1. Consider a simple model of a physiological calcium oscillator in cellular dynamics. The equations describing the dynamics of the system are given by the following equations: =1-( +1)x+xy = f(x, y) dy dt dx dt = =Bx - xy = g(x,y) where x (t) and y(t) represent the activity of activator and inhibitor variables, respectively, at time t. a) Find the nullcines of f and g. Show that there is only one steady state in this system. [6] b) Determine the condition for B required to keep the steady state you found in part a) stable by analysing the Jacobian evaluated at the steady state. [8] c) Show that a Hopf bifurcation occurs at a value a which you should determine by considering the eigenvalues of the Jacobian near the bifurcation point. [6] d) (This part of the question should not be handwritten) Now suppose = 2.5. Plot the phase portrait with the nullclines shown as dashed lines and the direction field as short arrows for 0 x, y 4. You MUST submit the code you used to demonstrate how you simulated the results, otherwise a mark of zero will be applied. Together with the working code, full marks will be awarded for a well-labelled and accurate plot. [12] e) By observing the phase portrait, state whether the steady state is stable or unstable. Briefly explain why.