Consider the predator prey type of system that's given above with a>0. The population x is prey. By itself, its rate of growth increases for small populations and then decreases for x>1. The predator is given by y, and it dies out when no prey is present. The parameter is given by a. You are asked to show that this system has a Hopf bifurcation.
i) Find the fixed point (x_{a},y_{a}) with both x_{a},y_{a} positive.
(0,0) is an equilibrium point. Another one occures when x = 0, a 3rd one occurs when y = 0. To find the 4th equilibrium point, solve 1+2x-x2-y and x - a simultaneously.