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

Authors: Wenjie Hu; Tao Dong; Hua Zhao
Journal: Complexity
 (2021)

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## Abstract

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. There 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. The 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. Then, 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. This 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.


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## Body

## 1. 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 [1–4]. The competition-cooperation relationship widely exists in the real enterprise clusters and has a major impact on the evolution of enterprise clusters [5–9]. 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 [11–15], which is described as follows:(1)x˙1t=r1x1t1−x1tK1−αx2t−τ−c22K2,x˙2t=r2x2t1−x2tK2+αx1t−τ−c12K1,where x1,x2 are the enterprise output; ri is the intrinsic growth; Kii=1,2 denotes the carrying capacity of market under nature unlimited conditions; cii=1,2 is the initial output of core enterprise; and τ is the production period. Let a1=r1/K,a2=r2/K,b1=r1α/K,b2=r2β/K, and system (1) can be rewritten as follows:(2)x˙1t=x1tr1−a1x1t−b1x2t−τ−d2,x˙2t=x2tr2−a2x2t+b2x1t−τ−d2.For this model, the dynamic behaviors including stability, Hopf bifurcation, and chaos have been widely studied [16–21].In practical enterprise clusters, organization structure has a major impact on the production efficiency of enterprise clusters. The efficiency of overall operation is one of the important factors of enterprise's success. Thus, 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 Figure1. In core and satellite structure, there are one core enterprise and at least two satellite enterprises. There 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.Figure 1
The enterprise cluster with one core enterprise and two satellite enterprises.Inspired by the discussion, in this paper, a competition-cooperation enterprise cluster model composed of a core enterprise and two satellite enterprises is proposed, which is shown in Figure1. In this enterprise cluster, there are one core enterprise and two satellite enterprises. There is a competitive relationship between two satellite enterprises. And there is a cooperative relationship between satellite enterprise and core enterprise. The model is described as follows:(3)x˙1t=x1tr1−a1x1t−b1x2t−c1yt−τ−d2,x˙2t=x2tr2−a2x2t−b2x1t−c1yt−τ−d2,y˙t=ytr3−a3yt+c2x1t−τ−d12+x2t−τ−d22,where xit is the satellite enterprise output; yt is the core enterprise output; ai is the self-regulation of enterprise i; ri is the intrinsic growth; bi is the completion rate of satellite enterprise; c1 is the completion rate between satellite enterprise and core enterprise; c2 is the rate of conversion of commodity into the reproduction of enterprise; d is the initial output of core enterprise; d1 is the initial output of satellite enterprisex1; d2 is the initial output of satellite enterprise x2; and τis the production period. The 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. There is a competitive relationship between two satellite enterprises. And there is a cooperative relationship between satellite enterprise and core enterprise.(2)
The boundedness of positive equilibrium is investigated. And there exists a upper limit of output of enterprise cluster model.(3)
The 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.The remainder of this paper is organized as follows. In Section2, 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.


