Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 639138, 14 pages

http://dx.doi.org/10.1155/2013/639138

Research Article

Permanence and Almost Periodic Solution for an Enterprise Cluster Model Based on Ecology Theory with Feedback Controls on Time Scales

Yuanhong Zhi, Zunling Ding, and Yongkun Li

Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China

Received 5 May 2013; Revised 2 September 2013; Accepted 2 September 2013

Academic Editor: Josef Diblík

Copyright © 2013 Yuanhong Zhi 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 present a model with feedback controls based on ecology theory, which effectively describes the competition and cooperation of enterprise cluster in real economic environments. Applying the comparison theorem of dynamic equations on time scales and constructing a suitable Lyapunov functional, sufficient conditions which guarantee the permanence and the existence of uniformly asymptotically stable almost periodic solution of the system are obtained.

1. Introduction

In recent years, a few researchers have presented some models about enterprise clusters based on ecology theory, which arouse growing interest in applying the methods of ecology and dynamic system theory to study enterprise clusters, for example [1–9] and references cited therein. In [1], two models from biology were given and explained by economic view, and sufficient conditions were obtained to guarantee the coexistence and stability of enterprise clusters. In [3], the developing strategy of enterprise clusters was analyzed based on the logistic model, and the suggestions of constructing cooperative relation and choosing generalization or specialization tactics for commodity were put forward. In addition, based on the theoretical model of ecological population science, Wang and Pan [6] made a detailed analysis to the equilibrium mechanism of enterprise clusters, including net model and center halfback model and drew a conclusion that the relationship of pierce competition and beneficial cooperation among enterprise clusters was the crucial factor for them to keep stability. More related research about enterprises cluster one can refer to the literatures [10–13]. Recently, the literature [5] considered the competition and cooperation system of two enterprises based on ecosystem:

where , represent the output of enterprises and , , are the intrinsic growth rate, denotes the carrying capacity of market under nature unlimited conditions, , are the competitive coefficients of two enterprises, , are the initial production of two enterprises. Accordingly, we consider now the equation with nonconstant coefficients, which can be obtained as a modified system (1) with variable coefficients (Letting , , , in system (1)):

In real world, the situation of enterprises is often distributed by unpredictable forces which can result in changes in enterprises' parameters such as intrinsic growth rates. So it is necessary to study models with control variables which are so-called disturbance functions [14–17].

As well known, both continuous and discrete systems are very important in implementation and applications, but it is troublesome to study the permanence and the existence of almost periodic solutions for continuous system and discrete system, respectively. It is very important to study that on time scales which can unify the continuous and discrete situations.

Motivated by the above statements, we consider the following competitive and cooperation model of a satellite enterprises and a dominant enterprise with feedback controls on time scale :

As pointed in [18, 19], periodic phenomenon and almost periodic phenomenon are widespread in nature and almost periodic phenomena is more frequent than periodic phenomenon. Hence, they have been the object of intensive analysis by numerous authors. In particular, there have been extensive results on existence of almost periodic solutions of differential equations in the literature. Some of these results can be found in [20–24]. Our main purpose of this paper is by using the comparison theorem of dynamic equations on time scales and constructing a suitable Lyapunov functional to study the permanence and the existence of almost periodic solutions of (3).

Remark 1. Let , ; if (the set of all real numbers), then (3) reduces to

If (the set of all integers), then (3) reduces to

Therefore, we only consider the permanence and existence of nonnegative almost periodic solution of system (3), then the permanence and almost periodic solution of (4) and (5) can be obtained as direct results of (3).

Let and denote the nonempty closed subset (time scales) of . Let be a continuous bounded function defined on , and we denote

Throughout this paper, we assume that

)  , , , , , , and are almost periodic functions on time scale such that

 where is the set of positively regressive functions from to .

2. Preliminaries

In this section, we will recall some definitions and lemmas which will be used in the proof of our main results.

Let be a nonempty closed subset (time scale) of . The forward and backward jump operators and the graininess (the set of all nonnegative real numbers) are defined, respectively, by

A point is called left-dense if and , left-scattered if , right-dense if and , and right-scattered if . If has a left-scattered maximum , then ; otherwise, . If has a right-scattered minimum , then ; otherwise .

A function is called rd-continuous provided it is continuous at right-dense points in (i.e., ) and its left-sided limits exist at left-dense points in . If is continuous at each right-dense point and each left-dense point, then is said to be a continuous function on . The set of continuous functions will be denoted by .

For and ; we define the delta derivative of , to be the number (if it exists) with the property that, for any , there exists a neighborhood of such that

for all . Let be right-dense continuous. If , then we define the delta integral by

A function is called regressive provided for all . The set of all regressive and rd-continuous functions will be denoted by . If , , then is a positively regressive function from to . If is a regressive function, then the generalized exponential function is defined by

with the cylinder transformation

Definition 2 (see [25, 26]). Let be two regressive functions, we define

Then the generalized exponential function has the following properties.

Lemma 3 (see [25, 26]). Assume that are two regressive functions; then

(i) and ; (ii); (iii); (iv); (v); (vi)If , then .

Definition 4 (see [27]). A time scale is called an almost periodic time scale if

Throughout this paper, we restrict our discussion on almost periodic time scales.

Definition 5 (see [27]). A function is called an almost periodic function if the -translation number set of :

is a relatively dense set in for all ; that is, for any given , there exists a number such that for any interval with length , there is a number in this interval such that

Definition 6 (see [27]). Let , where is an open set in and is said to be almost periodic in uniformly for or uniformly almost periodic for short. If for any compact set in , there exists a positive number such that any interval of length contains a number for which for all and , is called the -translation number of .

