# Direction and Stability of Hopf Bifurcation in a Delayed Solow Model with Labor Demand

Authors: Sanaa ElFadily; Abdelilah Kaddar; Khalid Najib
Journal: International Journal of Differential Equations
 (2019)

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## Abstract

This paper is concerned with a delayed model of mutual interactions between the economically active population and the economic growth. The main purpose is to investigate the direction and stability of the bifurcating branch resulting from the increase of delay. By using a second order approximation of the center manifold, we compute the first Lyapunov coefficient for Hopf bifurcation points and we show that the system under consideration can undergo a supercritical or subcritical Hopf bifurcation and the bifurcating periodic solution is stable or unstable in a neighborhood of some bifurcation points, depending on the choice of parameters.


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## Body

## 1. Introduction

The term “economic growth” is, generally, employed to describe the increase in the potential level of real output produced by an economy over a period of time. It is conventionally measured as the percent rate of increase in real gross domestic product (GDP). Several studies have analyzed national income and capital stock to explain an economy’s growth rate in terms of the level of labor force, saving, and productivity of capital. After the pioneering works of Harrod (1939) [1], Domar (1946) [2], Cassel (1924) (see Kelley (1967, [3]), and Solow (1956) [4], the literature on the modeling of mutual interactions between the economically active population and economic growth has increased considerably in recent years (see, for example, [5–9] and references cited in these publications). The original model of economic growth put forward by Solow (1956), based on an ordinary differential equation, was employed to describe the evolution of capital stock and, consequently, to explain an economy’s growth rate in terms of the labor force, the level of saving, and productivity of capital [4]. The aforementioned models play an important role in explaining phenomena related to economic growth, namely, development, population growth, unemployment, and savings and they have been developed and investigated in-depth in terms of the stability, bifurcations, oscillations, and chaotic behavior of solutions [5, 10–13].Recently, researchers in applied mathematics have proposed systems of differential equations to analytically study the relationship between economic growth and the population concerned [14–18]. They have proved the birth of branches of bifurcated periodic solutions from a positive equilibrium when the delay, namely, the time needed to build, plan, and install new equipment, increases and crosses some critical values [9, 15, 16].In [9], 2016, we divided the labor force population (the economically active population) into two subgroups: the unemployed and the employed persons. The national economy creates jobs to deal with persons who enter the job market every year (increasing of the number of unemployed persons). This creation reduces the unemployment rate that is calculated as follows: (1)p=UtUt+Lt100,where U(t) is the number of unemployed persons and L(t) is the number of employed persons.Empirical studies highlight that economic growth has particularly positive impact on job creation; see, for example, the literature review by Basnett and Sen [19]. To study this impact, we proposed the following model of the mutual interactions between the economically active population and the economic growth:(2)dKdt=sfK,L-δK,dLdt=γ1-LτgKτL,where K is the capital stock, s denotes the constant saving rate, δ is the depreciation rate of capital stock, f is the production function, g is the effective labor demand, τ is the time delay of the recruitment process that is the average time needed for expressing each identifying the jobs vacancy, analyzing the job requirements, reviewing applications, and screening and selecting the right candidate, and γ=r.p/1-p with r being the employment rate and p being the unemployed rate. A significant part of our contribution focused on the existence of the Hopf bifurcation phenomenon around the positive equilibrium of the system (2), when the delay τ passes through a critical value. Yet we have not discussed the direction of Hopf bifurcation and the stability of the resulting periodic solutions of this system.In this work, we investigate the direction and stability of the resulting bifurcating branch of the system (2) by using a second order approximation of the center manifold and computing the first Lyapunov coefficient for Hopf bifurcation points [20]. The results show that the system under investigation can undergo a supercritical Hopf bifurcation and the bifurcating periodic solution is stable in a neighborhood of some bifurcation points.The remainder of this paper is structured as follows. In Section2 we recall previous results. In Section 3 we prove our new result concerning the direction of Hopf bifurcation, that is, to ensure whether the bifurcating branch of periodic solution exists locally and to determine the properties of these bifurcating periodic solutions. Finally, in Section 4 we present our conclusions, some open problems, and future work.


