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A further generalization of an economic growth model is the main topic of this paper. The paper specifically analyzes the effects on the asymptotic dynamics of the Solow model when two time delays are inserted: the time employed in order that the capital is used for production and the necessary time so that the capital is depreciated. The existence of a unique nontrivial positive steady state of the generalized model is proved and sufficient conditions for the asymptotic stability are established. Moreover, the existence of a Hopf bifurcation is proved and, by using the normal form theory and center manifold argument, the explicit formulas which determine the stability, direction, and period of bifurcating periodic solutions are obtained. Finally, numerical simulations are performed for supporting the analytical results.

Most of the phenomena occurring in real-world complex systems, especially in the economics systems, have not an immediate effect but appear with some delay. Therefore time delays have been inserted into mathematical models and in particular in models of the applied sciences based on ordinary differential equations; see the recent book [

The introduction of a time delay into an ordinary differential equation could change the stability of the equilibrium (stable equilibrium becomes unstable) and could cause fluctuations, and Hopf bifurcation can occur. Indeed global existence of Hopf bifurcations has been proved in many delay mathematical models; see papers [

If on one hand, the stability and bifurcation analysis of ordinary differential equations with a single time delay is well outlined in the pertinent literature [

The present paper is concerned with a further generalization of the Solow model [

The rest of the paper is organized into four more sections which follow this introduction. Specifically, Section

Recently, Zak has proposed in [

The assumption that the growth of the amount of capital at time

It is worth stressing that for

The mathematical model (

The characteristic equation (

Let

The characteristic equation (

Setting

Let

Differentiating the characteristic equation (

Bearing the above analysis in mind and the Hopf bifurcation theorem for functional differential equations due to Hale and Verduyn Lunel (see p. 246, Theorem 1.1 of the book [

Let

Equation (

For every arbitrary

The inequality

If

For any

Let

If

From the discussion above, and recalling that for any

Let

equation (

if

if

In this section, we study the direction of bifurcations and the stability of bifurcating periodic solutions of (

For notational convenience, let

Let

This section is concerned with some numerical simulations of the mathematical model (

The first set of simulations refers to the following case:

The time evolution of the function

The dynamics depicted by Figure

The time evolution of the function

The second set of simulations refers to the following case:

The time evolution of the function

Finally, we would show some numerical simulations related to the evolution of

The time evolution of

In the present paper, a generalization of the Solow model by inserting two time delays has been considered. The delays, respectively, represent the time employed in order that the capital is used for production and the necessary time so that the capital is depreciated. Specifically, an asymptotic analysis has been performed referring to the stability analysis of the steady state and the conditions under which a Hopf bifurcation appears.

According to the analysis developed in this paper, the stability of the positive equilibrium changes as the time delays vary. Indeed if

The introduction of time delays can be also performed in the mathematical model developed in [

The Hopf bifurcation analysis developed in this paper must be revised if the mathematical models are not based on ordinary differential equations. Recently an increasing number of partial differential equation models for tumor growth or therapy have been developed; see the references section of paper [

Moreover thermostated integrodifferential equations have been proposed in papers [

It is worth stressing that also the Boltzmann equation with the one-dimensional Bhatnagar-Gross-Krook relaxation type operator [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The first author acknowledges the support by the FIRB project RBID08PP3J-Metodi matematici e relativi strumenti per la modellizzazione e la simulazione della formazione di tumori, competizione con il sistema immunitario, e conseguenti suggerimenti terapeutici. The authors acknowledge the financial support of MEDAlics-Research Center for Mediterranean Relations.