1550078481-Ordinary_Differential_Equations__Roberts_

448 Ordinary Differential Equations

while doubling y 0 causes By 6 to quadruple. Since the initial strengths of the

opposing forces, x 0 and y 0 , appear quadratically in equation (4) and, hence,

effect the outcome of the battle in a quadratic manner, equation ( 4) is known

as "Lanchest er's square law."

A "guerrilla force" is one which is invisible to its enemy. When the enemy

fir es into a region containing the guerrilla force, the enemy does not know when

a kill occurs, so the enemy is unable to concentrate its fire on the remaining

guerrillas. It is reasonable to assume t hat the combat loss rate of a guerrilla

force is j ointly proportional to the number of guerrillas and the number of t he

enemy, since the probability that the enemy kills a guerrilla increases as the

number of guerrillas in a given region increases and it increases as the number

of enemy firing into the region increases. Thus, the Lanchester model for a

conventional force, x, engaged in battle with a guerrilla force, y, is

(5)

dx

dt =f(t)-Ax-By

dy =g(t)-Cy-Dxy

dt

where A , B , C , and Dare nonnegative constants; where f(t) is the reinforce-

ment rate, Ax is the operational loss rate, and By is the combat lo ss rate of

the conventional force, x; and where g(t) is the reinforcement rate, Cy is the

operational lo ss rate, and Dxy is th e combat loss rate for the guerrill a force,

y.

Let us now consider a conventional force and a guerrilla force engaged in

a b attle in which no reinforcements occur, j(t) = g(t) = 0, and in which no

operational lo sses occur, A = C = 0. That is, let us consider the nonlinear

autonomous system

dx

dt =-By

(6)

dy

dt = -Dxy.

Dividing the second equation of (6) by the first, we see that

dy

dx

dy/dt

dx/dt

-Dx y

-By

D x

B

Multiplyi ng by B dx and integrating from (x 0 , y 0 ) to (x(t), y(t)), we find

f,y(t) 1 x(t)

B dy D xdx

Yo xo
or

B(y(t) - Yo)

D(x^2 (t) - x6)

2