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Research in Applied Mathematics: Vol. 1
Research Article
Research in Applied Mathematics
Vol. 1 (2017), Article ID 101269, 10 pages
doi:10.11131/2017/101269

Hopf Bifurcation in a Delayed Solow-Verhulst Model

A. Kaddar1, S. Sahbani2, and H. Talibi Alaoui2

1Faculty of Law, Economics and Social Sciences-Salé, Mohammed V University in Rabat, BP: 5295, Morocco

2Faculty of Sciences, Chouaib Doukkali University in El Jadida, BP. 20, 24.000, Morocco

Received 28 November 2016; Accepted 12 February 2017

Editor: Jianlong Qiu

Copyright © 2017 A. Kaddar 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

In this paper, we propose a mathematical study of the relationship between population dynamics and economic growth. To do this, the total population is divided into three disjoint classes: employed, unemployed and economically inactive population. On the one hand, the evolution of the number of individuals in each compartment is described by Verhulst model and on the other hand the economic growth is governed by the Solow equation. The resulting model is a system of differential equations with time delay. The dynamics, of this system, are studied in terms of local stability and of local Hopf bifurcation. Some numerical simulations are given to illustrate our theoretical results. Additionally we conclude with some remarks.

Research Article
Research in Applied Mathematics
Vol. 1 (2017), Article ID 101269, 10 pages
doi:10.11131/2017/101269

Hopf Bifurcation in a Delayed Solow-Verhulst Model

A. Kaddar1, S. Sahbani2, and H. Talibi Alaoui2

1Faculty of Law, Economics and Social Sciences-Salé, Mohammed V University in Rabat, BP: 5295, Morocco

2Faculty of Sciences, Chouaib Doukkali University in El Jadida, BP. 20, 24.000, Morocco

Received 28 November 2016; Accepted 12 February 2017

Editor: Jianlong Qiu

Copyright © 2017 A. Kaddar 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

In this paper, we propose a mathematical study of the relationship between population dynamics and economic growth. To do this, the total population is divided into three disjoint classes: employed, unemployed and economically inactive population. On the one hand, the evolution of the number of individuals in each compartment is described by Verhulst model and on the other hand the economic growth is governed by the Solow equation. The resulting model is a system of differential equations with time delay. The dynamics, of this system, are studied in terms of local stability and of local Hopf bifurcation. Some numerical simulations are given to illustrate our theoretical results. Additionally we conclude with some remarks.