Lemma 7 (see [27]). A is an almost periodic function if and only if for any sequence there exists a subsequence such that converges uniformly on as . Furthermore, the limit function is also an almost periodic function.

Definition 8. System (3) is said to be permanent if there exist positive constants , , , which are independent of the solutions of the system, such that any solution of system (3) satisfies

Consider the following nonlinear almost periodic differential system on time scales:

and the product system of (18)

Lemma 9 (see [28, 29]). Suppose that there exists a Lyapunov functional defined for satisfying the following conditions:

(i), where and is increasing}; (ii), where is a constant; (iii), where and .

In addition, if there exists a solution of system (18) such that for , where is any compact subset of  , then there exists a unique uniformly asymptotically stable almost periodic solution of system (18). In particular, if is periodic of period , then there exists a unique uniformly asymptotically stable periodic solution of (18) of period .

3. Persistence

We need the following lemma to obtain the permanence of system (3).

Lemma 10 (see [28, 29]). Let .

(i) If , then for

 In particular, if ,  , we have . (ii) If , then for

 In particular, if ,  , we have .

Proposition 11. Assume that (H ₁) holds; then every solution of system (3) satisfies

where

Proof. Let be any solution of system (3); it follows from the first equation of system (3) and the Bernoulli inequality for , that

By applying (i) of Lemma 10 to the differential inequality above, we have

For any positive constant small enough, it follows from (25) that there exists a large enough such that for all . Then the second equation and the third equation of system (3) lead to

That is,

By applying Lemma 10, it follows from (27) that

Letting in the above inequality leads to

By using similar arguments as those in the proofs of (28), it follows that

This completes the proof.

Proposition 12. Assume that (H ₁) holds; suppose further that

()  , then every solution    of system  (3)  satisfies

 where

Proof. Let be any positive solution of system (3). We first prove that

For any small enough constant , there exists a point and such that

By the first equation of system (3) and condition , we have

We assert

otherwise, we assume that there exists such that

then we get

Equation (38) implies , which is a contradiction; therefore (36) holds for .

Consequently,

then

Now, for any small enough , there exists a point such that ; from the third equation of system (3), we have

by using a similar argument as that in (40), we can get

Next we prove

Since then there exists such that for any and . By the second equation of system (3), we have

Applying (ii) of Lemma 10, we have

similar to the above proof, it is easy to obtain that

So the proof of Proposition 12 is complete.

Theorem 13. Assume that (H ₁)-(H ₂) hold; then system (3) is persistent.

We denote by the set of all solutions of system (3) satisfying

for all . It should be noticed that, from the proofs of Propositions 11 and 12, we know that the conditions of Theorem 13 hold and is an invariant set of system (3).

4. Existence of Uniformly Asymptotically Stable Almost Periodic Solution

According to Lemma 9, we first prove that there exists a bounded solution of system (3) and then construct an adaptive Lyapunov functional for system (3). The next result tells us that there exists a bounded solution of (3). To be precise, consider the following.

Proposition 14. Assume that (H ₁)-(H ₂) hold, then .

Proof. By the almost periodicity of , , , , , , , , there exists a sequence with as such that

Let be an arbitrary small number; it follows from Propositions 11 and 12 that there exists a positive number such that

Denote

For any positive integer , it is easy to see that there exist sequences

such that the sequences have subsequences, denoted by again, converging on any finite interval of as , respectively. Thus we have functions such that

Combining with

gives

We can easily see that is a solution of system (3) and

for . Since is an arbitrary small positive number, it follows that

for . This completes the proof.

Theorem 15. Assume that (H ₁)- (H ₂) hold. Suppose further that

() and , where ,

 then (3) has a unique almost periodic solution that is uniformly asymptotically stable and satisfies

 for all .

Proof. From Proposition 14, we know that system (3) has a bounded solution satisfying

Hence,

where

For all , we define the norm

Suppose that and are any two solutions of system (3) defined on ; then , , where .

Consider the product system of system (3)

We construct a Lyapunov function defined on as follows:

it is easy to see that the norm

and the norm

are equivalent; that is, there exist two constants ,   such that

thus

Let , , ; thus condition (i) of Lemma 9 is satisfied. In addition,

where

Therefore, condition (ii) of Lemma 9 is satisfied. Finally, calculating the along system (63), we can obtain

Let

Then

and (71) is transformed into

where , , , . Using the mean value theorem, we have

where lies between and lies between and . Substituting (75) into (63), we have

then

where

hence

where

From the condition , and , we see that the condition (iii) of Lemma 9 is satisfied. So all the conditions of Lemma 9 are satisfied. By Lemma 9, there exists a unique uniformly asymptotically stable almost periodic solution of system (3) which satisfies

and for all . This completes the proof.

Corollary 16. Assume that (H₁)–(H₃) hold. Suppose that the nonnegative coefficients , , , , , , , are periodic of period ; then system (3) has a unique uniformly asymptotically stable periodic solution of period .

5. An Example

In this section, we present an example to illustrate the feasibility and effectiveness of our results.

Consider the system (3) with the following coefficients

Let . Then system (3) is persistent and has a unique uniformly asymptotically stable almost periodic solution.

Proof. By calculation, we have that

So all conditions are satisfied and the results follow from Theorems 13 and 15. Therefore, system (3) is persistent and has a unique uniformly asymptotically stable almost periodic solution. This completes the proof.

Acknowledgments

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11361072 and the YNU Postdoctoral Science Foundation Project under Grant W4030002.

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