Mathematical Modelling of Leadership Dynamics and Financial Performance Using Coupled Ordinary Differential Equations

Leadership is widely known to be the key to organizational effectiveness and firm financial performance, but the situation is complicated by the fact that leadership style, leadership behavior, employee experience, organizational state, and financial performance are not fully mathematically correlated. In this paper, two basic contributions are made. First, a mathematical model based on differential equations is developed to describe the dynamics of leadership and organization, and a rigorous mathematical proof is provided for the existence of a repeatable solution. Secondly, the model is analyzed and interpreted analytically and numerically using numerical solutions obtained from the differential transform method. In this technique, the set of differential equations is converted into a more convenient form. Since the coupled system of differential equations is not analytically tractable, the Differential Transform Method and numerical simulations are used to obtain approximate analytical solutions of the main equations. Lastly, the results are explored in terms of the evolution of leadership and its impact on employee, contextual, and financial outcomes. The results show that adaptive leadership strategies lead to better employee outcomes and make organizations more resilient and improve long-term financial performance compared to a focus on short-term wins. Therefore, the model proposed in this study is designed to be a quantitative tool for tracking the leadership process and a mathematical basis for managing organizations in the era of modernism.