TITLE:
Modeling and Simulation of Player Distribution in a 4-3-3 System Using Reaction-Diffusion Equations
AUTHORS:
Aliou Sonko, Mamadou Moustapha Diop
KEYWORDS:
Football, 4-3-3 System, Reaction-Diffusion Equations, Mathematical Modeling, Parameter Calibration, Tracking Data, Empirical Validation, Tactical Analysis, Finite Differences, Semigroup Theory, Spatial Distribution, Collective Performance
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.3,
March
16,
2026
ABSTRACT: The tactical organization of players in a 4-3-3 system is a crucial issue for optimizing collective performance in football. This article proposes an innovative mathematical approach to model and simulate the spatial distribution of players using reaction-diffusion partial differential equations. Unlike existing multi-agent or stochastic approaches that treat space discretely, our continuous model simultaneously integrates three fundamental forces: tactical influence (attraction toward assigned zones), team cohesion (coordination between groups), and ball attractiveness (responsiveness to game situations). The existence and uniqueness of solutions are rigorously established through semigroup theory and the Lumer-Phillips theorem, ensuring the mathematical validity of the proposed framework. A finite-difference discretization with an implicit scheme enables a stable and efficient numerical implementation. The major contribution of this work lies in the systematic calibration of the model parameters using real tracking data (Metrica Sports), followed by a multidimensional quantitative validation. The results demonstrate excellent agreement between simulated and empirical densities (spatial correlation > 0.82, Wasserstein distance