Dynamic Analysis of a Competition-Cooperation Enterprise Cluster with Core-Satellite Structure and Time Delay

Core and satellite structure is one of the common structures in enterprise clusters. In core and satellite structure, there are one core enterprise and at least two satellite enterprises. +ere exist a competitive relationship between satellite enterprises and a cooperative relationship between satellite enterprise and core enterprise. However, the dynamic evolution of competition-cooperation enterprise clusters with core-satellite structure is not well understood. In this paper, a novel competition-cooperation enterprise cluster model with core-satellite structure is proposed.+e boundedness of the positive equilibrium is investigated. It is found that there exists upper bound of both core enterprise output and satellite enterprise output and the upper bound of core enterprise not only depends on its own production capacity but also depends on the production capacity of two satellite enterprises. +en, by selecting the production period as bifurcating parameter, the conditions of local stability and Hopf bifurcation are obtained. Once the production period passes a critical value, the output of both core enterprise and satellite enterprise loses stability and displays periodic fluctuations. +is may lead to the decline of efficiency of enterprise and resource mismatch. Furthermore, the fluctuation properties are studied. Finally, a numerical example is presented to show the effectiveness of theorem.


Introduction
Recently, enterprise cluster, as an effective form of industrial space organization, has gradually become a common phenomenon in the process of modern industry and internationalization [1][2][3][4]. e competition-cooperation relationship widely exists in the real enterprise clusters and has a major impact on the evolution of enterprise clusters [5][6][7][8][9]. To study the influence of competition and cooperation on the evolution of enterprise clusters, researchers propose a competition-cooperation enterprise cluster model [10] based on the ecology model [11][12][13][14][15], which is described as follows: where x 1 , x 2 are the enterprise output; r i is the intrinsic growth; K i (i � 1, 2) denotes the carrying capacity of market under nature unlimited conditions; c i (i � 1, 2) is the initial output of core enterprise; and τ is the production period. Let a 1 � (r 1 /K), a 2 � (r 2 /K), b 1 � (r 1 α/K), b 2 � (r 2 β/K), and system (1) can be rewritten as follows: For this model, the dynamic behaviors including stability, Hopf bifurcation, and chaos have been widely studied [16][17][18][19][20][21].
In practical enterprise clusters, organization structure has a major impact on the production efficiency of enterprise clusters. e efficiency of overall operation is one of the important factors of enterprise's success. us, it is necessary to consider the structure in the enterprise cluster model. Among the many organization structures, core and satellite structure is one of the common structures in enterprise clusters, which is described in Figure 1. In core and satellite structure, there are one core enterprise and at least two satellite enterprises.
ere exist competitive relationship among satellite enterprises and cooperative relationship between satellite enterprise and core enterprise. For example, in automobile enterprise cluster, the core enterprise produces the motor vehicle and the satellite enterprise produces automobile parts for core enterprises. To reduce the cost and ensure the stability of supply chain, the core enterprise has at least two satellite enterprises for the same automobile part. It is easy to see that there exists competition between the two satellite enterprises and there exists cooperation between satellite enterprise and core enterprise. However, few works investigate the dynamic evolution of enterprise cluster model with core-satellite structure.
Inspired by the discussion, in this paper, a competitioncooperation enterprise cluster model composed of a core enterprise and two satellite enterprises is proposed, which is shown in Figure 1. In this enterprise cluster, there are one core enterprise and two satellite enterprises.
ere is a competitive relationship between two satellite enterprises. And there is a cooperative relationship between satellite enterprise and core enterprise. e model is described as follows: where x i (t) is the satellite enterprise output; y(t) is the core enterprise output; a i is the self-regulation of enterprise i; r i is the intrinsic growth; b i is the completion rate of satellite enterprise; c 1 is the completion rate between satellite enterprise and core enterprise; c 2 is the rate of conversion of commodity into the reproduction of enterprise; d is the initial output of core enterprise; d 1 is the initial output of satellite enterprisex 1 ; d 2 is the initial output of satellite enterprise x 2 ; and τis the production period. e main contributions of this paper are as follows: (1) A competition-cooperation enterprise cluster model is composed of a core enterprise and two satellite enterprises. ere is a competitive relationship between two satellite enterprises. And there is a cooperative relationship between satellite enterprise and core enterprise. (2) e boundedness of positive equilibrium is investigated. And there exists a upper limit of output of enterprise cluster model.
(3) e production period plays a key role in dynamics of the proposed enterprise cluster. When it passes a critical value, the output of the enterprise cluster system loses its stability and displays a periodic fluctuation, which may cause a drop in productivity of the enterprise cluster system. e remainder of this paper is organized as follows. In Section 2, the boundedness analysis of positive equilibrium is given. In Section 3, the conditions of Hopf bifurcation are discussed. In Section 4, the normal form of Hopf bifurcation is given. In Section 5, an example is given to verify the theoretical analysis. In Section 6, we give the economic meaning.

