J. Math. Computer Sci., 29 (2023), 264–271 Online: ISSN 2008-949X Journal Homepage: www.isr-publications.com/jmcs Modeling the dynamics of business cycle with general in- vestment and variable depreciation rate of capital stock Sara Lasfar a,∗ , El Mehdi Warrak a , Khalid Hattaf a,b , Noura Yousfi a a Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences, Ben M’Sick Hassan II University of Casablanca, P.O Box 7955 Sidi Othman, Casablanca, Morocco. b Equipe de Recherche en Mod ´ elisation et Enseignement des Math ´ ematiques (ERMEM), Centre R ´ egional des M ´ etiers de l’Education et de la Formation (CRMEF), 20340 Derb Ghalef, Casablanca, Morocco. Abstract In this paper, we develop a business cycle model with general investment, variable depreciation rate of capital stock and two delays. The first delay describes the time lag between the decision of investment and its implementation, while the second one models the time lag for investment to be produced. The well-posedness and the existence of economic equilibrium are carefully investigated. Moreover, the stability of the economic equilibrium and the existence of Hopf bifurcation are established. The case when the two delays are equal is rigourously studied. Keywords: Business cycle, depreciation, time delays, stability, Hopf bifurcation. 2020 MSC: 34D20, 34C23, 91B55. ©2023 All rights reserved. 1. Introduction Business cycles are recurrent economic phenomena often called economic cycles and which are defined as a type of fluctuations in macroeconomic variables caused by the instability of endogenous economic factors. In the literature, several mathematical models have been proposed to understand these business cycles. In 1940, Kaldor [5] demonstrated that fluctuations mainly due to investment. Kalecki [6] intro- duced the idea of the existence of a delay between the investment decision and its implementation. In 1999, Krawiec and Sydlowski [7] incorporated Kalecki’s idea into Kaldor’s model presented in [5]. In 2017, Hattaf et al. [4] introduced a second delay in the model of [7] in order to describe the time for the investment to be productive. Then they proposed the following business cycle model:  dY dt = α[I(Y (t), K(t)) - γY (t)], dK dt = I(Y (t - τ 1 ), K(t - τ 2 )) - δK(t), where Y (t) and K(t) denote the gross product and capital stock at time t, respectively. The first delay τ 1 is the time lag between the decision of investment and its implementation. The second delay τ 2 is the ∗ Corresponding author Email address: sara.lasfar1995@gmail.com (Sara Lasfar) doi: 10.22436/jmcs.029.03.05 Received: 2022-05-08 Revised: 2022-08-05 Accepted: 2022-08-18