# Dynamics of a Market Share Model for Enterprises with Coopetition Strategy

Authors: Mingxia Zhao
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## 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 thresholdR0 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.


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## Body

## 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:
(1)dx1dt=r1x1(1-x1N1)+δ2r1x1x2N2-λ2r1x1x2N2,dx2dt=r2x2(1-x2N2)+δ1r2x1x2N1-λ1r2x1x2N1,
where xi is the production value of enterprise i; Ni is the limit production value of enterprise i; ri 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 Section2, 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 So(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 compartmentS1(t), the input of compartment S1(t) includes k1βSoC and pd21S2. k1βSoC denotes the market share from compartment So(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:
(2)dS1dt=k1βSoC-d12S1+pd21S2.In the same way, we can obtain the change over time in the size of the compartmentS2(t), which is as follows:
(3)dS2dt=k2βSoC-d21S2+pd12S1.For the market share of other enterprises compartmentSo(t), the output of compartment So(t) includes k1βSoC and k2βSoC, and the input of compartment So(t) includes (1-p)d12S1 and (1-p)d21S2. So the change over time in the size of the compartment So(t) is as follows:
(4)dSodt=(1-p)(d12S1+d21S2)-(k1+k2)βSoC.For the density of coopetition awareness programs compartmentC(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:
(5)dCdt=k1S1+k2S2-δC.So the mathematical model is described as the following ordinary differential equations:(6)dSodt=(1-p)(d12S1+d21S2)-(k1+k2)βSoC,dS1dt=k1βSoC-d12S1+pd21S2,dS2dt=k2βSoC-d21S2+pd12S1,dCdt=k1S1+k2S2-δC,
where
(7)So(t)+S1(t)+S2(t)=1.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:
(8)dSodt=0,dS1dt=0,dS2dt=0,dCdt=0.
It has an equilibrium P0=(So(0),S1(0),S2(0),C(0)).The positive coopetition equilibriumP*=(So*,S1*,S2*,C*) and coopetition free equilibrium P0=(So0,0,0,0) of system (6) are determined by equations
(9)(1-p)(d12S1+d21S2)-(k1+k2)βSoC=0,k1βSoC-d12S1+pd21S2=0,k2βSoC-d21S2+pd12S1=0,k1S1+k2S2-δC=0.Calculating system (9), if S1=S2=0, we can obtain that C=0 and So=So0=S, and system (6) has a coopetition free equilibrium P0=(So0,0,0,0); if S1≠0, S2≠0, we can obtain that
(10)So*=d12d21δ(1-p2)β(d12k2(k1p+k2)+d21k1(k1+k2p)),C*=d12k2(k2+k1p)+d21k1(k1+k2p)d12(k2+k1p)δS2*,S1*=d21(k1+k2p)d12(k1p+k2)S2*,S2*=d12(k1p+k2)d12(k1p+k2)+d21(k1+k2p)×(1-d12d21δ(1-p2)β(d12k2(k1p+k2)+d21k1(k1+k2p))).
For the positive coopetition equilibrium P* of system (6), we must make S2*>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:
(11)R0=β(d12k2(k1p+k2)+d21k1(k1+k2p))d12d21δ(1-p2).From the above analysis, we can conclude that whenR0<1, system (6) only has a coopetition free equilibrium P0=(So0,0,0,0); when R0>1, system (6) has a unique positive coopetition equilibrium P*=(So*,S1*,S2*,C*). Define the set
(12)X={d12k2(k2+k1p)+d21k1(k1+k2p)d12(k2+k1p)δ(So,S1,S2,C)∣So,S1,S2≥0,0≤C≤d12k2(k2+k1p)+d21k1(k1+k2p)d12(k2+k1p)δ},
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 equilibriumP0=(So0,0,0,0) 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 equilibriumP0 of system (6) is globally asymptotically stable when R0<1.Proof.