Boundedness of Positive Equilibrium
In this section, we investigate the boundedness of positive equilibrium. It can be seen that system (3) has more than three equilibria if any one of the enterprise output is zero. As it has no economic sense if one of the enterprise output is zero, we only study the property of positive equilibrium where all enterprise outputs are positive. Let E * � (x * 1 , x * 2 , y * ) be the positive equilibrium of system (3), where From the perspective of enterprise management, the output of enterprise cannot be negative.
Proof. First, we investigate the boundedness of x 1 (t) and According to Lemma 1, one has Similarly, one has If there exists t n , using the same method, one can obtain that x 1 (t) has upper limit at t � t n . us, one has In this same way, it follows that there exists We complete the proof. □ Remark 1. From eorem 1, one can see that there are upper bounds on the output of core enterprise and two satellite enterprises. Moreover, the upper bound of core enterprise not only depends on its own production capacity but also depends on the production capacity of two satellite enterprises.
has positive roots.

Direction of the Hopf Bifurcation
In this section, we study the properties of Hopf bifurcation by using the center manifold [23,[26][27][28]. Letting with where where According to Riesz representation theorem, there exists a function η(θ, μ) of bounded variation for θ ∈ [− 1, 0], such that Let where δ(θ) is Dirac delta function. By [21], we define (32) en, system (24) can be rewritten as where x t (θ) � x(t + θ). e adjoint operator A * of A is defined by where η T is the transpose of the matrix η.

⎧ ⎨ ⎩ (66)
So, Now, we compute E 1 and E 2 . From the definition of A in (31), one can obtain Following the method in [18][19][20][21], we have en, we can obtain which leads to and en, by (72) and (74), we can obtain the following properties of Hopf bifurcation:

Numerical Examples
In this section, a numerical example is presented to support our obtained results. Consider system (3) (17), we can obtain τ 0 � 4.15.
First, we chooseτ � 4.1 < τ 0 , and the outputs of three enterprises are shown in Figure 2. It is easy to see that output of enterprise is asymptotically stable.
Finally, we choose τ � 4.25 > τ 0 , and the outputs of three enterprises are shown in Figure 3; it is easy to see that the output of the enterprise displays a periodic fluctuation.

Economic Meaning
In this paper, a novel competition-cooperation enterprise cluster model with core-satellite structure is proposed. e boundedness of the positive equilibrium is investigated. It is found that there exists upper bound of both core enterprise output and satellite enterprise output and the upper bound of core enterprise not only depends on its own production capacity but also depends on the production capacity of two satellite enterprises. Moreover, it is found that the production period plays a key role on the evolution of enterprise cluster. From simulation, we can obtain that the critical value of the production period is τ 0 � 4.15. When the production period passes τ 0 � 4.15, the output of the enterprise cluster system loses its stability and displays a periodic fluctuation, which means too long production cycle will lead to capacity fluctuations. From the viewpoint of enterprise management, output fluctuation may affect the stable development of enterprises, worker employment, and production efficiency. us, it is important for enterprise cluster to control the production period in a suitable region.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.