## 2. 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∗=x1∗,x2∗,y∗ be the positive equilibrium of system (3), where(4)x1∗=a2r1−b1r2−ac21−b1c2y∗−d2a1a2−b1b2,x2∗=b2r1−a1r2−bc21−a1c2y∗−d2b1b2−a1a2,y∗=r3+c2x1∗−d12+x2∗−d22a3.From the perspective of enterprise management, the output of enterprise cannot be negative. Thus, the initial condition of enterprise output must satisfyx1θ≥0, x2θ≥0, and yθ≥0 for θ∈−τ,0. In respect of the boundedness of E∗=x1∗,x2∗,y∗, we have the following lemma.Lemma 1.
(see [21]). Let c>0,d>0.(1)
Ifdxt/dt≥xd−cx is satisfied, then limt⟶∞infxt≥d/c for all t≥0 and x0>0.(2)
Ifdxt/dt≤xd−cx is satisfied, then limt⟶∞supxt≤d/c for all t≥0 and x0>0.Theorem 1.
(1)
Supposexiθ≥0,i=1,2,θ∈−τ,0 and xi0>0,i=1,2,t>0; there exists ψi=ψixiθ>0,i=1,2 such that xit<ψi,i=1,2, where ψ1=maxsupx−τ≤θ≤01θ,r1/a1,ψ2=maxsup−τ≤θ≤0x2θ,r2/a2.(2)
Supposeyθ≥0,θ∈−τ,0 and y0>0,t>0; there exists ψ3=ψ3yθ>0, such that yt<ψ3, where ψ3=maxsupy−τ≤θ≤0θ,minr1−a1ψ,r2−a2ψ2.Proof.
First, we investigate the boundedness ofx1t and x2t. By (3), one can obtain x˙1t≥0 for all t>T. Then, we have(5)dx1tdt≤x1tr1−a1x1.
According to Lemma1, one has(6)limt⟶∞supx1≤r1a1.
Similarly, one has(7)limt⟶∞supx2≤r2a2.
If there existstnn=1∞ such that x˙1tn≥0 is local max, where x1t<x1tnt∈0,tn, using the same method, one can obtain that x1t has upper limit at t=tn. Thus, one has ψ1=maxsup−τ≤θ≤0x1θ,r1/a1. Similarly, we also can obtain ψ2=maxsup−τ≤θ≤0x2θ,r2/a2.
In this same way, it follows that there existsψ3>0, such that y0<ψ3 for t>0, where ψ3=maxsupy−τ≤θ≤0θ,minr1−a1ψ,r2−a2ψ2. We complete the proof.Remark 1.
From Theorem1, 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.