In order to investigate the asymptotic behavior of equilibriumP0=(So0,0,0,0) for system (6), we can reduce (6) to the following equations:
(13)dS1dt=k1βSoC-d12S1+pd21S2,dS2dt=k2βSoC-d21S2+pd12S1,dCdt=k1S1+k2S2-δC.
For the coopetition free equilibriumP0, we define the following Lyapunov function:
(14)L1=mS1+nS2+C.
Then the derivative of L1 along solutions of system (13) is
(15)dL1dt=mS1′+nS2′+C′=m(k1βSoC-d12S1+pd21S2)+n(k2βSoC-d21S2+pd12S1)+k1S1+k2S2-δC=A+B,
where
(16)A=mk1βSoC+nk2βSoC-δC,B=m(-d12S1+pd21S2)+n(-d21S2+pd12S1)+k1S1+k2S2.
Considering the following equations(17)-md12S1+npd12S1+k1S1=0,mpd21S2-nd21S2k2S2=0,
we have
(18)m=d12k2p+d21k1d12d21(1-p2),n=d12k2+d21k1pd12d21(1-p2).
Letting (18) generate into (15), we can obtain that
(19)dL1dt=A+B=(d12k2p+d21k1d12d21(1-p2)k1+d12k2+d21k1pd12d21(1-p2)k2)×βSoC-δC≤(R0-1)δC.
Therefore, whenR0<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 whenR0>1, the unique positive coopetition equilibrium P*=(So*,S1*,S2*,C*) of system (6) is globally attractive.For system (6), the first equation is independent of the last three equations. And as for So(t)+S1(t)+S1(t)=1, system (6) has the following limiting system:
(20)dS1dt=(1-S1-S2)k1βC-d12S1+pd21S2,dS2dt=(1-S1-S2)k2βC-d21S2+pd12S1,dCdt=k1S1+k2S2-δC.For system (20), we define the vector x¯=(S1,S2,C).Theorem 2.
WhenR0>1,pd21≥max{C(t)}k1β, and pd12≥max{C(t)}k2β, the positive coopetition equilibrium E*=(S1*,S2*,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).
Letf¯: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*=(S1*,S2*,C*),
(21)Df¯(E*)=(-d12-k1βC*pd21-k1βC*k1β(1-S1*-S2*)pd12-k2βC*-d21-k2βC*k1β(1-S1*-S2*)k1k2-δ).
Because pd21≥max{C(t)}k1β and pd12≥max{C(t)}k2β, f¯ is cooperative.
Note that, for∀α∈(0,1) and x¯>0,
(22)f1(αx¯)=α[(1-αS1-αS2)k1βC-d12S1+pd21S2]≥α[(1-S1-S2)k1βC-d12S1+pd21S2]=αf1(x¯).
In a similar way, we can obtain that
(23)f2(αx¯)≥αf2(x¯),f3(αx¯)=αf3(x¯).
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*=(S1*,S2*,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.
WhenR0>1, the unique positive coopetition equilibrium P*=(So*,S1*,S2*,C*) of system (6) is globally attractive with respect to (So(0),x¯(0))∈X.Proof.
LetΦ(t):R+4→R+4 be the solution semiflow of system (6), and let ω be omega limit set of Φ(So(0),x¯(0)), (So(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 thatω={P*}. If this were not true; then, ω={P0}, then we should prove limt→∞supSo=1, limt→∞supS1=0, limt→∞supS2=0, and limt→∞supC=0. Let
(24)M=(-d12pd21k1βpd12-d21k1βk1k2-δ).
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:
(25)ℛ0>1⟺s(M)>0,ℛ0<1⟺s(M)<0.
Sinces(M)>0, we can choose a small ε>0 such that s(M2)>0, (M2=M+εM1), where
(26)M1=(00k1β00k1β000).
It follows that there exists a t0 such that 1-S1-S2>1-ε, for t>t0. Thus, we have
(27)dS1dt>(1-ε)k1βC-d12S1+pd21S2,dS2dt>(1-ε)k2βC-d21S2+pd12S1,dCdt=k1S1+k2S2-δC.
Consider the following system:(28)dS1′dt=(1-ε)k1βC′-d12S1′+pd21S2′,dS2′dt=(1-ε)k2βC′-d21S2′+pd12S1′,dC′dt=k1S1′+k2S2′-δC′.
Since the matrix M2 has positive eigenvalue s(M2) with a positive eigenvector, it is easy to see that (S1′(t),S2′(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*=(So*,S1*,S2*,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 thresholdR0 on parameters.


