Suppose two similar countries Y and Z are engaged in an

Chapter , Problem 30

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Suppose two similar countries Y and Z are engaged in an arms race. Let y(t) and z(t) denote the size of the stockpiles of arms of Y and Z, respectively. We model this situation with the system of differential equations dy dt = h(y,z) dz dt = k(y,z). Suppose that all we know about the functions h and k are the two assumptions: If country Zs stockpile of arms is not changing, then any increase in size of Ys stockpile of arms results in a decrease in the rate of arms building in country Y. The same is true for country Z. If either country increases its stockpile, the other responds by increasing its rate of arms production. (a) What do the assumptions imply about h/y and k/z? (b) What do the assumptions imply about h/z and k/y? (c) What types of equilibrium points are possible for this system? Justify your answer. [Hint: Suppose you have an equilibrium point. What do your results in parts (a) and (b) imply about the Jacobian matrix at that equilibrium point?]