## 2. Previous Results

In this section, we recall the basic results on the local asymptotic stability and the local existence of periodic solutions (Hopf bifurcation) of the positive steady state(K∗,L∗) of the system (2).As in [9], we assume that the function g is continuously differentiable, satisfying the following hypotheses:(H1):  g(0)>0(H2):  g is a strictly monotone increasing and concave function(H3):  limK→+∞g(K)=LewhereLe is the maximal effective labor demand [5]. Moreover, as in the Solow model [4], we consider a Cobb-Douglas function [21]:(3)fK,L=AKαL1-α,where A is a positive constant that reflects the level of the technology and α∈(0,1) and 1-α are the output elasticities of capital and labor, respectively.Proposition 1 (see [9]).
System (2) always has two equilibria P0=(0,0) and P1=(0,g(0)) which exist for all parameter values. On the other hand, if hypotheses (H1),(H2), and (H3) hold, then system (2) also admits a unique positive equilibrium (K∗,L∗), where K∗ is the unique positive solution of(4)gK=δsA1/1-αK,and L∗ is determined by(5)L∗=δsA1/1-αK∗.Theorem 2 (see [9]).
IfM<1, then there exists τ0>0 such that, (i): forτ∈[0,τ0), the steady state (K∗,L∗) is locally asymptotically stable(ii): forτ>τ0,(K∗,L∗) is unstable(iii): forτ=τ0, a Hopf bifurcation of periodic solutions of system (2) occurs at (K∗,L∗) when τ=τ0
with(6)ω02=12γ2-1-αδ2+γ2-1-αδ22-4δ2γ21-α21-M21/2,and(7)τ0=1ω0arccos⁡ω02δα-1Mγ(ω02+δ1-α1-M2,and(8)M=K∗L∗-1g′K∗.