## 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 So(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 compartmentS1(t), the input of compartment S1(t) includes k1βSoC and pd21S2. k1βSoC denotes the market share from compartment So(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:
(2)dS1dt=k1βSoC-d12S1+pd21S2.In the same way, we can obtain the change over time in the size of the compartmentS2(t), which is as follows:
(3)dS2dt=k2βSoC-d21S2+pd12S1.For the market share of other enterprises compartmentSo(t), the output of compartment So(t) includes k1βSoC and k2βSoC, and the input of compartment So(t) includes (1-p)d12S1 and (1-p)d21S2. So the change over time in the size of the compartment So(t) is as follows:
(4)dSodt=(1-p)(d12S1+d21S2)-(k1+k2)βSoC.For the density of coopetition awareness programs compartmentC(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:
(5)dCdt=k1S1+k2S2-δC.So the mathematical model is described as the following ordinary differential equations:(6)dSodt=(1-p)(d12S1+d21S2)-(k1+k2)βSoC,dS1dt=k1βSoC-d12S1+pd21S2,dS2dt=k2βSoC-d21S2+pd12S1,dCdt=k1S1+k2S2-δC,
where
(7)So(t)+S1(t)+S2(t)=1.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:
(8)dSodt=0,dS1dt=0,dS2dt=0,dCdt=0.
It has an equilibrium P0=(So(0),S1(0),S2(0),C(0)).The positive coopetition equilibriumP*=(So*,S1*,S2*,C*) and coopetition free equilibrium P0=(So0,0,0,0) of system (6) are determined by equations
(9)(1-p)(d12S1+d21S2)-(k1+k2)βSoC=0,k1βSoC-d12S1+pd21S2=0,k2βSoC-d21S2+pd12S1=0,k1S1+k2S2-δC=0.Calculating system (9), if S1=S2=0, we can obtain that C=0 and So=So0=S, and system (6) has a coopetition free equilibrium P0=(So0,0,0,0); if S1≠0, S2≠0, we can obtain that
(10)So*=d12d21δ(1-p2)β(d12k2(k1p+k2)+d21k1(k1+k2p)),C*=d12k2(k2+k1p)+d21k1(k1+k2p)d12(k2+k1p)δS2*,S1*=d21(k1+k2p)d12(k1p+k2)S2*,S2*=d12(k1p+k2)d12(k1p+k2)+d21(k1+k2p)×(1-d12d21δ(1-p2)β(d12k2(k1p+k2)+d21k1(k1+k2p))).
For the positive coopetition equilibrium P* of system (6), we must make S2*>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:
(11)R0=β(d12k2(k1p+k2)+d21k1(k1+k2p))d12d21δ(1-p2).From the above analysis, we can conclude that whenR0<1, system (6) only has a coopetition free equilibrium P0=(So0,0,0,0); when R0>1, system (6) has a unique positive coopetition equilibrium P*=(So*,S1*,S2*,C*). Define the set
(12)X={d12k2(k2+k1p)+d21k1(k1+k2p)d12(k2+k1p)δ(So,S1,S2,C)∣So,S1,S2≥0,0≤C≤d12k2(k2+k1p)+d21k1(k1+k2p)d12(k2+k1p)δ},
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 equilibriumP0=(So0,0,0,0) 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 equilibriumP0 of system (6) is globally asymptotically stable when R0<1.Proof.
In order to investigate the asymptotic behavior of equilibriumP0=(So0,0,0,0) for system (6), we can reduce (6) to the following equations:
(13)dS1dt=k1βSoC-d12S1+pd21S2,dS2dt=k2βSoC-d21S2+pd12S1,dCdt=k1S1+k2S2-δC.
For the coopetition free equilibriumP0, we define the following Lyapunov function:
(14)L1=mS1+nS2+C.
Then the derivative of L1 along solutions of system (13) is
(15)dL1dt=mS1′+nS2′+C′=m(k1βSoC-d12S1+pd21S2)+n(k2βSoC-d21S2+pd12S1)+k1S1+k2S2-δC=A+B,
where
(16)A=mk1βSoC+nk2βSoC-δC,B=m(-d12S1+pd21S2)+n(-d21S2+pd12S1)+k1S1+k2S2.
Considering the following equations(17)-md12S1+npd12S1+k1S1=0,mpd21S2-nd21S2k2S2=0,
we have
(18)m=d12k2p+d21k1d12d21(1-p2),n=d12k2+d21k1pd12d21(1-p2).
Letting (18) generate into (15), we can obtain that
(19)dL1dt=A+B=(d12k2p+d21k1d12d21(1-p2)k1+d12k2+d21k1pd12d21(1-p2)k2)×βSoC-δC≤(R0-1)δC.
