Do a phase plane analysis of the nonlinear system below, in the first quadrant only. That is, sketch all nullclines and clearly indicate the equilibrium points. Use arrows to give the orientation on nullclines, as well as in each region. dx = 8y - x^2 - 12 dt dy = 2(2 - y)(y - 6) dt

(12 points) We are still considering the nonlinear system from question #2. You may refer to your work from that question to answer the following: Does there exist Yo > 0 such that if z(0) > 0 and y(0) > Yo, then y(t) > Yo for all t > 0? If so, give such a Yo and explain briefly. If not, justify your answer. Is it true that, if y(0) < sqrt(z(0) + 12), then we will have y(t) < sqrt(z(0) + 12) for all t >= 0? Solve the system in the special case where y(0) = 6 and z(0) != 6, that is, find y(t) and z(t) for all t > 0. (This requires techniques from previous units.)