Dynamics of a Market Share Model for Enterprises with Coopetition Strategy

Abstract

A deterministic model is used to study the change of the market share with coopetition strategy for enterprises. The model takes into consideration both coopetition enterprises and other enterprises, and the coopetition threshold 𝑅₀ is identified and global dynamics are completely determined by 𝑅₀. It shows that 𝑅₀ is a global threshold parameter in the sense that if 𝑅₀<1, the coopetition free equilibrium is globally stable and the market share of coopetition enterprises tends to zero, whereas if 𝑅₀>1, there is a unique coopetition equilibrium which is globally attractive with some conditions, and thus the market share of coopetition enterprises tends to a steady state value. By some sensitivity analysis of 𝑅₀ on parameters, we conclude that the size of the coopetition threshold 𝑅₀ and coopetition equilibrium depended on the cooperation competitiveness of coopetition enterprises.

1. Introduction

Coopetition (cooperation and competition) was put forward by Nalebuff and Brandenburger for the first time in 1996, and they discussed the importance of coopetition in business by using game theory as a theoretical frame [1]. In the same year, Bengtsson and Kock also called the phenomenon coopetition which contains competition and cooperation [2]. Coopetitive relationships are complex as they consist of two diametrically different types of logic interaction. Actors involved in coopetition are involved in a relationship that on the one hand consists of hostility due to conflicting interests and on the other hand consists of friendliness due to common interests. These two types of logic of interaction are in conflict with each other and must be separated in a proper way to make a coopetitive relationship possible. In 2000, Bengtsson and Kock studied the coopetition of corporate network [3]. The coopetition theory of Deutsch divides the target of interest subjects into cooperation, competition and independence; the relationship of the target partners affects their interaction results and then directly affects their action strategies [4]. Hausken introduced competition between groups that may induce cooperation to emerge in defection games despite considerable cost of cooperation [5]. Loebbecke et al. studied knowledge transfer and distribution which are based on coopetition [6].

With the development of the information, communication technology, networks, and virtual organizations, the coopetition phenomenon has appeared between enterprises. Resource heterogeneity determines that the supply chain of alliance partner enterprises is a symbiotic relationship with coopetition, that is, the coopetition relationships of the enterprise dominant in the supply chain network. Zhuo et al. [7] studied the supply chain alliance partner enterprise coopetition relations based on Volterra model. In their model, they considered the alliance which contains two enterprises. The following system (1) is the main model: 𝑑𝑥₁𝑑𝑡=𝑟₁𝑥₁𝑥1−¹𝑁₁+𝛿₂𝑟₁𝑥₁𝑥₂𝑁₂−𝜆₂𝑟₁𝑥₁𝑥₂𝑁₂,𝑑𝑥₂𝑑𝑡=𝑟₂𝑥₂𝑥1−²𝑁₂+𝛿₁𝑟₂𝑥₁𝑥₂𝑁₁−𝜆₁𝑟₂𝑥₁𝑥₂𝑁₁, (1) where 𝑥_𝑖 is the production value of enterprise 𝑖; 𝑁_𝑖 is the limit production value of enterprise 𝑖; 𝑟_𝑖 is the natural growth rate of enterprise 𝑖. 𝛿₁ represents the contribution rate from enterprise 1 to enterprise 2, and 𝛿₂ describes the contribution rate from enterprise 2 to enterprise 1. 𝜆₁ represents the competition rate from enterprise 1 to enterprise 2, and 𝜆₂ describes the competition rate from enterprise 2 to enterprise 1. In this model, they only considered the change of production value of enterprise about cooperation and competition and do not include the direction in which the production value has gone. Based on previous work, in this paper, we consider a deterministic model of two coopetition enterprises and other enterprises and study the change direction of the market share with coopetition strategy of these two coopetition enterprises.

The paper is organized as follows. In Section 2, we present and interpret the dynamical model that describes the market share of two coopetition enterprises and other enterprises, give the coopetition threshold of this model and the uniqueness of positive coopetition equilibrium when the coopetition threshold is larger than 1, and prove the stability of the coopetition equilibrium. And in Section 3, some numerical simulations are showed to illustrate the effectiveness of the proposed result. Section 4 gives a brief discussion about main results.