Therefore, whenR0<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 whenR0>1, the unique positive coopetition equilibrium P*=(So*,S1*,S2*,C*) of system (6) is globally attractive.For system (6), the first equation is independent of the last three equations. And as for So(t)+S1(t)+S1(t)=1, system (6) has the following limiting system:
(20)dS1dt=(1-S1-S2)k1βC-d12S1+pd21S2,dS2dt=(1-S1-S2)k2βC-d21S2+pd12S1,dCdt=k1S1+k2S2-δC.For system (20), we define the vector x¯=(S1,S2,C).Theorem 2.
WhenR0>1,pd21≥max{C(t)}k1β, and pd12≥max{C(t)}k2β, the positive coopetition equilibrium E*=(S1*,S2*,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).
Letf¯: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*=(S1*,S2*,C*),
(21)Df¯(E*)=(-d12-k1βC*pd21-k1βC*k1β(1-S1*-S2*)pd12-k2βC*-d21-k2βC*k1β(1-S1*-S2*)k1k2-δ).
Because pd21≥max{C(t)}k1β and pd12≥max{C(t)}k2β, f¯ is cooperative.
Note that, for∀α∈(0,1) and x¯>0,
(22)f1(αx¯)=α[(1-αS1-αS2)k1βC-d12S1+pd21S2]≥α[(1-S1-S2)k1βC-d12S1+pd21S2]=αf1(x¯).
In a similar way, we can obtain that
(23)f2(αx¯)≥αf2(x¯),f3(αx¯)=αf3(x¯).
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*=(S1*,S2*,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.
WhenR0>1, the unique positive coopetition equilibrium P*=(So*,S1*,S2*,C*) of system (6) is globally attractive with respect to (So(0),x¯(0))∈X.Proof.
LetΦ(t):R+4→R+4 be the solution semiflow of system (6), and let ω be omega limit set of Φ(So(0),x¯(0)), (So(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 thatω={P*}. If this were not true; then, ω={P0}, then we should prove limt→∞supSo=1, limt→∞supS1=0, limt→∞supS2=0, and limt→∞supC=0. Let
(24)M=(-d12pd21k1βpd12-d21k1βk1k2-δ).
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:
(25)ℛ0>1⟺s(M)>0,ℛ0<1⟺s(M)<0.
Sinces(M)>0, we can choose a small ε>0 such that s(M2)>0, (M2=M+εM1), where
(26)M1=(00k1β00k1β000).
It follows that there exists a t0 such that 1-S1-S2>1-ε, for t>t0. Thus, we have
(27)dS1dt>(1-ε)k1βC-d12S1+pd21S2,dS2dt>(1-ε)k2βC-d21S2+pd12S1,dCdt=k1S1+k2S2-δC.
Consider the following system:(28)dS1′dt=(1-ε)k1βC′-d12S1′+pd21S2′,dS2′dt=(1-ε)k2βC′-d21S2′+pd12S1′,dC′dt=k1S1′+k2S2′-δC′.
Since the matrix M2 has positive eigenvalue s(M2) with a positive eigenvector, it is easy to see that (S1′(t),S2′(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*=(So*,S1*,S2*,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 thresholdR0 on parameters.


## 3. Numerical Simulations

In this coopetition model, the coopetition thresholdR0 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.Takingβ=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 ofS1(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 ofS1(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 Figure2, 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 thresholdR0 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.Figure4(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) The coopetition thresholdR0 in terms of k1,k2. (b) The coopetition threshold R0 in terms of d12,d21.
(a)
(b)When the coopetition thresholdR0 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 S1* and S2* are increasing with respect to k1 or k2, and an increase in d12 or d21 will decrease S1* and S2*. 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 thresholdR0 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 S1* and S2* are increasing with respect to k1 or k2, and an increase in d12 or d21 will decrease S1* and S2*. 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.



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*Source: 863805-2013-11-26.xml*