Force Concentration - Mathematical Model

Mathematical Model

There is no battlefield where battle tactics can be reduced to a pure race of delivering damage ignoring all other circumstances. However, in some types of military warfare such as battle for air superiority, confrontation of armoured forces in World War II and battleship-based naval battles, the ratio of armed forces could become the dominant factor. In that case, equations stated in Lanchester's laws model the potential outcome of the conflict fairly well. Balance between the two opponent forces incline to the side of superior force by the factor of . For example, two tanks against one tank are superior by a factor of four.

This result could be understood if the rate of damage (considered as the only relevant factor in the model) is solved as a system of differential equations. The rate in which each army delivers damage to the opponent is proportional to the number of units – in the model each unit shoots at a given rate – and to the ability or effectiveness of each surviving unit to kill the enemy. The sizes of both armies decrease at different rates depending on the size of the other, and casualties of the superior army approach zero as the size of the inferior army approaches zero. This can be written in equations:

• is the number of units in the first army

• is the rate in which army 1 damages army 2 (affected by unit quality or other advantage)

• is a coefficient which describes army 1's ability to inflict damage per unit per time.

The above equations result in the following homogeneous second-order linear ordinary differential equations:

To determine the time evolution of and, these equations need to be solved using the known initial conditions (the initial size of the two armies prior to combat). See http://www.sosmath.com/tables/diffeq/diffeq.html for the general solution of second-order linear differential equations.

This model clearly demonstrates (see picture) that an inferior force can suffer devastating losses even when the superior force is only slightly larger, in case of equal per-unit qualitative capabilities: in the first example (see picture, top plot) the superior force starts only 40% larger, yet it brings about the total annihilation of the inferior force while suffering only 40% losses. Quality of the force may outweigh the quantitative inferiority of the force (middle plot) when it comes to battle outcomes.