## 3. Direction and Stability of the Hopf Bifurcation

In Theorem2, we obtained a condition under which system (2) undergoes Hopf bifurcation at τ0; that is, a family of periodic solutions bifurcate from the positive steady state point (K∗,L∗) at the critical value τ0. One interesting question here is to determine the direction of Hopf bifurcation, that is, to ensure whether the bifurcating branch of periodic solution exists locally for τ>τ0 or τ<τ0 and to determine the properties of bifurcating periodic solutions, for example, stability on the center manifold. In this section we use a second order approximation of the center manifold and we compute the first Lyapunov coefficient for Hopf bifurcation points to formulate an explicit algorithm about the direction and the stability of the bifurcating branch of periodic solutions of (2) (see [20]).Normalizing the delayτ by scaling t→t/τ and effecting the change x(t)=K(τt)-K∗, y(t)=L(τt)-L∗ and τ=τ0+μ, where μ∈R. Then the system (2) is transformed into(9)dxdt=τ0+μa1x+a2y+a11xy+a20x2+a02y2+a12xy2+a21x2y+a30x3+a03y3,dydt=τ0+μb1x1+b2y1+b11x1y1+y+b20x12+b21x12y1+x+b30y13,where (10)x1=xt-1;y1=yt-1;a1=α-1δ;a2=1-αδK∗L∗-1;b1=γg′K∗;b2=-γ;a20=12αα-1δK∗-1;a02=12αα-1δK∗L∗-2;a11=α1-αδL∗-1;a12=12α2α-1δL∗-1;a21=-12αα-12δK∗-1L∗-1;b11=γg′K∗L∗-1;b20=12γg′′K∗-2g′K∗2L∗-1;and (11)b21=12γg′′K∗L∗-1-2g′K∗L∗-2;b30=γ6g′′′K∗L∗-3-3g′K∗g′′K∗L∗-1L∗-3+1-6g′K∗3L∗-2. Hence, system (9) becomes a functional differential equation in C≔C([-1,0],R2) as(12)u˙t=Lμut+fμ,ut,where u=(x,y)T∈R2,Lμ:C→R2 is the linear operator and f:R×C→R2 is the nonlinear part which are given, respectively, by(13)Lμφ=τ0+μα-1δ1-αδL∗-1K∗00φ10φ20+00γg′K∗-γφ1-1φ2-1,and(14)fμ,φ=τ0+μQ1Q2,where (15)Q1=a11φ10φ20+a20φ120+a02φ220+a12φ10φ220+a21φ120φ20+a30φ130+a03φ230,and(16)Q2=b11φ1-1φ2-1+φ20+b20φ12-1+b21φ12-1φ2-1+φ10+b30φ1-1.Let(17)L≔Lμ:C-1,0,R2→R2.Using the Riesz representation theorem (see [22]), we obtain(18)Lφ=∫-10dηθφθ,where(19)ηθ,μ=τ0+μα-1δδθ1-αδL∗-1K∗δθγg′K∗δθ+1-γδθ+1,with δ(·) being the Dirac function. For φ=(φ1,φ2)∈C, define(20)Aμφθ=dφdθθforθ∈-1,0,∫-10dηθ,μφθforθ=0,and (21)Rμφ=0forθ∈-1,0,fμ,φforθ=0.Thus, the system (9) can be transformed into the following functional differential equation:(22)u˙t=Aμut+Rμut,where ut(θ)=u(t+θ),θ∈[-1,0].Now, forψ=(ψ1,ψ2)∈C([0,1],R2∗), let us consider the operator (23)A∗:C0,1,R2→R2,defined by(24)A∗ψs=-dψdss,fors∈0,1-∫-10ψ-sdηs,fors=0,and forφ∈C([-1,0],R2) and ψ∈C([0,1],R2∗) we define a bilinear inner product:(25)ψ,φ=ψ¯0φ0-∫-10∫0θψ¯ξ-θdηθφξdξ, where η(θ)=η(θ,0), and then A(0) and A∗ are adjoint operators.Suppose thatq(θ) and q∗(s) are eigenvectors of A(0) and A∗ corresponding to iω0τ0 and -iω0τ0, respectively. By a simple computation, we have(26)qθ=1,q1Teiθω0τ0,and (27)q∗s=κq2,1Teiθω0τ0, where q1=(iω0-α-1δ)/(1-α)δL∗-1K∗, and q2=(iω0+γ)/(1-α)δL∗-1K∗.Using the normalization condition, i.e.,q∗(s),q(θ)=1, we get(28)κ=-1-αδL∗-1K∗2iω0-γ+α-1δ-γτ0eiω0τ01-αδL∗-1K∗g′K∗+α-1δ+iω0.It is easy to check that q∗,q¯=0.Let(29)zt=q∗,ut,Wz,z¯,θ=utθ-Re⁡ztqθ,Wz,z¯=Wz,z¯,θ,and(30)f0z,z¯=f(0,Wz,z¯+Re⁡ztqθ,where ut is the solution of Eq. (22).Next, we compute the coordinates describing center manifoldC0 at μ=0.On the center