International Journal of Differential Equations
Volume 2019 (2019), Article ID 7609828, 8 pages
https://doi.org/10.1155/2019/7609828

Research Article

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

Sanaa ElFadily,¹ Abdelilah Kaddar,² and Khalid Najib³

¹Mohammadia School of Engineering, Mohammed V University in Rabat, Rabat, Morocco
²Laboratoire de Finance, Entrepreneuriat, et Développement, Faculté des Sciences Juridiques, Economiques et Sociales de Salé, Université Mohammed V de Rabat, Sala Al Jadida, Morocco
³Department of Applied Mathematics, National School of Mineral Industry, Rabat, Morocco

Correspondence should be addressed to Sanaa ElFadily; elfadilysanaa@gmail.com

Received 11 February 2019; Revised 24 April 2019; Accepted 8 May 2019; Published 2 June 2019

Academic Editor: Patricia J. Y. Wong

Copyright © 2019 Sanaa ElFadily et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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:

where is the number of unemployed persons and 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:

where is the capital stock, denotes the constant saving rate, is the depreciation rate of capital stock, is the production function, is the effective labor demand, 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 with being the employment rate and 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 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 Section 2 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 of the system (2).

As in [9], we assume that the function is continuously differentiable, satisfying the following hypotheses:

 ():   ():   is a strictly monotone increasing and concave function ():  

where is the maximal effective labor demand [5]. Moreover, as in the Solow model [4], we consider a Cobb-Douglas function [21]:

where is a positive constant that reflects the level of the technology and and are the output elasticities of capital and labor, respectively.

Proposition 1 (see [9]). System (2) always has two equilibria and which exist for all parameter values. On the other hand, if hypotheses , and hold, then system (2) also admits a unique positive equilibrium , where is the unique positive solution of

and is determined by

Theorem 2 (see [9]). If , then there exists such that,

 (i): for , the steady state is locally asymptotically stable (ii): for is unstable (iii): for , a Hopf bifurcation of periodic solutions of system (2) occurs at when

with

and

and

3. Direction and Stability of the Hopf Bifurcation

In Theorem 2, we obtained a condition under which system (2) undergoes Hopf bifurcation at ; that is, a family of periodic solutions bifurcate from the positive steady state point at the critical value . 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 or 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 delay by scaling and effecting the change , and , where Then the system (2) is transformed into

where

and

Hence, system (9) becomes a functional differential equation in as

where is the linear operator and is the nonlinear part which are given, respectively, by

and

where

and

Let

Using the Riesz representation theorem (see [22]), we obtain

where

with being the Dirac function. For , define

and

Thus, the system (9) can be transformed into the following functional differential equation:

where

Now, for , let us consider the operator

defined by

and for and we define a bilinear inner product:

where , and then and are adjoint operators.

Suppose that and are eigenvectors of and corresponding to and , respectively. By a simple computation, we have

and

where , and

Using the normalization condition, i.e., , we get

It is easy to check that

Let

and

where is the solution of Eq. (22).

Next, we compute the coordinates describing center manifold at

On the center manifold , we have

and this last equation is written as follows:

where

and

where and are local coordinates for center manifold in the direction of and

Note that is real if is real.

Thus, from (22) we have

which leads to

with

where

By and , we get

and

It follows from (19) and (30) that

The coefficients in (33) are

and

In order to compute , we need to compute and

For , we have

which on comparing the coefficients with (37) gives

From (38) and (45) and the definition of , we get

Since , we have

and

where , for and , are two vectors given by

and

Hence, the first Lyapunov coefficient is given by

where

For , we have and Thus, , and consequently, we obtain the following result on the stability of limit cycle bifurcating from the positive equilibrium.

Theorem 3. Assume that conditions of Theorem 2 hold. Then, the following is considered.
The direction of the Hopf bifurcation is determined by the sign of :

(1)if , then it is a supercritical bifurcation and the bifurcating periodic solutions existing for are stable(2)if , then it is a subcritical bifurcation and the bifurcating periodic solutions existing for 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:

Example 4. If , then system (2) has a unique positive equilibrium point , the critical value of time delay , and the first Lyapunov coefficient , and consequently an unstable periodic solution bifurcates from the positive equilibrium (see Figure 1).

(a) Number of employed workers for

(b) Capital stock evolution for = 1.5718

(c) Number of employed workers for = 1.572

(d) Capital stock evolution for = 1.572

Figure 1: For , an unstable periodic solution bifurcates from the positive equilibrium: (a and b) for , existence of bifurcating periodic solution and (c and d), for , absence of bifurcating periodic solution (see Theorem 3).

Example 5. If , then system (2) has a unique positive equilibrium point The critical value of time delay is and the first Lyapunov coefficient is Thus a stable periodic solution bifurcates from the positive equilibrium (see Figure 2).

(a) Number of employed workers for = 1.57176

(b) Capital stock evolution for = 1.57176

(c) Number of employed workers for = 1.570796

(d) Capital stock evolution for = 1.570796

Figure 2: For , an unstable periodic solution bifurcates from the positive equilibrium: (a and b) for , existence of bifurcating periodic solution and (c and d), for , absence of bifurcating periodic solution (see Theorem 3).

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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

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