E. A population of fish is living in an environment with limited resourom. This environment can :11:inr support the population it it contains no more than M sh {otherwise some sh would starve due to an inadequate supplyr of food. eta}. There is considerable evidence to support the theory that, for some sh species, there is a minimum population 111 such that the species will become extinct if the size of the population falls below 131.. Such a population can be modelled using a modied logistic equation: dP P m (a) Use the dierential equation to show that any solution is increasing if m. s: P s: M and decrmsingiferzm. (h) For the case where i: = 1, M = lililll and m = lil.il, draw a direction eld and use it to sketch several solutions for 1lirar'ious initial populations. What are the equilibrium solutions? {c) One can show that lrMm M{Pgm}e 'm(PgM} (a mew: _ {Pu M) is a solution with initial population PM} = Pg. Use this to show that, if Pl] s: m, then there is a time t at which PH) = {l {and so the population will be extinct]. so) =