1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 449

Hence,

(7) By(t) - Dx2(t) = Byo - Dx6 = K

2 2

where K is a constant. The graph of equation (7) is a one-parameter (K is the

parameter) family of parabolas with the y-axis as the axis of symmetry, with

vertex at (0, K/ B), and which opens upward. The trajectories in the first

quadrant defined by equation (7) are sketched in Figure 10.14. For K > 0,

the guerrilla force, y, wins the battle and at the end of the battle the number

of combatants in the y-force is K/ B. For K = 0, there is a tie-that is, for

some t > 0, x(t) = y(t*) = 0. For K < 0, the conventional force, x, wins the

battle and at the end of the battle the number of combatants in the x-force

is J-2K/D.

y

K/B

K>O K<O

,_,.._, K = 0 ,.-"--.

Figure 10.14 Phase-Plane Portrait of Lanchester's Combat Model

for a Conventional Force Versus a Guerrilla Force

with No Reinforcements and No Operational Losses

EXERCISES 10.4

1. Suppose two conventional forces are engaged in a battle in which each force
   reinforces its combatants at a constant rate and no operational lo sses occur.
   The Lanchester model for this battle is

(8)

dx

dt = r - By

dy

- = s-Dx

dt