# Local Hopf Bifurcation in a Competitive Model of Market Structure with Consumptive Delays

Authors: Xuhui Li
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

A competitive model of market structure with consumptive delays is considered. The local stability of the positive equilibrium and the existence of local Hopf bifurcation are investigated by analyzing the distribution of the roots of the associated characteristic equation. The explicit formulas determining the stability and other properties of bifurcating periodic solutions are derived by using normal form theory and center manifold argument. Finally, numerical simulations are given to support the analytical results.


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## Body

## 1. Introduction

The Lotka-Volterra predator-prey model is proposed by Lotka and Volterra to describe the dynamics between populations in ecology. And it has been extensively studied by many authors [1–7]. Recently, the Lotka-Volterra predator-prey model has been used in economics by many scholars [8–11]. In [8], Brander and Taylor proposed a general equilibrium model of renewable resource and population dynamics related to the Lotka-Volterra predator-prey model. They applied the model to the rise and fall of Easter Island and showed that plausible parameter values generate a “feast and famine” pattern of cyclical adjustment in population and resource stocks. In [9], Delfino and Simmons investigated the links between the health structure of the population and the productive system of an economy which is subject to infectious disease by combining the Lotka-Volterra model with Solow-Swan growth model. They analyzed the local dynamics and found that the epidemiological-economic stationary state is locally stable and an attractor for a wide range of initial conditions. In [10], Farmer derived a simple nonequilibrium model for price formation. He applied the model to several commonly used trading strategies and discussed how the model can be used to understand the long term evolution of financial markets. In [11], Kong analyzed the evolution of market structure with Lotka-Volterra model. And a model of market structure was established by simulating the relations of product in market:
(1)dx(t)dt=r1x(t)[1-x(t)N1-σ12y(t)N2],dy(t)dt=r2y(t)[1-y(t)N2-σ21x(t)N1],
where x(t) denotes the output of the product X at time t. y(t) denotes the output of the product Y at time t. r1 and r2  denote the growth rates of the products X and Y, respectively. N1 and N2 denote the production scale of the products X and Y, respectively. σ12 and σ21 are the competition rates between the products X and Y. In system (1), X and Y are products of the same type but produced by different manufacturers. Kong [11] obtained the conditions for the market structure coming into being by analyzing the stability of system (1).It is well known that time delays are universal in the market structure due to the consumption, competition, or other reasons. Therefore, it is necessary to incorporate time delays into system (1). Based on this consideration, we consider the following model with time delays:
(2)dx(t)dt=r1x(t)[1-x(t-τ1)N1-σ12y(t)N2],dy(t)dt=r2y(t)[1-y(t-τ2)N2-σ21x(t)N1],
where τ1 and τ2 are the consumption delays of the product X and Y, respectively. All the parameters in system (2) have the same meanings as in (1) and all of them are assumed to be positive.This paper is organized as follows. In Section2, we analyze the local stability of the positive equilibrium and the existence of the local Hopf bifurcation. In Section 3, we determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions by using normal form theory and center manifold argument. To support the analytical results, numerical simulations are included at last.