## 3. Bifurcation Analysis

In this section, the conditions of Hopf bifurcation for (3) with E∗=x1∗,x2∗,y∗ are presented. By (3), one can obtain the Jacobian matrix for (3) with E∗:(8)−a1x1∗−b1x1∗−2c1x1∗y∗−de−λτ−b2x2∗−a2x2∗−2c1x2∗y∗−de−λτ−2c2y∗x1∗−d1e−λτ−2c2y∗x2∗−d2e−λτ−a3y∗.Then, we have(9)λ3+e1λ2+e2λ+e3+e−2λτe4λ+e5=0,where(10)e1=a1x1∗+a2x2∗+a3x3∗,e2=a3y∗a2x2∗+a1x1∗+x1∗x2∗a1a2−b1b2,e3=−4c1c2y∗y∗−d1x1∗x1∗−d1−x2∗x2∗−d1,e4=4c1c2x1∗x2∗y∗x1∗−d1y∗−d1b1−a1+4c1c2x1∗x2∗y∗x2∗−d1y∗−d1b2−a1,e5=a3x1∗x2∗y∗a1a2−b1b2.Ifiωω>0 is a root of (9), one can obtain(11)−iω3−e1ω2+e2iω+e3iωe−2iωτ+e4e−2iωτ+e5=0.Separating the real and imaginary parts of (11), we have(12)e3ωsin2ωτ+e4cos2ωτ=e1ω2−e5,(13)e3ωcos2ωτ−e4sin2ωτ=ω3−e2ω.By (12) and (13), one can obtain(14)ω6+f1ω4+f2ω2+f3=0,where f1=e12−2e2,f2=e2−2e1e5−e32,f3=e52−e42.Letz = ω2; then, (14) becomes(15)z3+f1z2+f2z+f3=0.Lettingz∗=1/3−f1+f12−3f2, hz∗=z∗3+f1z∗2+f2z∗+f3, we get the following.Lemma 2.
(see [22–24]).(i)
Iff3<0, (15) has at least one positive root.(ii)
Iff3≥0 and f12−fc2≤0, (15) has no positive root.(iii)
Iff3≥0, f12−3f2>0, z∗>0, and hz∗≤0, (15) has positive roots.
Assume (14) has three positive roots ωk=zk, k = 1, 2, 3. By (12) and (13), we have(16)cos2ωτ=e3ω4+e1e4ω2−e2e3ω2+e4e5e3ω2+e42.Thus, denote(17)τkj=12ωkarcose3ω4+e1e4ω2−e2e3ω2+e4e5e3ω2+e42+2jπ2ωk,where k = 1, 2, 3; j = 0, 2, …. Define τ0=τk00=mink=1,2,3τk00,ω0=ωk0.Note that whenτ=0, (9) becomes(18)λ3+e1λ2+e2+e3λ+e4+e5=0.By using the Routh–Hurwitz criterion [16], one can obtain the condition that all roots of (18) have negative real parts.(19)H1e1e2+e4−e3+e5>0.Lemma 3.
(see [25]). Consider the exponential polynomial(20)Pλ,e−λτ,……,e−λτm=λn+p10λn−1+……+pn−10λ+pn0+p11λn−1+……+pn−11λ+pn1e−λτ+……+p1mλn−1+……+pn−1mλ+pnme−λτm,where τi≥0i=1,2,…,m and (pji=j=1,2,…,m are constants. As τ1,τ2,…,τm vary, the sum of the order of the zeros of Pλ,e−λτ,…,e−λτmon the open right half plane can change only if a zero appears on or crosses the imaginary axis.Lemma 4.
Ifh′z0≠0, then sgnRedλ/dτ−1≠0 when τ=τ0 and the sign of h′z0 is same as the sign of Redλ/dτ|τ=τ0.Proof.
Differentiating (9) with respect to τ yields(21)3λ2+2e1λ+e2+e3−2τe3λ+e4e−2λτdλdτ=2λe3λ+e4e−2λτ.
For convenience, we denoteω0 and τ0 by ω and τ; then, we have(22)dλdτ−1=3λ2+2e1λ+e2e2λτ2λe3λ+e4+e32λe3λ+e4−τλ.
Then, we get(23)Redλdτ−1τ=τ0=Re3λ2+2e1λ+e2e2λτ2λe3λ+e4τ=τ0+Ree32λe3λ+e4τ=τ0=1Ak−2ω0e2−3ω02ω03−e2ω0+4e1ω02e1ω02−e5−2e32ω02=1Ak6ω06+4fω04+2f2ω02=2Akz03z02+2f1z0+f2=2z0Akh′z0,where Ak=4e32ω04+4e42ω02>0.
Then, ifh′z0≠0, we have sgnRedλ/dτ−1≠0, and the sign of h′z0 is the same as the sign of Redλ/dτ|τ=τ0. We complete the proof.
Thus, from Lemmas2, 3, and 4, one has the following.Theorem 2.
Suppose that (H1) holds, and we have(1)
Iff3>0 and f12−3f2≤0, (3) is asymptotically stable at E∗.(2)
If eitherf3<0 or f3≥0 and f12−3f2>0z∗>0, hz∗≤0 holds, (3) is asymptotically stable at E∗ for τ∈0,τ0 and undergoes a Hopf bifurcation at E∗ when τ=τ0.


