(30 points) The equation dp dt =rP 1 - K + I (t ) , (3a) P(t = 0) = Pin (3b) describes the dynamics of a population of size P as a function of time t under the logistic growth model, which you hopefully studied in your elementary differential equations class. r is a positive parameter denoting the net growth rate of the population at low population values where competition for resources is negligible. K is a positive parameter denoting the carrying capacity. Pin is the positive value of the population size at the starting time. The function I(t) denotes the rate of immigration into the population as a function of time t. We will consider a model: I(t) = X(1 + a sin(yt)) (3c) where A, a, and y are some positive parameters (independent of time) which are further restricted so that I(t) 2 0 always. I don't believe the model (3) can be solved exactly, especially by elementary methods. So we will seek an approximate solution where we treat the effects of immigration as weak compared to the internal birth-death dynamics of the population. (a) (20 points) Nondimensionalize the problem intelligently given the above descrip- tion, choosing good scales so that the nondimensionalized population and time variables both vary on scales of order unity. Write the governing equations in terms of your nondimensional variables and parameters. (b) (10 points) Make precise the vague notion of the effects of immigration being weak. When what mathematical expression is small can one pursue a perturbation theory based on this "weak effect" to obtain an approximation for the dynamics of the population