## 2. Local Stability of the Positive Equilibrium and Existence of the Hopf Bifurcation

Considering the economical significance of the system (2), we are interested only in the positive equilibrium of (2).It is not difficult to verify that if conditions(H1):0<σ12<1 and (H2):0<σ21<1 hold, system (2) has a unique positive equilibrium E*(x*,y*), where
(3)x*=(1-σ12)N1(1-σ21)N2y*,y*=N2(1-σ21)1-σ12σ21.Letu1(t)=x(t)-x*, u2(t)=y(t)-y*, and u3(t)=y2(t)-y2*. We still denote u1(t), u2(t) by x(t), y(t), respectively. Then system (2) can be transformed to the following form:
(4)dx(t)dt=a12y(t)+b11x(t-τ1)+a13x(t)y(t)+a14x(t)x(t-τ1),dy(t)dt=a21x(t)+c22y(t-τ2)+a23x(t)y(t)+a24y(t)y(t-τ2),
where
(5)a12=-r1σ12N2x*,b11=-r1N1x*,a21=-r2σ21N1y*,c22=-r2N2y*,a13=-r1σ12N2,a14=-r1N1,a23=-r2σ21N1,a24=-r2N2.
The linearized system of system (4) is
(6)dx(t)dt=a12y(t)+b11x(t-τ1),dy(t)dt=a21x(t)+c22y(t-τ2).
The characteristic equation of system (6) at the positive equilibrium is
(7)λ2+A+Bλe-λτ1+Cλe-λτ2+De-λ(τ1+τ2)=0,
where
(8)A=-a12a21<0,B=-b11>0,C=-c22>0,D=b11c22>0.Case 1 (τ1=τ2=0).
Equation (7) becomes
(9)λ2+(B+C)λ+A+D=0.
Obviously, B+C>0. Therefore, the roots of (9) must have negative real parts if condition (H3):A+D>0 holds. Namely, the positive equilibrium E*(x*,y*) is locally stable if (H3):A+D>0 holds.Case 2 (τ1>0,τ2=0).
Equation (7) can be transformed into the following form:
(10)λ2+Cλ+A+(Bλ+D)e-λτ1=0.Letλ=iω1(ω1>0) be a root of (10); then we get
(11)Dsinτ1ω1-Bω1cos⁡τ1ω1=Cω1,Dcos⁡τ1ω1+Bω1sinτ1ω1=ω12-A.
From (11), we can have
(12)ω14+(C2-B2-2A)ω12+A2-D2=0.
It is easy to know that (12) has only one positive real root
(13)ω10=-(C2-B2-2A)+(C2-B2-2A)2-4(A2-D2)2,
if (H4):A2-D2<0 holds.The corresponding critical value of time delayτ1k is
(14)τ1k=1ω10arccos(D-BC)ω102-ADB2ω102+D2+2kπω10,k=0,1,2….
Next, we verify the transversality condition. Differentiating (10) with respect to τ1, we have
(15)[dλdτ1]-1=-2λ+Cλ(λ2+Cλ+A)+Bλ(D+Bλ)-τ1λ.
Thus,
(16)Re[dλdτ1]τ1=τ10-1=2ω102+C2-B2-2AB2ω102+D2.From (13) and (16), it is easy to verify that Re[dλ/dτ1]τ1=τ10-1>0. Thus, the transversality condition is satisfied. In conclusion, we have the following results.Theorem 1.
If condition(H4):A2-D2<0 holds, then the positive equilibrium E*(x*,y*) of system (2) is asymptotically stable for τ1∈[0,τ10) and unstable when τ1>τ10; system (2) undergoes Hopf bifurcation at τ1=τ10.Case 3 (τ1=0,τ2>0).
Equation (7) becomes
(17)λ2+Bλ+A+(Cλ+D)e-λτ2=0.
Let λ=iω2(ω2>0) be a root of (17); then we get
(18)Dsinτ2ω2-Cω2cos⁡τ2ω2=Bω2,Dcos⁡τ2ω2+Cω2sinτ2ω2=ω22-A,
which follows that
(19)ω24+(B2-C2-2A)ω22+A2-D2=0.
Similar to Case 1, we can know that (19) has only one positive root
(20)ω10=-(B2-C2-2A)+(B2-C2-2A)2-4(A2-D2)2
if (H4):A2-D2<0 holds. The corresponding critical value of time delay τ2k is
(21)τ2k=1ω20arccos(D-BC)ω202-ADC2ω202+D2+2kπω20,k=0,1,2….
