1550078481-Ordinary_Differential_Equations__Roberts_

446 Ordinary Differential Equations

The Lanchester model for two conventional forces engaged in battle is

(1)

dx

dt = f(t) - Ax - By

dy = g(t) - Cy - Dx

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 loss rate of
the x-force; and where g(t) is the reinforcement rate, Cy is the operational
loss rate, and Dx is the combat loss rate of the y-force. Here, B is the com-
bat effectiveness coefficient for the y-force and D is the combat effectiveness
coefficient of the x-force.

First, let us consider the simplest case of system (1)- the case in which the
battle takes place so rapidly that no reinforcements arrive, J(t) = g(t) = 0,

and no operational losses occur, A = C = 0. Thus, we wish to consider the

linear autonomous system

(2)

dx

-=-By

dt

dy = -Dx.

dt

Solving -By = 0 and -Dx = 0 simultaneously, we see that the origin is
the only critical point of system (2). The coefficient matrix of the linear
a utonomous system (2)

A= ( _i -~)

has eigenvalues A= ±VBJ5. (Verify this fact.) Since one eigenvalue is positive

and the other is negative, the origin is a saddle point of system (2).

The trajectories, (x(t), y(t)), of system (2) satisfy the first-order differential
equation

(3)

dy

dx

dy/dt

dx/dt

-Dx

-By·

Let xo > 0 and Yo > 0 be the number of combatants of the two forces at the

start ( t = 0) of the battle. Separating variables in equation ( 3) and integrating

from the initial point (x 0 , y 0 ) to (x(t), y(t)), we find

1

y(t) 1 x(t)

B ydy = D xdx

Yo xo

and, therefore,