Spatial distributed Lanchester model considering nonlinear dynamics

Nikita D. Borisov, Vladimir V. Nefedov

Abstract


The paper proposes a spatially distributed Lanchester model for simulating the dynamics of military conflicts, accounting for diffusion and reaction processes in a two-dimensional domain with obstacles. The model describes the evolution of two troop groups, denoted as U(x, y, t) and V(x, y, t), through a system of nonlinear partial differential equations with diffusion coefficients and time-modulated reaction terms. Interaction occurs after the loss of communication with the central command post, when the main strike and force concentration have already been completed. In other words, the modeled event represents a local interaction implemented according to a pre-agreed tactical plan. The study examines the impact of three types of obstacles—square, circular, and smooth convex—modeling impassable areas that affect the spatial dynamics of wave fronts. The numerical solution of the system is implemented using the alternating direction method and the finite element method with time discretization based on the Crank-Nicolson scheme, ensuring stability through adaptive mesh refinement near obstacles. Initial conditions reflect the distribution of troops along the front line with homogeneous Dirichlet boundary conditions applied. The model's stability is confirmed by Lyapunov exponent analysis, and sensitivity to parameters is investigated to assess their impact on system dynamics. The results of numerical experiments illustrate the influence of obstacle geometry on troop concentrations, with a quantitative analysis of the proportions of areas dominated by forces over time.

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