2. Mathematical Modeling and Analysis

2.1. Model Formulation

In our model, we classify the market quantity of all enterprises into three compartments: 𝑆₁(𝑡) describes the market share of the coopetition enterprise 1, 𝑆₂(𝑡) denotes the market share of the coopetition enterprise 2, and 𝑆_𝑜(𝑡) represents the market share of other enterprises at time 𝑡. Let 𝐶(𝑡) denote the density of coopetition awareness programs between two enterprises at time 𝑡. There are some assumptions about this dynamical model. (i) It is assumed that the market quantity of all enterprises is kept constant. (ii) The increase and decrease of the market share of these two coopetition enterprises only depended on the cooperation and competition between these two enterprises. (iii) The growth rate of density of coopetition awareness programs is assumed to be proportional to the market share of the coopetition enterprise 1 and enterprise 2. There are also some other assumptions about this dynamical model in the flowchart. More details can be found in Figure 1.

Figure 1 

Coopetition diagram of two enterprises.

For the market share of coopetition enterprise 1 compartment 𝑆₁(𝑡), the input of compartment 𝑆₁(𝑡) includes 𝑘₁𝛽𝑆_𝑜𝐶 and 𝑝𝑑₂₁𝑆₂. 𝑘₁𝛽𝑆_𝑜𝐶 denotes the market share from compartment 𝑆_𝑜(𝑡) to compartment 𝑆₁(𝑡) through the cooperation between enterprise 1 and enterprise 2, and 𝑝𝑑₂₁𝑆₂ represents the market share from compartment 𝑆₂(𝑡) to compartment 𝑆₁(𝑡) through the competition between enterprise 1 and enterprise 2. Due to the competition between enterprise 1 and enterprise 2, the output of 𝑆₁(𝑡) is 𝑑₁₂𝑆₁. So the change of 𝑆₁(𝑡) is as follows: 𝑑𝑆₁𝑑𝑡=𝑘₁𝛽𝑆_𝑜𝐶−𝑑₁₂𝑆₁+𝑝𝑑₂₁𝑆₂. (2)

In the same way, we can obtain the change over time in the size of the compartment 𝑆₂(𝑡), which is as follows: 𝑑𝑆₂𝑑𝑡=𝑘₂𝛽𝑆_𝑜𝐶−𝑑₂₁𝑆₂+𝑝𝑑₁₂𝑆₁. (3)

For the market share of other enterprises compartment 𝑆_𝑜(𝑡), the output of compartment 𝑆_𝑜(𝑡) includes 𝑘₁𝛽𝑆_𝑜𝐶 and 𝑘₂𝛽𝑆_𝑜𝐶, and the input of compartment 𝑆_𝑜(𝑡) includes (1−𝑝)𝑑₁₂𝑆₁ and (1−𝑝)𝑑₂₁𝑆₂. So the change over time in the size of the compartment 𝑆_𝑜(𝑡) is as follows: 𝑑𝑆_𝑜=𝑑𝑑𝑡1−𝑝₁₂𝑆₁+𝑑₂₁𝑆₂−𝑘₁+𝑘₂𝛽𝑆_𝑜𝐶. (4)

For the density of coopetition awareness programs compartment 𝐶(𝑡), the input of the compartment 𝐶(𝑡) contains 𝑘₁𝑆₁+𝑘₂𝑆₂, which is the density of coopetition awareness programs discharged by the coopetition enterprises 1 and 2. The output of the compartment 𝐶(𝑡) is the depletion of coopetition awareness programs, which is 𝛿𝐶. So the change over time in the size of the compartment 𝐶(𝑡) is as follows: 𝑑𝐶𝑑𝑡=𝑘₁𝑆₁+𝑘₂𝑆₂−𝛿𝐶. (5)

So the mathematical model is described as the following ordinary differential equations: 𝑑𝑆_𝑜=𝑑𝑑𝑡1−𝑝₁₂𝑆₁+𝑑₂₁𝑆₂−𝑘₁+𝑘₂𝛽𝑆_𝑜𝐶,𝑑𝑆₁𝑑𝑡=𝑘₁𝛽𝑆_𝑜𝐶−𝑑₁₂𝑆₁+𝑝𝑑₂₁𝑆₂,𝑑𝑆₂𝑑𝑡=𝑘₂𝛽𝑆_𝑜𝐶−𝑑₂₁𝑆₂+𝑝𝑑₁₂𝑆₁,𝑑𝐶𝑑𝑡=𝑘₁𝑆₁+𝑘₂𝑆₂−𝛿𝐶, (6) where 𝑆_𝑜(𝑡)+𝑆₁(𝑡)+𝑆₂(𝑡)=1. (7)

