Development of the Model. This is a mathematical model for the build-up of conflict between two nations. Each nation is ready to defend itself against the other, and each considers that the possibilities of attack are real. Let x(t) and y(t) be the war potential of nations A and B respectively. We can measure this potential in terms of the level of armament of each country. By t we denote the time. Assume that the rate of increase of x in time depends linearly on y, that is, x 0 (t) = ky(t) for some constant k > 0. Suppose that the cost of increasing and maintaining armaments has a restraining effect so that we can assume that x 0 (t) = ky(t) x(t) for some constant > 0. Finally, we add a constant term g 0 which represents the underlying grievance felt by nation A towards nation B, and obtain the following differential equation: x 0 (t) = ky(t) x(t) + g Similar arguments yield the following differential equation for y: y 0 (t) = lx(t) y(t) + h where l > 0, > 0, h 0 are constants. Altogether, we obtain the following system of ordinary differential equations for the war potential of two nations: x 0 (t) = x(t) + ky(t) + g y 0 (t) = lx(t) y(t) + h (1) 1 where k, l, , , g, h are constants. System (1) can be written in matrix form as follows: x 0 (t) y 0 (t) = k l x(t) y(t) + g h

a) Find the equilibria for arbitrary k, l, , , g, h.

b) Choose k, l, , , g, h such that lk > 0. Find numerically the solution of (1) choosing an arbitrary initial condition.

c) Give a mathematical interpretation of the result.

SOLVE ALL USING MATLAB