In the 1930's L. F. Richardson proposed that an arms race between two countries could be modeled by a system of differential equations. One arms race which can be reasonably well described by differential equations is the US-Soviet Union arms race between 1945 and 1960. If $x$ represents the annual Soviet expenditures on armaments (in billions of dollars) and $y$ represents the corresponding US expenditures, it has been suggested[1] that $x$ and $y$ obey the following differential equations $${dx\over dt} = -0.45x +10.5,$$ $${dy\over dt} = 8.2 x - 0.8y -142.$$

(a) Find the nullclines and equilibrium points for these differential equations.
Vertical ($x$-)nullcline: $x =$
Horizontal ($y$-)nullcline: $y =$

(b) Find the equilibrium points.
Equilibria =
(Enter the points as comma-separated (x,y) pairs, e.g., (1,2), (3,4).)

(c) By sketching the nullclines and figuring out what directions trajectories go in each region, be sure you can determine what you expect typical trajectories in the phase plane will do.

Then, consider the following historical data. In 1948, the arms budgets $(x,y)$ were (20,20). What do you expect to happen to $x$ and $y$ in the long run if we start from this initial condition?
$x \to$
$y \to$
(Enter a numerical value, or infinity if the quantity diverges.)

(d) The actual expenditures for the USSR and US for the five years following 1948 are shown in the table below.

Is this evolution of the values for $x$ and $y$ what the model predicts? ? the data are reasonably consistent with the model the data leave a region the model says they shouldn't the data show a trajectory that isn't at all consistent with the model

[1] R. Taagepera, G. M. Schiffler, R. T. Perkins and D. L. Wagner, Soviet-American and Israeli-Arab Arms Races and the Richardson Model (General Systems, XX, 1975).