## 4. Direction of the Hopf Bifurcation

In this section, we study the properties of Hopf bifurcation by using the center manifold [23, 26–28]. Letting x1=x1−x1∗,x2=x2−x2∗,x3=x3−x3∗,x¯it=xiτt,τ=τ0+μ, (3) can be transformed into a FDE as(24)x˙t=Lμxt+fμ,xt,with(25)Lμϕ=τ0+μB1φ0+B2φ−1,where(26)B1=b11b120b21b22000b33,B2=00b1300b23b31b320,(27)fμ,φ=τ0+μf1f2f3,where(28)f1=b14φ120+b15φ10φ20+b16φ32−1+b17φ10φ3−1+b18φ10φ32−1,f2=b24φ10φ20+b25φ220+b26φ32−1+b27φ20φ3−1+b28φ20φ32−1,f3=b34φ12−1+b35φ22−1+b36φ320+b37φ1−1φ30+b38φ2−1φ30+b39φ12−1φ30+b39φ22−1φ30,b11=−a1x1∗,b12=−b1x1∗,b13=−2c1x1∗y∗−d,b14=−a1,b15=−b1,b16=−c1x1∗,b17=−c1,b18=−2c1d1b21=−b2x2∗,b22=−a2x2∗,b23=−2c1x2∗y∗−d,b24=−b2,b25=−a2,b26=−c1x2∗,b27=−2c1d1b28=−c1b31=−2c2y∗x1∗−d1,b32=−2c2y∗x2∗−d2,b33=−−a3y∗,b34=c2y∗,b35=c2y∗,b36=−a3,b37=−2c2d1,b38=−2c2d2,b39=c2.According to Riesz representation theorem, there exists a functionηθ,μ of bounded variation for θ∈−1,0, such that(29)Lμφ=∫−10dηθ,μφθ,φ∈C.Let(30)ηθ,μ=τ0+μB1δθ+B2δθ+1,where δθ is Dirac delta function.By [21], we define(31)Aμφ=dφdθ,θ∈−1,0,∫−10dηθ,μφθ,θ=0,(32)Rμφ=0,θ∈−1,0,fμ,φ,θ=0.Then, system (24) can be rewritten as(33)x˙t=Aμxt+Rμxt,where xtθ=xt+θ.The adjoint operatorA∗of A is defined by(34)A∗μψ=−dψsdθ,s∈0,1,∫−10dηTt,0ψ−t,s=0,where ηT is the transpose of the matrix η.Forφ∈C1−1,0 and ψ∈C10,1, we define(35)ψ,φ=ψ¯0⋅φ0−∫θ=−10∫ξ=0θψ¯ξ−θdηθφξdξ,where ηθ=ηθ,0. We know that ±iτ0ω0 is an eigenvalue of A0, so ±iτ0ω0 is also an eigenvalue of A∗0. We can get(36)qθ=1αβeiτ0ω0θ,−1<θ≤0.By [21], we have(37)Aq0=iτ0ω0q0.Hence, one can obtain(38)α=−b21iω0+b33+b31e−2iω0τ0iω0+b22iω0+b33+b32b23e−2iω0τ0,(39)β=b31iω0+b22−b21b32e−iω0τ0iω0+b22iω0+b33+b32b23e−2iω0τ0.Assume that the eigenvectorq∗ of A∗is(40)q∗s=1α∗β∗eiτ0ω0s.By [21], we have(41)A∗q0=−iτ0ω0q∗0.Hence, we obtain(42)α∗=b13b21+b23iω0−b11b12b23e−iω0τ0+b13iω0−b22,β∗=iω0−b11iω0−b22−b12b21eiω0τ0iω0−b22b13+b21.Let(43)q∗,q=1.Then, we can compute(44)q∗,q=q∗0⋅q0−∫θ=−10∫ξ=0θ−q∗Tξ−θdηθqξdξ=1ρ¯1+αα∗+ββ∗−∫θ=−10θρ¯τ0eiw0τ0θb11+b21α¯∗+b12α+b22αα¯∗+b33ββ¯∗δθ+b31β¯∗+b32αβ¯∗+b13β+b23α¯∗βδθ+1dθdξ=1ρ¯1+αα∗+ββ∗+τ0ρ¯e−iτ0ω0b31β¯∗+b32αβ¯∗+b13β+b23β¯∗β=1.Hence, we obtain(45)ρ¯=1+αα∗+ββ∗+τ0e−iτ0ω0b31β¯∗+b32αβ¯∗+b13β+b23α¯∗β.Assume thatxt is a solution of (33) with μ=0; we define(46)zt=q∗,xt,Wt,θ=xt−2Reztqθ.OnC0, one has(47)Wt,θ=Wz,z¯,t,where(48)Wz,z¯,t=W20θz22+W11θzz¯2+W02θz¯22+….In fact,z and z¯ are local coordinates for C0 in q and q∗. As μ=0, we