A modification of the Lanchester spatially distributed model is considered, which describes a zero-sum game between two military forces in a two-dimensional domain. The model was first developed in 1916 to simulate a combat between two military forces during World War I. The model takes into account directional strikes given by the velocity vector and the initial displacement of troop concentrations using a step function. The system incorporates diffusion terms, nonlinear responses, Gaussian white noise, and a topographic obstacle (an arbitrarily shaped lake) that restricts the troop displacement. The numerical solution is implemented using the finite element method with domain triangulation to allow for irregular domains. The simulation results demonstrate an increase in spatial heterogeneity due to noise and obstacles as well as a significant impact of directional movement and initial displacement on the conflict dynamics. A stability analysis confirms that system is stable, including an analysis for the finite element method. In addition, a sensitivity analysis to parameters is performed and the computational error is estimated. The results are visualized to illustrate the evolution of troop densities over time. The proposed approach has the advantages over existing methods that it better handles irregular domains and the model is scalable.
Borisov et al. (Sun,) studied this question.