All parameters are assumed to be nonnegative in system (6). 𝑘₁ and 𝑘₂ represent the rate with which cooperation awareness programs are being implemented from enterprises 1 and 2, respectively. 𝛿 denotes the depletion rate of coopetition awareness programs. 𝑑₂₁ represents the competition awareness programs rate from enterprise 1 to enterprise 2, and 𝑑₁₂ represents the competition awareness programs rate from enterprise 2 to enterprise 1. 𝑝 (0<𝑝<1) is the proportion of the market share from enterprise 1 to enterprise 2 or from enterprise 2 to enterprise 1 due to the competition between enterprise 1 and enterprise 2, and 𝛽 is the transmission rate of enterprise cooperation.

For system (6), if there is no coopetition, system (6) will become the following system: 𝑑𝑆_𝑜𝑑𝑡=0,𝑑𝑆₁𝑑𝑡=0,𝑑𝑆₂𝑑𝑡=0,𝑑𝐶𝑑𝑡=0. (8) It has an equilibrium 𝑃₀=(𝑆_𝑜(0),𝑆₁(0),𝑆₂(0),𝐶(0)).

The positive coopetition equilibrium 𝑃^∗=(𝑆^∗_𝑜,𝑆^∗₁,𝑆^∗₂,𝐶^∗) and coopetition free equilibrium 𝑃⁰=(𝑆⁰_𝑜,0,0,0) of system (6) are determined by equations 1−𝑝𝑑₁₂𝑆₁+𝑑₂₁𝑆₂−𝑘₁+𝑘₂𝛽𝑆_𝑜𝑘𝐶=0,₁𝛽𝑆_𝑜𝐶−𝑑₁₂𝑆₁+𝑝𝑑₂₁𝑆₂𝑘=0,₂𝛽𝑆_𝑜𝐶−𝑑₂₁𝑆₂+𝑝𝑑₁₂𝑆₁𝑘=0,₁𝑆₁+𝑘₂𝑆₂−𝛿𝐶=0. (9)

Calculating system (9), if 𝑆₁=𝑆₂=0, we can obtain that 𝐶=0 and 𝑆_𝑜=𝑆⁰_𝑜=𝑆, and system (6) has a coopetition free equilibrium 𝑃⁰=(𝑆⁰_𝑜,0,0,0); if 𝑆₁≠0, 𝑆₂≠0, we can obtain that 𝑆^∗_𝑜=𝑑₁₂𝑑₂₁𝛿1−𝑝²𝛽𝑑₁₂𝑘₂𝑘₁𝑝+𝑘₂+𝑑₂₁𝑘₁𝑘₁+𝑘₂𝑝,𝐶^∗=𝑑₁₂𝑘₂𝑘₂+𝑘₁𝑝+𝑑₂₁𝑘₁𝑘₁+𝑘₂𝑝𝑑₁₂𝑘₂+𝑘₁𝑝𝛿𝑆^∗₂,𝑆^∗₁=𝑑₂₁𝑘₁+𝑘₂𝑝𝑑₁₂𝑘₁𝑝+𝑘₂𝑆^∗₂,𝑆^∗₂=𝑑₁₂𝑘₁𝑝+𝑘₂𝑑₁₂𝑘₁𝑝+𝑘₂+𝑑₂₁𝑘₁+𝑘₂𝑝×𝑑1−¹²𝑑₂₁𝛿1−𝑝²𝛽𝑑₁₂𝑘₂𝑘₁𝑝+𝑘₂+𝑑₂₁𝑘₁𝑘₁+𝑘₂𝑝. (10) For the positive coopetition equilibrium 𝑃^∗ of system (6), we must make 𝑆^∗₂>0, and it means that 1−𝑑₁₂𝑑₂₁𝛿(1−𝑝²)/𝛽(𝑑₁₂𝑘₂(𝑘₁𝑝+𝑘₂)+𝑑₂₁𝑘₁(𝑘₁+𝑘₂𝑝))>0 and 𝛽(𝑑₁₂𝑘₂(𝑘₁𝑝+𝑘₂)+𝑑₂₁𝑘₁(𝑘₁+𝑘₂𝑝))/𝑑₁₂𝑑₂₁𝛿(1−𝑝²)>1. So we can define the coopetition threshold by the following equation: 𝑅₀=𝛽𝑑₁₂𝑘₂𝑘₁𝑝+𝑘₂+𝑑₂₁𝑘₁𝑘₁+𝑘₂𝑝𝑑₁₂𝑑₂₁𝛿1−𝑝². (11)

