1550078481-Ordinary_Differential_Equations__Roberts_

414 Ordinary Differential Equations

particle would be forced back into region I and farther away from ( x, y).
The same holds true for a particle which reaches line L2 from inside region I.

At L 2 , dy / dt = 0 but dx / dt > 0 so t he particle moves back into region I and

away from ( x, y). Hence, a particle in region I remains in region I for all
time and moves away from (x, y) as time increases. Tha t is, x -; oo and
y -; oo as t -; oo. So if the initial arms expenditures are such that (xo, Yo)
falls in region I, a runaway arms race results. This example illustrates that
Rich ardson 's arms race model predicts an unstable arms race can result even

when both nations have a feeling of "goodwill" toward the other nation (r < 0

and s < 0). A runaway arms race occurs if the initial arms exp enditures by

bot h nations a re sufficiently large so that (xo, Yo) li es in region I.

y

Region dx/dt dy/dt II

r

I

I + +

II +

III

IV +

IV

x

F igure 10.3 Graph of Lines L1 and L2 for r < 0, s < 0 and C /A < B / D

In region III, dx/dt < 0 and dy/dt < 0. So a particle in region III tends to

move to the left and down in this region, away from (x*, y*) and toward the

origin (0, 0). When the initial arms expenditures (xo, Yo) lie in region III, it

can b e shown that mutual disarma ment (total disarmament) always res ults-

that is , as t -; oo, x -; 0 and y -; 0.

Looking at the arrows in Figure 10.3, we see that a particle in region II

will move to the right and down. Thus, a p a rticle in region II, (i) will move

toward the segment of line L 2 b etween regions I and II, (ii) will move toward

the equilibrium point (x*, y*), or (iii) will move toward the segment of line L 1

b etween regions II and III. If a p article which starts in region II moves to the

portion of the line L 2 between regions I and II, the particle will proceed into

region I and a runaway arms race will result, because on L 2 , dx/dt > 0 and

dy/dt = 0. If a particle in region II moves to t he equilibrium point (x*, y*), it

will remain there. Whereas, if a particle which starts in region II moves to the

portion of the line L 1 between regions II and III, the particle will proceed into

region III and mutual disarmament will resu lt, b ecause on L 1 , dx/dt = 0 and

dy/dt < 0. So when the initial arms expenditures li e in region II, a runaway

arms race may occur, a stable arms race may occur, or mutual disarmament

may occur.