recently introduced in population dynamics in relation to population ecology; we refer the reader to [24–36] and the reference therein. However, to this day, only a little work has been done on the existence of positive periodic solutions to the generalized periodic-species Gilpin-Ayala type competition models in periodic environments with deviating arguments of the form and impulses. Motivated by this, in this paper, we mainly consider the following-species Gilpin-Ayala type competition models in periodic environments with deviating arguments of the form and impulses:

with initial conditions

whererepresents the density of the ith speciesat time,is the intrinsic growth rate of the ith speciesat this time,may be negative while;,,denote the competitive coefficient between the ith speciesand jth species, and,,,,are continuous-periodic functions with,,,,,  ,,,corresponding to the time delays with;are positive constants. Assume that,,are constants and there exists an integersuch that,,where.

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

 , with the norm defined by; , with the norm defined by; ; ; , with the norm defined by; , with the norm defined by.

The previously mentioned spaces are all Banach spaces. We also denote

and make the following assumptions:

   , and,.   satisfiesand.   is a real sequence with, andis an-periodic function.   ,are constants such that,.

To conclude this section, we summarize in the following definition and lemma that will be needed in our arguments.

Definition 1. A functionis said to be a positive solution of the system (9) and (10), if the following conditions are satisfied:

(a)is absolutely continuous on each.(b)for each,andexist, and.(c)satisfies the first equation of the system (9) and (10) for almost everywhere (for short a.e.) inand satisfiesfor,.

Under the previously mentioned hypotheses ()–(), we consider the following nonimpulsive Lotka-Volterra competitive systems:

with initial conditions

where

By a solutionof the system (12) and (13), it means an absolutely continuous functiondefined onthat satisfies (12) and (13).

The following lemma will be used in the proofs of our results. The proof of Lemma 2 is similar to that of Theoremin [24].

Lemma 2. Suppose that ()–() hold. Then

(i)ifis a solution of (12) and (13) on, thenis a solution of (9) and (10) on;(ii)ifis a solution of (9) and (10) on, thenis a solution of (12) and (13) on.

Proof. (i) It is easy to see thatis absolutely continuous on every interval,,

On the other hand, for any,

thus,

It follows from (15)–(17) thatis a solution of the system (9) and (10). Similarly, ifis a solution of the system (12) and (13), we can prove thatis a solution of the system (9) and (10).

(ii) Sinceis absolutely continuous on every interval,,and in view of (16), it follows that for any

which implies thatis continuous on. It is easy to prove thatis absolutely continuous on. Similar to the proof of, we can check thatare solutions of (8) on. Similarly, ifis a solution of the system (9) and (10), we can prove thatis a solution of the system (12) and (13). The proof of Lemma 2 is completed.

In the following section, we only discuss the existence of a periodic solution for the system (12) and (13).

The paper is organized as follows. In the next section, we give some definitions and lemmas. In Section 3, we derive a necessary and sufficient condition ensuring at least one positive periodic solution of the system, by using the Krasnoselskii fixed-point theorem in the cone of Banach space. In Section 4, as applications, we consider some particular cases of the system which have been investigated extensively in the references mentioned previously.

2. Preliminaries

We will first make some preparations and list a few preliminary results. Letwith the norm,It is easy to verify thatis a Banach space.

We define an operatoras follows:

where

It is clear that,,.

In view of, for, we define

Defineas a cone inby

We easily verify thatis a cone in. For convenience of expressions, we define an operatorby

The proof of the main result in this paper is based on an application of the Krasnoselskii fixed-point theorem in cones. Firstly, we need to introduce some definitions and lemmas.

Definition 3. Letbe a real Banach space andbe a closed, nonempty subset of.is said to be a cone if

(1)for all, and,(2)imply.

Lemma 4 (see [37–39]). Letbe a cone in a real Banach space. Assume thatandare open subsets ofwithwhere,Letbe a continuous and completely continuous operator satisfying

(1), for any;(2)The fact that there existssuch that, for anyand.

Then,has a fixed point inThe same conclusion remains valid if (1) holds for for anyand (2) holds for anyand.

Lemma 5. Assume that (H ₁)–(H ₄) hold. Then the solutions of the system (12) and (13) are defined onand are positive.

Proof. By Lemma 2, we only need to prove that the solutionsof (12) and (13) are defined onand are positive on. From (12), we have that for anyand

Therefore,are defined onand are positive on. The proof of Lemma 5 is complete.

Lemma 6. Assume that (H ₁)–(H ₄) hold. Thenis well defined.

Proof. In view of the definitions ofand, for any, we have

Therefore,. Furthermore, for any, it follows from (20) that

On the other hand, for any, we obtain

Therefore,The proof of Lemma 6 is complete.

Lemma 7. The operatoris continuous and completely continuous.

