1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 409

fear is modelled as the nonnegative constant A times the amount, y, currently
being spent per year by the second nation for arms. The term -Cx repre-
sents the rate of decrease in yearly expenditures for arms by the first nation
due to the resistance of its people to increased spending for arms. Thus, the
rate of decrease in expenditures due to societal resistance is modelled as the
nonpositive constant -C times the amount, x, currently being spent per year
by the first nation for arms. The constant term r represents the underlying
"grievance" which the first nation feels toward the second nation. If r is pos-
itive, then the first nation has a grievance against the second nation which
causes it to increase arms expenditures. If r is negative, the first nation has a
feeling of goodwill toward the second nation and, therefore, t ends to decrease
its expenditures for arms. The terms of the second equation of system (1)
can be interpreted in an analogous manner. The system of differential equa-
tions (1) for Richardson's arms race model asserts that the rate of change in
the amount spent on arms per year by one nation is increased in proportion
to the amount the other nation is currently spending on arms per year, is
decreased in proportion to the amount it is currently spending on arms per
year, and is decreased or increased based on a feeling of goodwill or grievance
against the other nation.
Appropriate initial conditions for Richardson's arms race model (1) is the
amount spent on arms per year by both nations at a particular time. Suppose
at time t = to the first nation spends x(to) = xo for arms per year and the
second nation spends y(to) =Yo· Thus, the initial value problem to be solved
is

(2)

dx

dt = -Cx + Ay + r; x(to) = xo

dy

dt

Bx - Dy+ s; y(to) =YO·

Or, written in ma trix-vector notation

Using the techniques of chapter 9, a general solution to this initial value

problem can be found (i) by computing the eigenvalues and eigenvectors of the

associated homogeneous system; (ii) by writing the complementary solution

(the solution of the associated homogeneous system); (iii) by calculating a

particular solution to (2'); (iv) by writing the general solution of system (2'),

which is the sum of the complementary solution and the particular solution;

and (v) by determining values for the arbitrary constants appearing in the

general solution which satisfy the given initial conditions. Our general solution

would then depend on the six parameters A, B, C , D , r, and s and on the

initial conditions xo and Yo·