Similar to Case2, we can get Re[dλ/dτ2]τ2=τ20-1>0. That is, the transversality condition is satisfied. Therefore, we have the following results.Theorem 2.
If condition(H4):A2-D2<0 holds, then the positive equilibrium E*(x*,y*)of system (2) is asymptotically stable for τ2∈[0,τ20) and unstable when τ2>τ20; system (2) undergoes Hopf bifurcation at τ2=τ20.Case 4 (τ1>0,τ2∈(0,τ20)).
We consider (7) with τ2 in its stable interval, regarding τ1 as a parameter.
Letλ=iω(ω>0) be a root of (7). Separating real and imaginary parts, then we get
(22)(Bω-Dsinτ2ω)cos⁡τ1ω-Dcos⁡τ2ωsinτ1ω=-Cωcos⁡τ2ω,(Bω-Dsinτ2ω)sinτ1ω+Dcos⁡τ2ωcos⁡τ1ω=ω2-A-Cωsinτ2ω,
which follows that
(23)ω4+(C2-B2-2A)ω2+A2-D2-2×(Cω3-(AC+BD)ω)sinτ2ω=0.
Suppose the following. (H5): equation (23) has at least finite positive roots. And all the roots of (23) are denoted by ω1,ω2,…,ωn. For every fixed ωi(i=1,2,…,n), the corresponding critical value of time delay {τ1i(j)∣j=0,1,2,…} is
(24)τ1i(j)=1ωiarccos[(D-BC)ωi2-AD]cos⁡τ2ωiB2ωi2+D2-2BDωisinτ2ωi+2jπωi,i=1,2,…n,j=0,1,2,….
Let τ10*=min⁡{τ1i(j)∣i=1,2,…n,j=0,1,2,…},  ω10*=ω1i0.
Differentiating (7) with respect to τ1, we obtain
(25)[dλdτ1]-1=-2λ+Be-λτ1+Ce-λτ2(1-τ2λ)-τ2De-λ(τ1+τ2)λ(λ2+Cλ+A)-τ1λ.
Thus,
(26)Re[dλdτ1]τ1=τ10*-1=Δ1Δ3+Δ2Δ4Δ32+Δ42,
where
(27)Δ1=Ccos⁡τ2ω10*-τ2Cω10*sinτ2ω10*+τ2Dsinτ2ω10*sinτ10*ω10*+(B-τ2Dcos⁡τ2ω10*)cos⁡τ10*ω10*,Δ2=2ω10*-Csinτ2ω10*-τ2Cω10*cos⁡τ2ω10*+τ2Dsinτ2ω10*cos⁡τ10*ω10*-(B-τ2Dcos⁡τ2ω10*)sinτ10*ω10*,Δ3=C(ω10*)2cos⁡τ2ω10*,Δ4=(ω10*)3-Aω10*-C(ω10*)2sinτ2ω10*.Obviously, if condition(H6):Δ1Δ3+Δ2Δ4≠0, then the transversality condition is satisfied. Therefore, we have the following results.Theorem 3.
If conditions(H5) and (H6) hold and τ2∈(τ20,0), then the positive equilibrium E*(x*,y*) of system (2) is asymptotically stable for τ1∈[0,τ10*) and unstable when τ1>τ10*; system (2) undergoes Hopf bifurcation at τ1=τ10*.Case 5 (τ2>0,τ1∈(0,τ10)).
We consider (7) with τ1 in its stable interval, regarding τ2 as a parameter.
Letλ=iω(ω>0) be a root of (7). Separating real and imaginary parts, then we get
(28)(Cω-Dsinτ1ω)cos⁡τ2ω-Dcos⁡τ1ωsinτ2ω=-Bωcos⁡τ1ω,(Cω-Dsinτ1ω)sinτ2ω+Dcos⁡τ1ωcos⁡τ2ω=ω2-A-Bωsinτ1ω,
which follows that
(29)ω4+(B2-C2-2A)ω2+A2-D2-2(Bω3-(AB+CD)ω)sinτ1ω=0.
Similar to Case 4, suppose the following. (H7): equation (23) has at least finite positive roots, which are denoted by ω1,ω2,…,ωn. The corresponding critical value of time delay for every ωi is
(30)τ2i(j)=1ω2iarccos[(D-BC)ω2i2-AD]cos⁡τ1ω2iC2ω2i2+D2-2CDω2isinτ1ω2i+2jπω2i,i=1,2,…,n,j=0,1,2,….
Let τ20*=min⁡{τ2i(j)∣i=1,2,…,n,j=0,1,2,…},  ω20*=ω2i0.
Next, we give the following assumption:(31)(H8):[dRe(λ)dτ2]τ2=τ20*-1≠0.
Hence, we have the following theorem.Theorem 4.
If conditions(H7) and (H8) hold and τ1∈[0,τ10), the positive equilibrium E*(x*,y*) of system (2) is asymptotically stable for τ2∈[0,τ20*) and unstable when τ2>τ20*; system (2) undergoes Hopf bifurcation at τ2=τ20*.


