Show that, if P is a twice differentiable function of t and satisfies the logistic differential equation then dP dt P(-2) = kP = kP kP M' 2P dr = kP (1-27) (1-2). M b. (6 marks) Suppose a population grows according to the logistic growth model and that the initial population size is positive but less than half the carrying capacity of the environment. Use the result from part a. of this question to show the growth rate of the population is maximal when the population size is exactly half the carrying capacity.