From the above analysis, we can conclude that when 𝑅₀<1, system (6) only has a coopetition free equilibrium 𝑃⁰=(𝑆⁰_𝑜,0,0,0); when 𝑅₀>1, system (6) has a unique positive coopetition equilibrium 𝑃^∗=(𝑆^∗_𝑜,𝑆^∗₁,𝑆^∗₂,𝐶^∗). Define the set 𝑋=𝑆_𝑜,𝑆₁,𝑆₂,𝐶∣𝑆_𝑜,𝑆₁,𝑆₂𝑑≥0,0≤𝐶≤¹²𝑘₂𝑘₂+𝑘₁𝑝+𝑑₂₁𝑘₁𝑘₁+𝑘₂𝑝𝑑₁₂𝑘₂+𝑘₁𝑝𝛿, (12) which is invariant with respect to system (6).

2.2. Stability of the Coopetition Equilibrium

In this section, we will prove that the coopetition free equilibrium 𝑃⁰=(𝑆⁰_𝑜,0,0,0) is globally asymptotically stable. In order to prove global stability of the coopetition free equilibrium, the Lyapunov function will be used. The Lyapunov function is a powerful tool for the stability analysis of autonomous differential system, and it has been used for some epidemiological models with constant inflow and bilinear incidences or nonlinear incidences [8–16]. In the following, we will prove global stability of the coopetition free equilibrium by using a Lyapunov function.

Theorem 1. The coopetition free equilibrium 𝑃⁰ of system (6) is globally asymptotically stable when 𝑅₀<1.

Proof. In order to investigate the asymptotic behavior of equilibrium 𝑃⁰=(𝑆⁰_𝑜,0,0,0) for system (6), we can reduce (6) to the following equations: 𝑑𝑆₁𝑑𝑡=𝑘₁𝛽𝑆_𝑜𝐶−𝑑₁₂𝑆₁+𝑝𝑑₂₁𝑆₂,𝑑𝑆₂𝑑𝑡=𝑘₂𝛽𝑆_𝑜𝐶−𝑑₂₁𝑆₂+𝑝𝑑₁₂𝑆₁,𝑑𝐶𝑑𝑡=𝑘₁𝑆₁+𝑘₂𝑆₂−𝛿𝐶. (13)
For the coopetition free equilibrium 𝑃⁰, we define the following Lyapunov function: 𝐿₁=𝑚𝑆₁+𝑛𝑆₂+𝐶. (14) Then the derivative of 𝐿₁ along solutions of system (13) is 𝑑𝐿₁𝑑𝑡=𝑚𝑆^′₁+𝑛𝑆^′₂+𝐶^′=𝑚𝑘₁𝛽𝑆_𝑜𝐶−𝑑₁₂𝑆₁+𝑝𝑑₂₁𝑆₂+𝑛𝑘₂𝛽𝑆_𝑜𝐶−𝑑₂₁𝑆₂+𝑝𝑑₁₂𝑆₁+𝑘₁𝑆₁+𝑘₂𝑆₂−𝛿𝐶=𝐴+𝐵, (15) where 𝐴=𝑚𝑘₁𝛽𝑆_𝑜𝐶+𝑛𝑘₂𝛽𝑆_𝑜𝐶−𝛿𝐶,𝐵=𝑚−𝑑₁₂𝑆₁+𝑝𝑑₂₁𝑆₂+𝑛−𝑑₂₁𝑆₂+𝑝𝑑₁₂𝑆₁+𝑘₁𝑆₁+𝑘₂𝑆₂. (16)
Considering the following equations −𝑚𝑑₁₂𝑆₁+𝑛𝑝𝑑₁₂𝑆₁+𝑘₁𝑆₁=0,𝑚𝑝𝑑₂₁𝑆₂−𝑛𝑑₂₁𝑆₂𝑘₂𝑆₂=0, (17) we have 𝑑𝑚=¹²𝑘₂𝑝+𝑑₂₁𝑘₁𝑑₁₂𝑑₂₁1−𝑝²,𝑑𝑛=¹²𝑘₂+𝑑₂₁𝑘₁𝑝𝑑₁₂𝑑₂₁1−𝑝². (18)
Letting (18) generate into (15), we can obtain that 𝑑𝐿₁=𝑑𝑑𝑡=𝐴+𝐵₁₂𝑘₂𝑝+𝑑₂₁𝑘₁𝑑₁₂𝑑₂₁1−𝑝²𝑘₁+𝑑₁₂𝑘₂+𝑑₂₁𝑘₁𝑝𝑑₁₂𝑑₂₁1−𝑝²𝑘₂×𝛽𝑆_𝑜≤𝑅𝐶−𝛿𝐶₀−1𝛿𝐶. (19)
Therefore, when 𝑅₀<1, 𝑑𝐿₁/𝑑𝑡<0, and the equality 𝑑𝐿₁/𝑑𝑡=0 holds if and only if 𝑅₀=1. Thus the coopetition free equilibrium 𝑃⁰ is globally asymptotically stable in 𝑋 by LaSalle’s invariance principle [17]. This completes the proof.