## 3. Direction and Stability of the Hopf Bifurcation

In this section, we will use normal form theory and center manifold argument introduced by Hassard et al. [12] to determine the direction of Hopf bifurcation and stability of the bifurcating periodic solutions of system (2) with respect to τ1 for τ2∈(0,τ20). Without loss of generality, we assume that τ2*<τ10*, where τ2*∈(0,τ20).Letτ1=τ10*+μ,  μ∈R. Then μ=0 is the Hopf bifurcation value of system (2). Rescaling the time t→t/τ1, then system (2) can be written as
(32)u˙(t)=Lμut+F(μ,ut),
where
(33)Lμϕ=(τ10*+μ)(A′ϕ(0)+C′ϕ(-τ2*τ10*)+B′ϕ(-1)),F(μ,ut)=(τ10*+μ)(F1,F2)T,
with
(34)ϕ(θ)=(ϕ1(θ),ϕ2(θ))T∈C([-1,0],R2),A′=(0a12a210),B′=(b11000),C′=(000c22),F1=a13ϕ1(0)ϕ2(0)+a14ϕ1(0)ϕ1(-1),F2=a23ϕ1(0)ϕ2(0)+a24ϕ2(0)ϕ2(-τ2*τ10*).
By the Riesz representation theorem, there exists a 2×2 matrix function η(θ,μ):[-1,0]→R2 whose elements are of bounded variation, such that
(35)Lμϕ=∫-10dη(θ,μ)ϕ(θ),ϕ∈C([-1,0],R2).
In fact, we can choose
(36)η(θ,μ)={(τ10*+μ)(A′+B′+C′),θ=0,(τ10*+μ)(B′+C′),θ∈[-τ2*τ10*,0),(τ10*+μ)B′,θ∈(-1,-τ2*τ10*),0,θ=-1.
For ϕ∈C([-1,0],R2), we define
(37)A(μ)ϕ={dϕ(θ)dθ,-1≤θ<0,∫-10dη(θ,μ)ϕ(θ),θ=0,R(μ)ϕ={0,-1≤θ<0,F(μ,ϕ),θ=0.
Then system (32) is equivalent to the following operator equation:
(38)u˙(t)=A(μ)ut+R(μ)ut.
The adjoint operator A* of A is defined by
(39)A*(μ)ϕ={-dφ(s)ds,0<s≤1,∫-10dηT(s,μ)φ(-s),s=0,
associated with a bilinear form
(40)〈φ,ϕ〉=φ-(0)ϕ(0)-∫θ=-10∫ξ=0θφ-(ξ-θ)dη(θ)ϕ(ξ)dξ,
where η(θ)=η(θ,0).From the analysis above, we can see that±iτ10*ω10* are the eigenvalues of A(0) and they are also eigenvalues of A*(0). We assume that ρ(θ)=(1,ρ2)Teiτ10*ω10*θ are the eigenvectors of A(0) belonging to the eigenvalue +iτ10*ω10* and ρ*(θ)=D(1,ρ2*)Teiτ10*ω10*s are the eigenvectors of A*(0) belonging to the eigenvalue -iτ10*ω10*.By a simple computation, we can obtain(41)ρ2=a21iω10*-c22e-iτ2*ω10*,ρ2*=-iω10*+b11e-iτ10*ω10*a21.
In addition, from (40), we can get
