Problem 5 (20pts) Consider the toggle switch model given by a a = X, Ý y. 1+ y2 1 + x2 Determine the number and stability of all equilibria as a function of a > 0. = Hint Show that equilibrium points satisfy either y = x or y = 1/x. For both of these, find the number of positive equilibria (you might find Descartes' rule of sign helpful in the former) and how their existence depends on the value of a > 0. Following this, sketch the nullclines in each case separately, linearize the dynamics, and use the implicit function theorem to conclude stability properties of the equilibria by analyzing the sign of the trace and determinant of the system matrix obtained via the linearization.