Next, we will show that when 𝑅₀>1, the unique positive coopetition equilibrium 𝑃^∗=(𝑆^∗_𝑜,𝑆^∗₁,𝑆^∗₂,𝐶^∗) of system (6) is globally attractive.

For system (6), the first equation is independent of the last three equations. And as for 𝑆_𝑜(𝑡)+𝑆₁(𝑡)+𝑆₁(𝑡)=1, system (6) has the following limiting system: 𝑑𝑆₁=𝑑𝑡1−𝑆₁−𝑆₂𝑘₁𝛽𝐶−𝑑₁₂𝑆₁+𝑝𝑑₂₁𝑆₂,𝑑𝑆₂=𝑑𝑡1−𝑆₁−𝑆₂𝑘₂𝛽𝐶−𝑑₂₁𝑆₂+𝑝𝑑₁₂𝑆₁,𝑑𝐶𝑑𝑡=𝑘₁𝑆₁+𝑘₂𝑆₂−𝛿𝐶. (20)

For system (20), we define the vector 𝑥=(𝑆₁,𝑆₂,𝐶).

Theorem 2. When 𝑅₀>1,𝑝𝑑₂₁≥max{𝐶(𝑡)}𝑘₁𝛽, and 𝑝𝑑₁₂≥max{𝐶(𝑡)}𝑘₂𝛽, the positive coopetition equilibrium 𝐸^∗=(𝑆^∗₁,𝑆^∗₂,𝐶^∗) of system (20) is globally asymptotically stable with respect to 𝑥(0)∈𝑋.

