408 Ordinary Differential EquationsDuring World War I , Richardson served with an ambulance convoy in
France. While serving in the war, he began to contemplate and write about
the causes of war and how to avoid war. During the mid-1930s, Richardson
developed his model for a two nation arms race. Later, he extended his model
to include n nations and applied this model to the contemporary situation in
Europe. Hoping to avoid an impending war, Richardson submitted an article
on the subject to an American journal and requested its immediate accep-
tance. The editors rejected the article. Shortly after the outbreak of World
War II, Richardson retired. Early in his retirement he continued his research
on the causes of war, but later he abandoned those efforts and returned to his
studies in meteorology. He died on September 30 , 1953.
One idea underlying Richardson's two nation arms race model is the concept
of "mutual fear." One nation begins to arm itself. The nation's stated reason
for doing so is for the purpose of self-defense. A second nation, which fears
the first nation, then begins to arm itself, also for the purpose of self-defense.
Due to the "mutual fear" of each nation for the other, the armaments of both
nations continue to increase indefinitely with time. Since no nation has infinite
resources, this situation cannot actually occur. When expenditures for arms
become too large a part of a nation's budget, the people of that nation force
the government to limit the amount spent on arms and thereby "damp" and
perhaps even "limit" the a rms race.
Richardson selected the amount of money spent on arms per year by a
nation as the dependent variable in his arms race model. Stated verbally the
assumptions of his model are- Arms expenditures increase because of mutual fear.
- Societies resist ever-increasing expenditures for arms.
- Considerations which are independent of the amounts being spent for
arms contribute to the rate of change of the amount spent on arms.
If we let x(t) ~ 0 and y(t) ~ 0 represent the amounts spent on arms
per year by two nations in some standard monetary unit, then according to
Richardson's model, exp enditures for arms p er year must satisfy the following
system of differential equations
dx
dt = Ay-Cx+ r
(1)
dy
dt =Bx - Dy+ swhere A , B , C, and D are nonnegative real constants and r and s are real
constants (negative, zero, or positive). The term Ay represents the rate of
increase in yearly expenditures for a rms by the first nation due to its fear of
the second nation. That is, the rate of increase in yearly expenditures due to