Solving Linear and Nonlinear Delayed Differential Equations Using the Lambert <i>W</i> Function for Economic and Biological Problems

Abstract

Studies of the dynamics of linear and nonlinear differential equations with delays described by mathematical models play a crucial role in various scientific domains, including economics and biology. In this article, the Lambert function method, which is applied in the research of control systems with delays, is proposed to be newly applied to the study of price stability by describing it as a differential equation with a delay. Unlike the previous work of Jankauskienė and Miliūnas “Analysis of market price stability using the Lambert function method” in 2020 which focuses on the study of the characteristic equation in a complex space for stability, this study extends the application of this method by presenting a new solution for the study of price dynamics of linear and nonlinear differential equation with delay used in economic and biological research. When examining the dynamics of market prices, it is necessary to take into account the fact that goods or services are usually supplied with a delay. The authors propose to perform the analysis using the Lambert W function method because it is close to exact mathematical methods. In addition, the article presents examples illustrating the applied theory, including the results of the study of the dynamics of the nonlinear Kalecki’s business cycle model, which was not addressed in the previous work, when the linearized Kalecki’s business cycle model is studied as a nonhomogeneous differential equation with a delay.

Keywords

• differential delay equations
• delayed arguments
• Lambert W function
• market price
• nonlinear differential delay equations
• Kalecki’s business cycle model