Dyamics of the Battlefield: The Lanchester Model

Transcription

1 Dyamics of the Battlefield: The Lanchester Model Douglas Lewit Final project for Math-374 with Professor Anuj Mubayi of Northeastern Illinois University The Lanchester Model British engineer, Frederick W. Lanchester, born on October 23, 1868 and died March 8, 1946, was famous for his contributions to automotive engineering and operations research. Along with Harry Ricardo and Henry Royce, Lanchester was one of the three great automotive engineers of England. Around the time of World War I, July 28, 1914 to November 11, 1918, many mathematicians and engineers, including Lanchester, became fascinated by the dynamics of the battlefield. Various mathematical models were proposed in an effort to explain--and to predict--how military forces interacted on the battlefield. During World War I these mathematical investigations were mainly academic, although during World War II the United States government actually applied these models to make important decisions about the Battle of Iwo Jima in which the American forces seized control of the Japanese island of Iwo Jima. Outnumbered and outgunned by the Americans, the Japanese were defeated even before the battle began although the American forces suffered

2 many casualties and injuries. There are in fact several variations of the Lanchester model. The most commonly encountered of these is the so-called "direct fire model", a linear system of two autonomous, coupled linear differential equations in which the rate of attrition of one army is directly proportional to the firepower of the opposing army. Ironically, in such a model the initial numbers of soldiers has a greater effect on the outcome of the battle than the relative firepower of the two armies. This particular model is often described by Lanchester's square law, a mathematical statement that asserts that if Army X is times larger than army Y, then army Y must have more than times the firepower of Army X in order to defeat Army X. We will explore an example of this in which Army X begins with 400 soldiers (the Red army) and Army Y begins with 200 soldiers (the Green army). This is traditionally a deterministic model. Although the model is essentially continuous, we will analyze the model using both continous and discrete methods. (1.1) (1.2)

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4 (1.3) (1.4) (1.5) (1.6)

5 (1.7) (1.8) (1.9) (1.10) (1.11) (1.12) (1.13) (1.14) (1.15) (1.16)

6 (1.17) (1.18) (1.19) (1.20) (1.21) (1.22) (1.23) (1.24) (1.25)

7 (1.26) (1.27) This is very interesting! According to the Lanchester model that I used, dx/dt = -2Y, X(0) = 400, and dy/dt = -X, Y(0) = 200, there are about 283 survivors in the Red Army--army X, at the time when the Green Army--army Y, is completely wiped out. This isn't exactly the same as the discrete model above, but it's "still in the ball park" so to speak. According to the discrete model above, there are 300 surviving soldiers in the Red Army at the time when the Green Army's population is reduced to zero (ie. killed off completely). One

8 should always remember that there is going to be some loss of accuracy when discrete models are used to represent continuous functions, or when continuous models are used to represent discrete functions. Now what would happen if each green soldier was armed with 4 weapons instead of only 2? Would that give them enough of an advantage to win the battle? According to the Lanchester model, if army X has twice as many soldiers as army Y, then army Y requires four times the firepower just to break even so that neither army wins. This is called Lanchester's Square Law. Was Lanchester correct? Let's use Maple to find out! How much firepower does the Green army (Army Y) need in order to just break even and tie with the Red Army (Army X)? (2.1) (2.2) (2.3) 2 (2.4) (2.5) (2.6)

9 (2.7) (2.8) (2.9) (2.10) (2.11) (2.12) (2.13) (2.14) (2.15) (2.16) (2.17)

10 0 (2.18) (2.19) Both X(t) and Y(t) are exponential decay functions. This explains

11 why the red and green curves of the animated plot decay to the horizontal axis. There are two ways to interpret this. First, the armies experience mutual annihilation. Nobody survives the military conflict! The other interpretation is that the fighting continues infinitely or for eternity! Since human beings cannot fight indefinitely this interpretation is not realistic at all. But mathematically it makes sense. (A good example of how mathematical theory is sometimes in conflict with our common sense observations of reality.) Now let's see what the discrete simulation looks like! (2.20) (2.21)

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