Proof. We will use the theory of cooperate system to prove the global stability; therefore, we only need to verify the assumption in Corollary 3.2 in [18] for system (20).
Let 𝑓∶𝑋→𝑋 be defined by the right-hand side of system (20), 𝑓=(𝑓₁,𝑓₂,𝑓₃). Clearly 𝑓 is continuously differentiable, 𝑓(0)=0, 𝑓(𝑥)≥0, and 𝐷𝑓(𝑥) is Irreducible for all 𝑥∈𝑋. Calculating the jacobian matrix of system (20) at the unique positive coopetition equilibrium 𝐸^∗=(𝑆^∗₁,𝑆^∗₂,𝐶^∗), 𝐷𝑓𝐸^∗=−𝑑₁₂−𝑘₁𝛽𝐶^∗𝑝𝑑₂₁−𝑘₁𝛽𝐶^∗𝑘₁𝛽1−𝑆^∗₁−𝑆^∗₂𝑝𝑑₁₂−𝑘₂𝛽𝐶^∗−𝑑₂₁−𝑘₂𝛽𝐶^∗𝑘₁𝛽1−𝑆^∗₁−𝑆^∗₂𝑘₁𝑘₂.−𝛿 (21) Because 𝑝𝑑₂₁≥max{𝐶(𝑡)}𝑘₁𝛽 and 𝑝𝑑₁₂≥max{𝐶(𝑡)}𝑘₂𝛽, 𝑓 is cooperative.
Note that, for ∀𝛼∈(0,1) and 𝑥>0, 𝑓₁(𝛼𝑥)=𝛼1−𝛼𝑆₁−𝛼𝑆₂𝑘₁𝛽𝐶−𝑑₁₂𝑆₁+𝑝𝑑₂₁𝑆₂≥𝛼1−𝑆₁−𝑆₂𝑘₁𝛽𝐶−𝑑₁₂𝑆₁+𝑝𝑑₂₁𝑆₂=𝛼𝑓₁(𝑥). (22) In a similar way, we can obtain that 𝑓₂(𝛼𝑥)≥𝛼𝑓₂(𝑥),𝑓₃(𝛼𝑥)=𝛼𝑓₃(𝑥). (23) Thus 𝑓(𝛼𝑥)≥𝛼𝑓(𝑥), which is sublinear on 𝑋. By Lemma 2 and Corollary 3.2 in [18], we can conclude that the positive coopetition equilibrium 𝐸^∗=(𝑆^∗₁,𝑆^∗₂,𝐶^∗) of system (20) is globally asymptotically stable with respect to 𝑥(0)∈𝑋.

Next, by a similar proof to that of Theorem 3.1 in [19], we will prove the following theorem.

Theorem 3. When 𝑅₀>1, the unique positive coopetition equilibrium 𝑃^∗=(𝑆^∗_𝑜,𝑆^∗₁,𝑆^∗₂,𝐶^∗) of system (6) is globally attractive with respect to (𝑆_𝑜(0),𝑥(0))∈𝑋.

Proof. Let Φ(𝑡)∶𝑅⁴₊→𝑅⁴₊ be the solution semiflow of system (6), and let 𝜔 be omega limit set of Φ(𝑆_𝑜(0),𝑥(0)), (𝑆_𝑜(0),𝑥(0))∈𝑋. By Lemmas 1 and 1.2.1 in [20], 𝜔 is an internal chain transitive set for Φ(𝑡). Obviously, for system (6), there are only two equilibria 𝑃⁰ and 𝑃^∗ when 𝑅₀>1. By Theorems 1 and 3, it is not difficult to verify that Φ(𝑡) satisfies the condition of Theorem 1.2.2 in [20]; thus, 𝜔 should be either 𝑃⁰ or 𝑃^∗.
Next, we prove that 𝜔={𝑃^∗}. If this were not true; then, 𝜔={𝑃⁰}, then we should prove lim_𝑡→∞sup𝑆_𝑜=1, lim_𝑡→∞sup𝑆₁=0, lim_𝑡→∞sup𝑆₂=0, and lim_𝑡→∞sup𝐶=0. Let 𝑀=−𝑑₁₂𝑝𝑑₂₁𝑘₁𝛽𝑝𝑑₁₂−𝑑₂₁𝑘₁𝛽𝑘₁𝑘₂−𝛿. (24) Obviously, 𝑀 is irreducible and has nonnegative off-diagonal elements. Define 𝑠(𝑀)=max{Re𝜆∶𝜆asaneigenvalueof𝑀}, so 𝑠(𝑀) is a simple eigenvalue of 𝑀 with a positive eigenvector [21]. By Theorem 2 of van den Driessche and Watmough [22], there hold two equivalences: ℛ₀(𝑀)ℛ>1⟺𝑠>0,₀<1⟺𝑠(𝑀)<0. (25)
Since 𝑠(𝑀)>0, we can choose a small 𝜀>0 such that 𝑠(𝑀₂)>0, (𝑀₂=𝑀+𝜀𝑀₁), where 𝑀₁=00𝑘₁𝛽00𝑘₁𝛽000. (26) It follows that there exists a 𝑡₀ such that 1−𝑆₁−𝑆₂>1−𝜀, for 𝑡>𝑡₀. Thus, we have 𝑑𝑆₁>()𝑘𝑑𝑡1−𝜀₁𝛽𝐶−𝑑₁₂𝑆₁+𝑝𝑑₂₁𝑆₂,𝑑𝑆₂>()𝑘𝑑𝑡1−𝜀₂𝛽𝐶−𝑑₂₁𝑆₂+𝑝𝑑₁₂𝑆₁,𝑑𝐶𝑑𝑡=𝑘₁𝑆₁+𝑘₂𝑆₂−𝛿𝐶. (27)
Consider the following system: 𝑑𝑆^′₁=()𝑘𝑑𝑡1−𝜀₁𝛽𝐶^′−𝑑₁₂𝑆^′₁+𝑝𝑑₂₁𝑆^′₂,𝑑𝑆^′₂=()𝑘𝑑𝑡1−𝜀₂𝛽𝐶^′−𝑑₂₁𝑆^′₂+𝑝𝑑₁₂𝑆^′₁,𝑑𝐶^′𝑑𝑡=𝑘₁𝑆^′₁+𝑘₂𝑆^′₂−𝛿𝐶^′. (28) Since the matrix 𝑀₂ has positive eigenvalue 𝑠(𝑀₂) with a positive eigenvector, it is easy to see that (𝑆^′₁(𝑡),𝑆^′₂(𝑡),𝐶^′(𝑡))→(∞,∞,∞), 𝑡→∞. Using the comparison principle of Smith and Waltman [21], we also know that (𝑆₁(𝑡),𝑆₂(𝑡),𝐶(𝑡))→(∞,∞,∞), 𝑡→∞, which leads to a contradiction. Consequently the unique positive coopetition equilibrium 𝑃^∗=(𝑆^∗_𝑜,𝑆^∗₁,𝑆^∗₂,𝐶^∗) is globally attractive.