(42)D=[1+ρ2ρ-2*+b11e-iτ10*ω10*+c22e-iτ2*ω10*ρ2ρ-2*]-1,
such that 〈q*,q〉=1 and 〈q*,q-〉=0.Next, we get the following coefficients by using a similar computation process to [13]:
(43)g20=2τ10*D-[(-τ2*τ10*)a13q(2)(0)+a14q(1)(-1)i+ρ-2*(a23q(2)(0)+a24q(2)(0)q(2)(-τ2*τ10*))],g11=τ10*D-[+q-(2)(0)q(2)(-τ2*τ10*)))a13(q-(2)(0)+q(2)(0))+a14(q-(1)(-1)+q(1)(-1))+ρ-2*(+q-(2)(0)q(2)(-τ2*τ10*)))a23(q-(2)(0)+q(2)(0))+a24(q(2)(0)q-(2)(-τ2*τ10*)+q-(2)(0)q(2)(-τ2*τ10*)))],g02=2τ10*D-[(-τ2*τ10*)a13q-(2)(0)+a14q-(1)(-1)+ρ-2*(a23q-(2)(0)+a24q-(2)(0)q-(2)(-τ2*τ10*))],g21=2τ10*D-[+12W20(2)(-τ2*τ10*)q-(2)(0)))a13(W11(1)(0)q(2)(0)+12W20(1)(0)q-(2)(0)i+W11(2)(0)+12W20(2)(0))+a14(W11(1)(0)q(1)(-1)+12W20(1)(0)q-(1)(-1)+W11(1)(-1)+12W20(1)(-1))ii+ρ-2*(+12W20(2)(-τ2*τ10*)q-(2)(0))a23(W11(1)(0)q(2)(0)+12W20(1)(0)q-(2)(0)+W11(2)(0)+12W20(2)(0))+a24(W11(2)(0)q(2)(-τ2*τ10*)+12W20(2)(0)q-(2)(-τ2*τ10*)+W11(2)(-τ2*τ10*)q(2)(0)+12W20(2)(-τ2*τ10*)q-(2)(0)))],
with
(44)W20(θ)=ig20q(0)τ10*ω10*eiτ10*ω10*θ+ig¯02q¯(0)3τ10*ω10*e-iτ10*ω10*θ+E1e2iτ10*ω10*θ,W11(θ)=-ig11q(0)τ10*ω10*eiτ10*ω10*θ+ig¯11q¯(0)τ10*ω10*e-iτ10*ω10*θ+E2.
And E1, E2 satisfy the following equations, respectively:
(45)(2iω10*-b11e-2iτ10*ω10*-a12-a212iω10*-c22e-2iτ2*ω10*)E1=2(G11G21),(b11a12a21c22)E2=-(H11H21),
with
(46)G11=a13q(2)(0)+a14q(1)(-1),G21=a23q(2)(0)+a24q(2)(0)q(2)(-τ2*τ10*),H11=a13(q-(2)(0)+q(2)(0))+a14(q-(1)(-1)+q(1)(-1)),H21=a23(q-(2)(0)+q(2)(0))+a24(q(2)(0)q-(2)(-τ2*τ10*)+q-(2)(0)q(2)(-τ2*τ10*)).
Then, we can get the following coefficients:
(47)C1(0)=i2τ10*ω10*(g11g20-2|g11|2-|g02|23)+g212,μ2=-Re{C1(0)}Re{λ′(τ10*)},β2=2Re{C1(0)},T2=-Im⁡{C1(0)}+μ2Im⁡{λ′(τ10*)}τ10*ω10*.
In conclusion, we have the following results for system (2).Theorem 5.
Ifμ2>0(μ2<0), then the Hopf bifurcation is supercritical (subcritical); if β2<0(β2>0), then the bifurcating periodic solutions are stable (unstable); if T2>0(T2<0), then the period of the bifurcating periodic solution increases (decreases).


