Lanchester’s equations and their solutions, as continuous differential equations, have been studied for years. This article introduces a new approach with the use of the discrete form of Lanchester’s equations, using dynamical systems or difference equations. It begins with Lanchester’s square law and develops a generalized analytical solution for the discrete model that can be built by knowing only the kill rates and the initial force sizes of the combatants. It then forms the condition of parity (a draw) to develop a simple relationship of these variables to determine who wins the engagement. This article illustrates these models and their solutions using historic combat examples. It also illustrates that current counter-insurgency combat models can be built and solved using various forms of difference equations.
TopHistory is filled with examples of the unparalleled heroism and complexities of war. Specific battles like Bunker Hill, the Alamo, Gettysburg, Little Big Horn, Iwo Jima, the Battle of Britain, and the Battle of the Bulge are a part of our culture and heritage. Campaigns like the Cuban Revolution, Vietnam, Panama, and now the conflicts in Afghanistan and Iraq are a part of our personal history. Powers (2008) and Fox (2008) have suggested that we can model the Global War on Terror (GWOT) conflict with Lanchester’s equations.
Although combat is fought over continuous time, there are typically discrete starting, pause, and stopping points. Often models of combat employ discrete time simulation. For years, Lanchester’s differential equation models were the norm for computer simulations of combat. The diagram of simple combat as modeled by Lanchester is illustrated in Figure 1. We investigate and illustrate the use of a discrete version of these discrete equations. We use models of discrete dynamical systems via difference equations to model conflicts and gain insights by examining the models of “directed fire” historic conflicts such as Nelson’s Battle at Trafalgar, the Alamo, and Iwo Jima. We employ difference equations that allow for a complete numerical and graphical solution to be analyzed and do not require the mathematical rigor of differential equations. We further investigate the analytical form of the “direct fire” solutions to provide a solution template, where applicable, to be used in modeling efforts.
Figure 1. Change diagram of combat modeled by Lanchester
Lanchester’s equations stated that “under conditions of modern warfare” that combat between two homogeneous forces could be modeled from the state condition of a similar diagram (Taylor, 1980). We will call this state diagram (Figure 1), the change diagram.
We will use the paradigm, Future = Present + Change, to build our mathematical models using discrete dynamical systems. This will be paramount as eventually models will be built that cannot be solved analytically but can be analyzed by numerical (iteration) methods and graphs.
We begin by defining the following variables:
x(n) = the number of combatants in the X-force after period n.