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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.

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) [

Recently, researchers in applied mathematics have proposed systems of differential equations to analytically study the relationship between economic growth and the population concerned [

In [

Empirical studies highlight that economic growth has particularly positive impact on job creation; see, for example, the literature review by Basnett and Sen [

In this work, we investigate the direction and stability of the resulting bifurcating branch of the system (

The remainder of this paper is structured as follows. In Section

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

As in [

(

(

(

where

System (

If

(i): for

(ii): for

(iii): for

with

In Theorem

Normalizing the delay

Now, for

and for

Suppose that

Using the normalization condition, i.e.,

Let

Next, we compute the coordinates describing center manifold

On the center manifold

Note that

Thus, from (

By

In order to compute

For

For

Assume that conditions of Theorem

The direction of the Hopf bifurcation is determined by the sign of

if

if

In this section, we study how the dynamics of the model (

If

For

Number of employed workers for

Capital stock evolution for

Number of employed workers for

Capital stock evolution for

If

For

Number of employed workers for

Capital stock evolution for

Number of employed workers for

Capital stock evolution for

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 [

For further research, we suggest a study of the Bautin bifurcation for the case when the first Lyapunov coefficient equals zero.

No data were used to support this study.

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