When two countries are in an arms race, the rate at which each one spends money on arms

Chapter 0, Problem 54

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When two countries are in an arms race, the rate at which each one spends money on arms is determined by its own current level of spending and by its opponents level of spending. We expect that the more a country is already spending on armaments, the less willing it will be to increase its military expenditures. On the other hand, the more a countrys opponent spends on armaments, the more rapidly the country will arm. If $x billion is the countrys yearly expenditure on arms, and $y billion is its opponents, then the Richardson arms race model proposes that x and y are determined by differential equations. Suppose that for some particular arms race the equations are dx dt = 0.2x + 0.15y + 20 dy dt = 0.1x 0.2y + 40. (a) Explain the signs of the three terms on the right side of the equations for dx/dt. (b) Find the equilibrium points for this system. (c) Analyze the direction of the trajectories in each region. (d) Are the equilibrium points stable or unstable? (e) What does this model predict will happen if both countries disarm? (f) What does this model predict will happen in the case of unilateral disarmament (one country disarms, and the other country does not)? (g) What does the model predict will happen in the long run?