## 4. Numerical Simulation

In this section, some numerical simulations were given to support the analytical results obtained in the previous sections. Letr1=2,  r2=1.5,  N1=1,  N2=2, σ12=0.4, and σ21=0.6; then we have the following system:
(48)dx(t)dt=2x(t)[1-x(t-τ1)-0.2y(t)],dy(t)dt=1.5y(t)[1-y(t-τ2)2-0.6x(t)].Obviously, conditions(H1) and (H2) hold. By a simple calculation, we obtain that (48) has a unique positive equilibrium E*(0.7894,1.0526). Then we have A+D=0.9471>0. Namely, condition (H3) holds.Firstly, by computation we can easily getA2-D2=-1.4637<0; that is, condition (H4) holds. Further, we have ω10=1.4150, τ10=1.0241. By Theorem 1, we obtain the corresponding waveform and the phase plots are shown in Figures 1 and 2. Similarly, we have ω20=0.7029, τ20=1.8631. From Theorem 2, we obtain the corresponding waveform and the phase plots are shown in Figures 3 and 4.Figure 1
E
* is asymptotically stable for  τ1=0.95<τ10=1.0241 with initial values “0.25;0.92.”Figure 2
E
* is unstable for  τ1=1.10>τ10=1.0241 with initial values “0.25;0.92.”Figure 3
E
* is asymptotically stable for  τ2=1.65<τ20=1.8631 with initial values “0.25;0.92.”Figure 4
E
* is unstable for  τ2=2.05>τ20=1.8631 with initial values “0.25;0.92.”Secondly, we can obtainω10*=1.3398, τ10*=1.0931 when τ1>0,τ2=0.5∈(0,τ20). From Theorem 3, we know that if we let τ2 be in its stable interval and regard τ1 as a parameter, then E*(0.7894,1.0526) is asymptotically stable for τ1∈[0,τ10*) and unstable when τ1>τ10* and a Hopf bifurcation occurs. This is can be illustrated by Figures 5 and 6. In addition, we have λ′(τ10*)=-0.0221-0.9139i and from (47) we get C1(0)=-0.4057+1.9027i, μ2=-18.3575<0, β2=-0.8114<0, and T2=12.7546>0. From Theorem 5 we know that the direction of the Hopf bifurcation is subcritical and the bifurcating periodic solutions are stable.Figure 5
E
* is asymptotically stable for  τ2=0.5,τ1=0.86<τ10*=1.0931 with initial values “0.25;0.92.”Figure 6
E
*is unstable for τ2=0.5,τ1=1.15>τ10*=1.0931 with initial values “0.25;0.92.”Lastly, forτ2>0,τ1=0.15∈(0,τ10), we have ω20*=0.7119, τ20*=1.8062. That is, when τ2 increases from zero to τ20*=1.8062, E*(0.7894,1.0526) is asymptotically stable. Then it will loses its stability and a Hopf bifurcation occurs once τ2>τ20*=1.8062. This property can be illustrated by Figures 7 and 8.Figure 7
E
* is asymptotically stable for  τ1=0.15,τ2=1.65<τ20*=1.8062 with initial values “0.25;0.92.”Figure 8
E
* is unstable for τ1=0.15,τ2=1.95>τ20*=1.8062 with initial values “0.25;0.92.”


## 5. Conclusion

A competitive model of market structure with consumptive delays is studied in this paper. By analyzing the distribution of the roots of the associated characteristic equation, we obtained the conditions for the local stability of the model and the existence of the Hopf bifurcation. The main results are given in Theorems1–4, which show that the consumptive delays play important roles in the model. It is proved that when some conditions are satisfied, then Hopf bifurcation occurs when the delay passes through the corresponding critical value. In reality, the occurrence of Hopf bifurcation means that the coexistence of the two products in system (2) changes from the positive equilibrium to a limit cycle, which is not welcome in reality. Furthermore, the explicit formulas determining the stability and the direction of the bifurcating periodic solutions are given by using the normal form theory and center manifold argument. The main results are given in Theorem 5. From the view of economy, if the bifurcating periodic solutions are stable, the two products of the same type may coexist in an oscillatory mode. This is valuable from the view of economics. However, our study is restricted only to the theoretical analysis of such economic phenomena. It may be helpful for field investigation or experimental studies on the real situation.



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