1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 445

b. Suppose in system (19) c1 = c2 = c3 = 3 and aij = l. Is the critical

point of system (19) stable or unstable?

7. Suppose in system (19) that nation 3 is a pacifist nation. This situation
   can be represented by setting a31 = a32 = 0.

(i) Determine if the critical point of system (19) is stable or unstable.

(ii) For r1 = r2 = 2 and r3 = -1 so lve the system initia l value problem

consisting of system (19) and the initial conditions y 1 (0) = 4 ,

Y2(0) = 4 , y3(0) = 2 on the interval [0,3]. Display y 1 (t), y 2 (t),

and y3(t) on the same graph. Produce phase-plane graphs of y 2

versus Y1, y3 versus Y1, and y3 versus Y2.

b. Do parts (i) and (ii) of part a. for c 1 = c2 = c3 = 2 and a 12 = a 13 =

a21 = a23 = l.

10 .4 Lanchester's Combat Models

In 191 6, during World War I , F. W. Lanchester authored the book Aircraft

in Warfare: The Dawn of the Fourth Arm. In the text, Lanchester described

some simple mathematical models for the then emerging art of a ir warfare.

More recently, these models h ave been extended to general combat situations

and a re referred to as Lanchester's combat models.

In the elementary combat models, two forces are engaged in combat. Let

x(t) and y(t) denote the number of combatants in the "x-force" and "y-force"

respectively. The principle underlying Lanchester's combat models is that the

rate of change of the number of combatants is equal to the reinforcement

rate minus the operational loss rate minus the combat loss rate. The

r einforcement rate is the rate at which new combatants enter or withdraw

from the battle. The operational loss rate refers to noncombat losses due

to such things as disease, accident, desertion, etc. Lanchester proposed that

the operational loss rate be modelled as being proportional to the number of

combatants. This assumption appears to be too simplistic. The combat loss

rate is the rate at which combatants are killed in battle.

A "conventional force" is one which operates in the open and one whose

members are a ll within the "kill range" of the enemy. As soon as the con-

. ventional force suffers a loss, the enemy concentrates its fire on the remaining
conventiona l combatants. Thus, the combat loss rate of a conventional force
is proportional to the number of the enemy. The constant of proportionality
is called the "combat effectiveness coefficient."