So far all our analyses are focused on the mathematical models and their dynamic behavior, such as the coopetition threshold, the uniqueness of positive coopetition equilibrium, the global stability of the coopetition free equilibrium, and the global attraction of the positive coopetition equilibrium. We want to seek out how the coopetition strategy influences the market share of the coopetition enterprises, so in the next section, we will present some numerical simulations about the global stability of the coopetition equilibrium and give some sensitivity analysis of the coopetition threshold 𝑅₀ on parameters.

3. Numerical Simulations

In this coopetition model, the coopetition threshold 𝑅₀ is calculated and shown to be a threshold for the dynamics of the coopetition model. The main purpose is to let the market share of the two enterprises coexist by making the threshold 𝑅₀ to be more than 1, so we must know how the coopetition threshold depends on the model parameter values. In the following result, we will show that the coopetition threshold 𝑅₀ is a global threshold parameter for the extinction and persistence of the coopetition.

Taking 𝛽=0.025, 𝑝=0.4, 𝑑₁₂=0.7, 𝑑₂₁=0.6, and 𝛿=3 and using MATLAB ODE solver, we run numerical simulations for the case with 𝑅₀<1 (see Figure 2) and the case with 𝑅₀>1 (see Figure 3) to demonstrate the conclusions in Theorems 1 and 3.

Figure 2 

The phase diagram of 𝑆1(𝑡) and 𝑆2(𝑡) with 𝑘1=4 and 𝑘2=5. This gives 𝑅0≈0.8858<1 and the coopetition makes the market share of the enterprise not coexist.

Figure 3 

The phase diagram of 𝑆1(𝑡) and 𝑆2(𝑡) with 𝑘1=5 and 𝑘2=6. This gives 𝑅0≈1.3180<1 and the coopetition makes the market share of the enterprise coexist.

From Figure 2, we can see that the trajectory of system (6) will tend to the coopetition free equilibrium with different initial conditions when the coopetition threshold 𝑅₀ is less than 1, so it is concluded that the coopetition free equilibrium is stable when the coopetition threshold 𝑅₀<1. From Figure 3, we can also conclude that the coopetition equilibrium is stable when the coopetition threshold 𝑅₀ is more than 1. For the coopetition enterprises, the main purpose is keeping mutual benefit equilibrium in the cooperation and competition. In this situation, we must make the coopetition threshold 𝑅₀ more than 1.

From the expression of the coopetition threshold 𝑅₀ of system (6), it is easy to see that all parameter values are directly or indirectly contained in 𝑅₀. So some sensitivity analyses of the coopetition threshold 𝑅₀ about parameters are given in the following section.

