A predatorprey model has the form dN N NP ) y (1) WWW\"? _A+N where p, K, 3/, A are all positive constants. Here N is the prey density and P is the (assumed constant) predator density. (a) (3 marks) Show how to rewrite (1) as du aru Eu(1u) +u (2) where 1' = pt, u = N/K and a, are positive constants that you should find. (b) (5 marks) Show that for > a: there are exactly two equilibrium points for (2). (c) (5 marks) Show that the necessary and sufficient condition on or, ,6 for there to be exactly three distinct equilibrium points is 1 > H > 2%} 1 and a > [3 and determine their local stability in this case. (d) (4 marks) Suppose that P = p_ and the prey rests in a positive equilibrium point. If an influx of l' . Kp A 2 . . predator then fixes P > 4 1 + E , explain carefully what happens to the prey denSIty. l' yNP (e) (3 marks) If the predation term A+N is replaced by yNP A+N2 ' what is the maximum number of positive equilibria