have(49)z˙t=q∗,x˙t=q∗,Aμxt+Rμxt=q∗,Axt+q∗,Rxt=iτ0w0z+q¯∗0⋅f0,Wt,0+2Reztq0.Rewrite (39) as(50)z˙t=iτ0ω0z+gz,z¯,where(51)gz,z¯=g20z22+g11zz¯+g02z¯22+g21z2z¯2+….By (37) and (40), one has(52)W˙=x˙t−z˙q−z¯˙q¯˙=AW−2Req¯∗0⋅fz,z¯qθ,θ∈−2τ,0,AW−2Req¯∗0⋅fz,z¯qθ+f0z,z¯,θ=0.Let(53)W˙=AW+Hz,z¯,θ,where(54)Hz,z¯,θ=H20θz22+H11θzz¯2+H02θz¯22+….Following the method in [18–21], one can obtain(55)A−2iw0W20θ=−H20θ,AW11θ=−H11θ,A+2iw0W02θ=−H02θ.Sincext=xt+θ=Wz,z¯,θ+zq+z¯⋅q¯, one has(56)xt=W1z,z¯,θW2z,z¯,θW3z,z¯,θ+z1q1q2eiω0θ+z¯1q¯1q¯2e−iω0θ.Thus, one can obtain(57)φ10=z+z¯+W2010z22+W1110zz¯+W0210z¯22,φ20=zq1+z¯q¯1∗+W2010z22+W1110zz¯+W0210z¯22.So,(58)φ10φ20=q1z2+q¯1z¯2+q1+q2zz¯+W112+12W202+W111q1+12W201q¯z2z¯.By (27) and (43), one can obtain(59)fφ,μ=K11z2+K12zz¯+K13z¯2K21z2+K22zz¯+K23z¯2K31z2+K32zz¯+K33z¯2,where(60)K11=b14+αb15+b16β2e−2iω0τ0,K12=2b14+b15α+b15α¯+b17βe−iω0τ0+2ββ¯b16+β¯b17eiω0τ0,K13=b14+α¯b15+b16β¯2e2iω0τ0+b17β¯eiω0τ0,K14=2b14W1110+b15W1120+b14W2010+12b15W2020+b17W113−1+12b17W203−1+β2b18e−2iω0τ0+2ββ¯b18+b15αW1110+b17βW1110+2βb16e−iω0τ0W113−1+12β¯b17W2010eiω0τ0+β¯b16W203−1eiω0τ0,K21=b25α2+b27βαe−iω0τ0+b24α+b26β2e−2iω0τ0,K22=2b25αα¯+b27αβ¯eiω0τ0+b24α+b27α¯βe−iω0τ0+b24α¯+2b26ββ¯,K23=b25α¯2+b27eiω0τ0α¯β¯+b24α¯+b26β¯2e2iω0τ0,K24=2b28αββ¯+2b5αW1120+b27αW113−1+b24W1110α+b28α¯β2e−2iω0τ0+b25α¯W2020+12b27α¯W203−1+12b24α¯W2010+2b26e−iω0τ0W113−1+12b27eiω0τ0β¯W2020+b26β¯W203−1e−iω0τ0+b24W1120+12b24W2020,K31=b35e−2iw0τ0α2+b38e−iw0τ0αβ+b36β2+b37e−iw0τ0β+b34e−2iw0τ0,K32=2b35αα¯+b38αβ¯e−iw0τ0+b38α¯βeiw0τ0+2b36ββ¯+b37βeiw0τ0+2b34,K33=b35α¯2eiw0τ0+b38α¯β¯eiw0τ0+b36β¯2+b37β¯eiw0τ0+b34e2iw0τ0,K34=b39α2βe−2iw0τ0+2b39αα¯β+b38αW022−1e−iw0τ0+2b35e−iw0τ0αW112−1+b35αW202−1eiw0τ0+2b36βW1130+b37βW111−1+b38βW112−1+2b39β+b39β¯e−2iw0τ0+b37W1130e−iw0τ0+12b37W203eiw0τ0++2b34W111−1e−iw0τ0+b34W201−1eiw0τ0.Asq∗0=1/ρ1,q¯1∗,q¯2∗T, one has(61)gz,z¯=1ρ1,q¯1∗,q¯2∗K11z2+K12zz¯+K13z¯2K21z2+K22zz¯+K23z¯2K31z2+K32zz¯+K33z¯2.Then, we have(62)g20=2ρK11+α¯∗K21+β¯∗K31,g11=1ρK12+α¯∗K22+β¯∗K32,g02=2ρK13+α¯∗K23+β¯∗K33.Next, we computeW20θ and W11θ. According to the expression of g21, we have Hz,z¯,θ=−2Req¯∗0⋅fz,z¯qθ.