Development of Lanchester-Type Spatial Models with Obtaining Localized Solutions for the Interaction of Two Groups

This paper presents an advanced mathematical framework for modeling combat dynamics between two opposing forces using nonlinear reaction-diffusion equations. Extending classical Lanchester models, we incorporate spatially dependent diffusion coefficients to capture modern battlefield complexities. A robust numerical scheme based on the Alternating Direction Implicit (ADI) method is developed, ensuring stability and second-order accuracy in spatiotemporal discretization. The model integrates logistic growth, combat attrition, and tactical diffusion processes, validated through analytical benchmarks. Simulations reveal intricate pattern formation, transient dynamics, and boundary adherence, demonstrating applicability to military strategy optimization. The complete formulation and numerical implementation are thoroughly discussed, providing insights into nonlinear system behavior under varying tactical conditions.