manifoldC0, we have(31)z˙t=q∗,u˙t=q∗,A0ut+R0ut=A∗0q∗,ut+q∗,R0ut=iω0τ0zt+q∗0f0z,z¯,and this last equation is written as follows:(32)z˙t=iω0τ0zt+gz,z¯,where(33)gz,z¯=q∗0f0z,z¯=g20z22+g11zz¯+g02z¯22+⋯,and(34)Wz,z¯=W20θz22+W11θzz¯+W02θz¯22+⋯,where z and z¯ are local coordinates for center manifold C0 in the direction of q and q¯∗.Note thatW is real if ut is real.Thus, from (22) we have(35)W˙=u˙t-qz˙-q¯z¯˙,which leads to(36)W˙=A0W+Hz,z¯,θ,with(37)Hz,z¯,θ=H20θz22+H11θzz¯2+H02θz¯22+⋯,where(38)H20=-A0-2iω0τ0W20θ;H11=-A0W11θ;H02=-A0+2iω0τ0W02θ.Byut=(x,y)T=W(z,z¯)+zq(θ)+z¯q¯(θ) and q(θ)=(1,q1)Teiθω0τ0, we get(39)xt=z+z¯+12W200z2+W110zz¯+12W020z¯2+⋯,xt-1=ze-iθω0τ0+z¯eiθω0τ0+12W20-1z2+W11-1zz¯+12W020z¯2+⋯,yt=q1z+q1¯z¯+12W200z2+W110zz¯+12W020z¯2+⋯,and(40)yt-1=q1ze-iθω0τ0+q1¯z¯eiθω0τ0+12W20-1z2+12W11-1zz¯+12W02-1z¯2+⋯It follows from (19) and (30) that(41)gz,z¯=q¯∗f0z,z¯=q¯∗f00,ut.The coefficients in (33) are(42)g20=2τ0κ¯q¯2a02q12+a11q1+a20+b11eiτ0ω0q1eiτ0ω0+1+b20ei2τ0ω0,g02=2τ0κ¯q¯2a02q¯12+a11q¯1+a20+b11e-iτ0ω0q¯1e-iτ0ω0+1+b20e-i2τ0ω0,g11=τ0κ¯(q¯22a02q1q¯1+a11q¯1+a11q1+2a20+b11q¯1+q1+q¯1eiτ0ω0+q1e-iτ0ω0+2b20,and(43)g21=2τ0κ¯(q¯2a02q¯1W2200+2q1W2110+a1112W2200+W1110q1+W2110+12W1200q¯1+a122q¯1q1+q12+a20W120+2W111+a02q¯1W220+2q1W211+a21q¯1+2q1+b1112e-iτ0ω0W220-1+W111-1q1eiτ0ω0+eiτ0ω0W211-1+12W120-1q¯1e-iτ0ω0+12e-iτ0ω0W2200+eiτ0ω0W2110+W111-1q1+12W120-1q¯1+b202eiτ0ω0W111-1+e-iτ0ω0W120-1+b212q1+e2iτ0ω0q¯1+2eiτ0ω0q1+eiτ0ω0q¯1+3b30eiτ0ω0.In order to computeg21, we need to compute W11(θ) and W20(θ).Forθ∈[-1,0), we have(44)Hz,z¯,θ=q¯∗0f0q0-q∗0f¯0q¯0=-gz,z¯qθ-g¯z,z¯q¯θ=-g20qθ+g¯02q¯θz22-g11qθ+g¯11q¯θzz¯+⋯,which on comparing the coefficients with (37) gives(45)H20θ=-g20qθ+g¯02q¯θ,H11θ=-g11qθ+g¯11q¯θ.From (38) and (45) and the definition of A, we get(46)W˙20=2iτ0ω0W20θ+g20qθ+g¯02q¯θ.Since q(θ)=q(0)eiθτ0ω0, we have(47)W20=ig20eiτ0ω0θτ0ω0q0+ig¯02e-iτ0ω0θ3τ0ω0q¯0+E1e2iτ0ω0θ,and(48)W11=-ig11eiτ0ω0θτ0ω0q0+ig¯11e-iτ0ω0θτ0ω0q¯0+E2,where Ek=(Ek1,Ek2))T∈R2, for k=1 and k=2, are two vectors given by (49)E11=1-αδL∗-1K∗b110eiτ0ω0q1eiτ0ω0+1+b200e2iτ0ω0+γe-2iτ0ω0+2iτ0ω0a02q12+a11q1+a20-γe-2iτ0ω0+2iτ0ω0α-1δ-2iτ0ω0-1-αδγL∗-1K∗g′K∗e-2iτ0ω0,E12=α-1δ-2iτ0ω0b110eiτ0ω0q1eiτ0ω0+1+b200e2iτ0ω0-γg′K∗e-2iτ0ω0a02q12+a11q1+a201-αδγL∗-1K∗g′K∗e-2iτ0ω0+α-1δ-2iτ0ω0)γe-2iτ0ω0+2iτ0ω0,E21=(2a02q1q¯1+a11q¯1+q1+2a20+L∗-1K∗(b11q¯1+q1+eiτ0ω0q¯1+e-iτ0ω0q¯1+2b20γ1-L∗-1K∗g′K∗,and(50)E22=γ2a02q0q1+a11q0+a11q1+22a20+α-1δ(b11q¯1+q1+eiτ0ω0q¯1+e-iτ0ω0q1+2a20γδ1-α1-L∗-1K∗. Hence, the first Lyapunov coefficient is given by(51)l1τ=Re⁡c1ωτ+σIm⁡c1ω2τ2,where c1=g21/2+g112/λ+g022/2(2λ-λ¯)+g20g11(2λ+λ¯)/2λ2.Forτ=τ0, we have ω=ω0 and σ=0. Thus, l1(τ0)=1/2ω0τ0Re⁡g21-1/ω0τ0Im⁡g20g11, and consequently, we obtain the following result on the stability of limit cycle bifurcating from the positive equilibrium.Theorem 3.
Assume that conditions of Theorem2 hold. Then, the following is considered.
The direction of the Hopf bifurcation is determined by the sign ofl1(τ0): (1)
ifl1(τ0)<0, then it is a supercritical bifurcation and the bifurcating periodic solutions existing for τ>τ0 are stable(2)
ifl1(τ0)>0, then it is a subcritical bifurcation and the bifurcating periodic solutions existing for τ<τ0 are unstable.


