Dynamic analysis of competitive marketing strategies using differential game models and Runge–Kutta solutions

Abstract

This study presents a comprehensive mathematical and computational analysis of a competitive market model, incorporating Pontryagin’s Maximum Principle, the Hamiltonian formulation, and the Runge–Kutta (RK4) numerical method. The proposed model accounts for market share dynamics, pricing strategies, and advertising efforts, capturing real-world competitive interactions through logistic growth functions and nonlinear incidence rates. By formulating the problem as a differential game, we analyze the optimal control strategies of firms aiming to maximize their long-term market positions while minimizing operational costs. The results indicate that firms implementing aggressive pricing strategies initially experience rapid market share growth but later face diminishing returns due to market saturation and competitive responses. Conversely, firms adopting moderate strategies achieve sustained growth and long-term stability. Numerical simulations reveal the impact of damping effects, demonstrating that firms must balance short-term profitability with sustainable competitive positioning to maintain market dominance. The study also explores the role of advertising intensity, highlighting the nonlinear relationship between promotional expenditures and market gains. These findings provide valuable insights for firms seeking to refine their strategic decision-making in highly competitive environments. The research also emphasizes the importance of numerical optimization techniques in addressing real-world economic challenges, paving the way for future studies that integrate stochastic modeling, adaptive learning, and hybrid computational methods to enhance predictive accuracy in competitive market dynamics.