Figure 4(a) shows that increasing 𝑘₁ or 𝑘₂ can increase the coopetition threshold 𝑅₀. However, in Figure 4(b), an increase in 𝑑₁₂ or 𝑑₂₁ will decrease the coopetition threshold 𝑅₀. When the coopetition awareness programs rate (𝑘₁,𝑘₂) or the competition awareness programs rate (𝑑₁₂,𝑑₂₁) reaches a certain level, it will make the coopetition threshold 𝑅₀ more than 1, and then the coopetition enterprises will keep mutual benefit equilibrium in the cooperation and competition.

(a)

(b)

Figure 4 

(a) The coopetition threshold 𝑅0 in terms of 𝑘1,𝑘2. (b) The coopetition threshold 𝑅0 in terms of 𝑑12,𝑑21.

When the coopetition threshold 𝑅₀ is more than 1, we want to know the market share of these two coopetition enterprises at the equilibrium state. From the expression of the coopetition equilibrium 𝑃^∗, we also know that 𝑆^∗₁ and 𝑆^∗₂ are increasing with respect to 𝑘₁ or 𝑘₂, and an increase in 𝑑₁₂ or 𝑑₂₁ will decrease 𝑆^∗₁ and 𝑆^∗₂. In the same way, if the coopetition enterprises want to have enough market share, the coopetition awareness programs rate (𝑘₁,𝑘₂) and the competition awareness programs rate (𝑑₁₂,𝑑₂₁) must also reach a certain level.

4. Conclusion and Discussion

Coopetition is different competition and cooperation, which contains two components of competition and cooperation by using game theory. In network organizations, enterprises often cooperate with other enterprises in order to compete more effectively. In order to seize the opportunity from cooperation, enterprises will also seek attractive partners with a competitive attitude. In fact, cooperation and competition are a relative force, which gives the positive and negative effect in the market [7]. So in this paper, we propose a deterministic model of two coopetition enterprises and other enterprises to describe the change of market share of these two coopetition enterprises. It is found that the model has the coopetition threshold 𝑅₀ and two nonnegative equilibria, the coopetition free equilibrium and the coopetition equilibrium. The coopetition free equilibrium exists without any condition whereas the coopetition equilibrium exists provided 𝑅₀>1. Through the analysis of the model, it has been found that the global asymptotic behavior of system (6) is completely determined by the size of the coopetition threshold 𝑅₀; that is, the coopetition free equilibrium is globally asymptotically stable if 𝑅₀<1 while a coopetition equilibrium exists uniquely and is globally attractive if 𝑅₀>1.

By some sensitivity analysis of the coopetition threshold 𝑅₀ on parameters, we find that increasing the coopetition awareness programs rate (𝑘₁,𝑘₂) or decreasing the competition awareness programs rate (𝑑₁₂,𝑑₂₁) can increase the coopetition threshold 𝑅₀. From the expression of the coopetition equilibrium 𝑃^∗, we also know that 𝑆^∗₁ and 𝑆^∗₂ are increasing with respect to 𝑘₁ or 𝑘₂, and an increase in 𝑑₁₂ or 𝑑₂₁ will decrease 𝑆^∗₁ and 𝑆^∗₂. So we can conclude that the size of the coopetition equilibrium depended on the cooperation competitiveness of these two coopetition enterprises. This conclusion is well explained that there is a symbiotic relationship between enterprises (which contains both cooperation and competition, and both are paid and rewarding) and the coopetition enterprise can achieve mutual benefit equilibrium in the cooperation and competition.

The cooperative competition relationship between enterprises is involved in many aspects. In this paper, we only study the basic rule of coopetition between enterprises, and there also exist some assumptions on this dynamical model. We do not consider the situation without these assumptions, and the information sharing and blockade of coopetition enterprise can not be also taken into account in this model. So we need to continue research in the future. Furthermore, many researches have been performed to study spatial dynamics [23] which can be used to model the relationships between enterprises.

Conflict of Interests

The author declares that she has no conflict of interests.

Acknowledgments

This work was supported by the Ministry of Education of Humanities and Social Science Research Funds for Young (11YJC630090 and 10YJC630114) and the Shanxi Soft Science Research Funds (2011041034-01 and 2012041009-03).