Proof. By using a standard argument one can show thatis continuous on. Now, we show thatis completely continuous. Letbe any positive constant anda bounded set. For any, by (20), we have

Therefore, for any, we obtain

which implies thatis a uniformly bounded set. On the other hand, in view of the definitions ofand, we have

Again, from (20), we obtain

which implies that, for any, is also uniformly bounded. Hence,is a family of uniformly bounded and equi-continuous functions. By the well-known Ascoli-Arzela theorem, we know that the operatoris completely continuous. The proof of Lemma 7 is complete.

Lemma 8. Assume that (H ₁)–(H ₄) hold. The existence of positive-periodic solution of the system (12) and (13) is equivalent to that of nonzero fixed point ofin.

Proof. Assume thatis a periodic solution of (12) and (13). Then, we have

Integrating the previous equation over, we can have

Therefore, we have

which can be transformed into

Thus,is a periodic solution for system (12) and (13).

If, andwith, then for anyderivative the two sides of (20) about,

Hence,is a positive-periodic solution of (12) and (13). Thus we complete the proof of Lemma 8.

3. Existence of Periodic Solution of the System

Now, we are at the position to study the existence of positive periodic solutions of system (9) and (10). We mainly apply the Krasnoselskii fixed-point theorem in the cone of Banach space under some conditions to prove the main Theorem 9.

Theorem 9. Assume (H ₁)–(H ₄). System (9) and (10) has at least one positive-periodic solution if and only if the condition

holds.

Proof (sufficiency). Let

by condition (37), we know that. Choose a constantsuch that. Letand

For any,, from (20), we obtain

Hence, for any,, we have

which implies that condition (1) in Lemma 4 is satisfied.

On the other hand, we choosesuch that. Letand suppose. We show that, for anyand,. Otherwise, there existand, such that. Let; since, it follows that

which is a contradiction. This proves that condition (2) in Lemma 4 is also satisfied. By Lemmas 4 and 8, system (12) and (13) has at least one positive omega-periodic solution. From Lemma 2, system (9) and (10) has at least one positive-periodic solution.

(Necessity). Suppose that (37) does not hold. Then, there exists at least ansuch that

If system (12) and (13) has a positive-periodic solution, then we have

Integrating the previous equation overwe can have

which is a contradiction. The proof of Theorem 9 is complete.

4. Applications

In this section, we apply the results obtained in the previous section to some-species competition systems which are mentioned in Section 1. If we consider the environmental or biological factors, the assumption of the periodic oscillation of the parameters and impulse functions seems realistic and reasonable in view of any seasonal phenomena which they might be subject to, for example, mating habits, availability of food, weather conditions, and so forth.

Application 1. We consider the following three classes of periodic-species competition systems with impulses:

which are special cases of system (9), whereare the same as in ()–(). We denote

from Theorem 9, we have the following results.

Corollary 10. Assume that (H ₁)–(H ₄) hold. The systems (46) and (47) have at least one positive-periodic solution if and only if the following condition

holds.

Corollary 11. Assume that (H ₁)–(H ₄) hold. The system (48) has at least one positive-periodic solution if and only if the following condition

holds.

Application 2. We consider the following-species Gilpin-Ayala type competition systems with impulses:

which is a special case of system (9), whereare-periodic,are constants such that, and,,,. We denote

from Theorem 9, we have the following result.

Corollary 12. Assume that (H ₁)–(H ₄) and the condition

hold, the system (52) has at least one positive-periodic solution.

Application 3. We study the following periodic nonlinear-species competitive Lotka-Volterra models with impulses:

which is a special case of system (9), whereare-periodic, and,,,. We denote

from Theorem 9, we have the following result.

Corollary 13. Assume that (H ₁)–(H ₄) and the condition

hold, the system (55) has at least one positive-periodic solution.

Application 4. We study the following generalized periodic-species Gilpin-Ayala type competition models in periodic environments with deviating arguments of the form and impulses:

which is a special case of system (9), whereare-periodic,

are constants such that, and,,,,,. We denote

from Theorem 9, we have the following result.

Corollary 14. Assume that (H ₁)–(H ₄) and the condition

hold, the system (58) has at least one positive-periodic solution.

Application 5. We study the following two-species competitive systems with impulses:

which is a special case of system (9), whereare-periodic. We denote

from Theorem 9, we have the following result.

Corollary 15. Assume that (H ₁)–(H ₄) and the condition

hold, the system (61) has at least one positive-periodic solution.

Remark 16. We apply the main results obtained in the previous section to study some examples which have some biological implications; from the previous corollaries, we see that, under the appropriate conditions, the impulsive perturbations do not affect the existence of periodic solution of systems. However, if the impulsive perturbations are unbounded, some properties of the solution of systems could be changed significantly, which will be our further work.

Acknowledgments

The research was supported by the National Natural Science Foundation of China (10971229, 11161015, 11371367 and 11361012), the China Postdoctoral Science Foundation (2012M512162 and 2013T60934), Hunan Provincial Natural Science-Hengyang United Foundation of China (11JJ9002, 12JJ9001, 13JJ4098), the Education Foundation of Hunan province (nos. 12C0541, and 13C084), the Science Foundation of Hengyang Normal University (no. 11B36), and the Construct Program of the Key Discipline in Hunan province.

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