## 4. Numerical Simulations

In this section, we study how the dynamics of the model (2) change when the time delay and the output elasticities of capital vary. Let us consider the following examples where we suppose the parameters of the model take the following values: (52)s=0.3;A=1;δ=0.2;γ=1;andgK=300eK1+eK.Example 4.
Ifα=0.5, then system (2) has a unique positive equilibrium point E∗=(675;300), the critical value of time delay τ0=1.5719, and the first Lyapunov coefficient l1τ=τ0=0.0056151>0, and consequently an unstable periodic solution bifurcates from the positive equilibrium (see Figure 1).Figure 1
Forα=0.5, an unstable periodic solution bifurcates from the positive equilibrium: (a and b) for 1.5718<τ0, existence of bifurcating periodic solution and (c and d), for 1.572>τ0, absence of bifurcating periodic solution (see Theorem 3).
(a)
Number of employed workers forτ=1.5718 (b)
Capital stock evolution forτ = 1.5718 (c)
Number of employed workers forτ = 1.572 (d)
Capital stock evolution forτ = 1.572Example 5.
Ifα=0.1, then system (2) has a unique positive equilibrium point E∗=(470,73;300). The critical value of time delay is τ0=1.57079 and the first Lyapunov coefficient is l1|τ=τ0=-0.026613<0. Thus a stable periodic solution bifurcates from the positive equilibrium (see Figure 2).Figure 2
Forα=0.5, an unstable periodic solution bifurcates from the positive equilibrium: (a and b) for 1.57176>τ0, existence of bifurcating periodic solution and (c and d), for 1.570796<τ0, absence of bifurcating periodic solution (see Theorem 3).
(a)
Number of employed workers forτ = 1.57176 (b)
Capital stock evolution forτ = 1.57176 (c)
Number of employed workers forτ = 1.570796 (d)
Capital stock evolution forτ = 1.570796


## 5. Conclusion

In this paper, we considered the dynamic behavior of a delayed model of mutual interactions between the economically active population and the economic growth. The direction of the Hopf bifurcation and the stability of the bifurcated periodic solution of this model are informed by a second order approximation of the center manifold [20]. From some numerical simulations we conclude that, for some parameters, the Hopf bifurcation can appear and our proposed model can undergo a supercritical or subcritical Hopf bifurcation and the bifurcating periodic solution is stable or unstable in a neighborhood of some bifurcation points, depending on the choice of parameters. Our theoretical and experimental results would highlight that the choice of time delay influences the dynamic behavior of the economic system, which provides an effective way to control the evolution of the active population and economic growth of the system. These results could also help decision makers to better understand the fluctuations in economic growth.For further research, we suggest a study of the Bautin bifurcation for the case when the first Lyapunov coefficient equals zero.



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