Battle-outcome-prediction conditions for variable-coefficient Lanchester-type equations for area fire†

Abstract

New important battle-outcome-prediction conditions are developed for combat between two homogeneous military forces modelled by variable-coefficient Lanchester-type equations for area fire. Such conditions are very significant in modern operations research for developing important insights into the dynamics of combat. However, similar differential-equation models do arise in other fields of science and technology such as mathematical ecology and epidemiology, and consequently our new mathematical results may also find application there. These new important “simple approximate” battle-outcome-prediction conditions depend on not only the combat-attrition model but also the battle-termination model, and they are developed for two different types of battle-termination conditions (fixed-force-level-breakpoint battles and fixed-force-ratio-breakpoint battles). They are sufficient (but not necessary) to determine the outcome of battle without having to explicitly compute the force-level trajectories, and a generalization of Lanchester’s famous linear law to variable-coefficient combat is involved in their development. Certain integrability properties of the Lanchester attrition-rate coefficients figure prominently in these results, and an important physical interpretation (relating to logistics considerations) is given for these properties.