Theoretical analysis of the general linear model for lanchester ‐ type combat between two homogeneous forces

This paper studies Lanchester‐type combat between two homogeneous forces modeled by the so‐called general linear model with continuous replacements/withdrawals. It demonstrates that this model can be transformed into a simpler canonical form, which is also shown to arise from fixed‐force‐level‐breakpoint battles modeled by Lanchester‐type equations for modern warfare. Analytical expressions for the force levels for the general variable coefficient linear model with continuous replacements/withdrawals are constructed out of so‐called general Lanchester functions for the model without replacements/withdrawals, for which all solutions are shown to be nonoscillatory in the strict sense. These force‐level results are unfortunately so complicated and opaque that the constant‐coefficient version of the model must be studied before any insights into the dynamics of combat may be analytically obtained. Thus, fairly complete results are given for the general linear model with constant attrition‐rate coefficients and constant rates of replacement/withdrawal. However, the expressions for the force levels are still so complicated that we have not been able to develop battle‐outcome prediction conditions directly from them alone but have had to establish general results on the qualitative behavior of solutions. A significant result (and one that greatly complicates the prediction of battle outcome) is that all solutions to the model with replacements/withdrawals are no longer necessarily nonoscillatory in the strict sense, i.e., both sides force levels can take on negative values if the force‐on‐force attrition equations are not “turned off” at the right time. Thus, this paper shows that the addition of continuous replacements/withdrawals to a Lanchester‐type model may significantly change the qualitative behavior of the force‐level trajectories. Battle‐outcome prediction conditions are nevertheless given, and important insights into the dynamics of combat are briefly indicated.