Please solve all questions thoroughly!

When two countries are in an arms race, the rate at which each country spends money is determined by its own current level of spending and by its opponent's level of spending. It is assumed 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 country's opponent spends on armaments, the more rapidly a country will arm. Suppose that two adversarial countries, Alphaland and Betaland are in an arms race. If Alphaland spends $c billion on arms each year, and Betaland spends Sy billion, then the Richardson arms race model proposes that the yearly amount each country spends on armaments, r and y, are determined by the system of differential equations shown below: dr dt -0.22 +0.15y + 20 dy dt 0.1.0 - 0.2y + 40. (1) The following statements all refer to the first order linear system shown above. De- termine if each of the following statements is True or False. (a) The equation for dx/dt indicates that for a fixed value of Betaland's arma- ment expenditures, as Alphaland increases it armament expenditures, its rate of arms spending decreases. (b) The equation for det/dt indicates that the more Alphaland spends on arma- ments, the less Betaland will increase its future spending. () The equation for der/dt indicates that the more Betaland spends on arms, the more Alphaland will arm itself. (d) The equation for dx/dt indicates that if both countries are unarmed initially, (x = y = 0), then Alphaland will begin to arm itself. (e) Disarmament (r = y = 0) is an equilibrium solution for this system. (2) Sketch a phase plane portrait for this problem and use it to answer the following questions. On your phase portrait you will want to sketch the two lines, dx/dt = 0 and dy/dt = 0. These lines (called nullclines) will divide the phase plane into 4 regions. Indicate whether x and y are increasing or decreasing in each region on your phase portrait. Upload your phase portrait. (a) If x = 500 and y = 0, is I increasing or decreasing? Is y increasing or decreas- ing? (b) If r = 500 and y = 500, is r increasing or decreasing? Is y increasing or decreasing? (c) If r = 0 and y = 500, is : increasing or decreasing? Is y increasing or decreas- ing? (d) If r = 0 and y = 0, is r increasing or decreasing? Is y increasing or decreasing? (3) Find the equilibrium point(s), that is dx/dt = 0 and dy/dt = 0. (4) Select the true statement about the equilibrium point(s). (a) There is only one equilibrium point and it is stable. (b) There is only one equilibrium points and it is unstable. (c) There is only one equilibrium point and it is a saddle. (a) There is more than one equilibrium point, and at least one of the points is a stable equilibrium. (e) None of the above