(63)=−g20z22+g11zz¯+g02z¯22+…qθ−g¯20z¯22+g¯11zz¯+g¯02z22+…qθ,Comparing the coefficients of (63), one has(64)H20θ=−g20qθ−g¯02q¯θ,θ∈−1,0,(65)H11θ=−g11qθ−g¯11q¯θ,θ∈−1,0.Substituting (64) and (65) into (55), one can obtain(66)W˙20θ=2iτ0ω0W20θ+g20qθ+g¯20q¯θ,W˙11θ=+g11qθ+g¯11q¯θ.So,(67)W20θ=ig20τ0ω0q0eiτ0ω0θ−g¯023iτ0ω0q¯0e−iτ0ω0θ+E1e2iτ0ω0θ,W11θ=g11iτ0ω0q0eiτ0ω0θ−g¯11iτ0ω0q¯0e−iτ0ω0θ+E2.Now, we computeE1 and E2. From the definition of A in (31), one can obtain(68)∫−10dηθW20θ=2iτ0ω0W200−H200,(69)∫−10dηθW11θ=−H110.From (50), (68), and (69), we have(70)H20θ=−g20qθ−g¯02q¯θ+K11,K21,0T,H11θ=−g11qθ−g¯11q¯θ+K12,K22,0T.Following the method in [18–21], we have(71)iω0I−∫−10eiω0θdηθq0=0−iω0I−∫−10e−iω0θdηθq¯0=0.Then, we can obtain(72)2iω0I−∫−10e2iτ0ω0θdηθE1=K11K210T,which leads to(73)2iω0+b11b12b13e−iω0τ0b212iω0+b22b23e−iω0τ0−b31e−iω0τ0−b32e−iω0τ02iω0+b33E1=K11K21K31,and(74)−b11−b12−b13−b21−b22−b23b31b32−b33E2=K12K22K32.Then, by (72) and (74), we can obtain the following properties of Hopf bifurcation:(75)μ2=−ReC10Reλ′τ0,T2=−ImC10+μ2Imλ′0ω0,β2=−2ReC10.Theorem 3.
From (75), one has(1)
The directions of the Hopf bifurcation is determined bysignμ2: if μ2>0μ2<0, then the Hopf bifurcation is forward (backward) and the bifurcating periodic solutions exist for τ>τ0τ<τ0.(2)
The stability of the bifurcating periodic solutions is determined bysignβ2: the bifurcating periodic solutions are stable (unstable) if β2<0β2>0.(3)
The period of the bifurcating periodic solutions is determined bysignT2: the period increases (decreases) if T2>0T2<0.


## 5. Numerical Examples

In this section, a numerical example is presented to support our obtained results. Consider system (3) with the following parameters: r1=r2=r3=1, c1=0.05, c2=0.081, d1=d2=d=0.01, a1=0.2, a2=0.3, a3=0.4, b1=0.1, and b2=0.1. The positive equilibrium of (3) is1.8155,0.9077,3.3149. By (17), we can obtain τ0=4.15.First, we chooseτ=4.1<τ0, and the outputs of three enterprises are shown in Figure 2. It is easy to see that output of enterprise is asymptotically stable.Figure 2
τ=4.05<τ0. The positive equilibrium of system (3) is asymptotically stable.Finally, we chooseτ=4.25>τ0, 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.Figure 3
τ=4.25>τ0. The bifurcation periodic solution is stable.


## 6. Economic Meaning

In this paper, a novel competition-cooperation enterprise cluster model with core-satellite structure is proposed. The 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 isτ0=4.15. When the production period passes τ0=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. Thus, it is important for enterprise cluster to control the production period in a suitable region.



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*Source: 6644292-2021-06-02.xml*