3) A logistic growth model can be used model population growth when there are limited resources to support the population. For instance, the model can be used to track the population growth of a species of fish in a pond, where the limited space and availability of food will place constraints on how the fish population can increase in time. The differential equation that models logistic growth is dN dt =KN (1- 2) where N(t) is the population at time t, k is the growth rate, and L is a constant. For the purpose of this problem, we will take L = 20 and k = 1/4. a) (5 points) The direction field for the differential equation is shown below. i) Sketch the solution to the initial value problem - dN dt = KN (1 - 2 ) , N(0) = 2 for t 2 0. ii) From the direction field, what do you expect the maximum population to be? That is, make a conjecture about 1-+ 00 lim NV(t) based on the direction field. Write your answer in the box. lim N(t) = 1-+00 b) (3 points) Substitute the values & = = and [ = 20 into the equation and show that the differential equation can be 80 brought into the form - "N(20 - N) - dN = dt.Problem 3 continued... c) (4 points] Evaluate the indefinite integral [$41le- 80 d] {3 points) Use your work from part (c) to solve the initial value problem '\\i(2l} 'V] em = dt, Nm) = 2. Emlicitlgir some for N as a function of: in your final answer. Note: This will involve some algebra. Be sure to show all your steps. e] [2 points) Use your solution to the IVP from Part c to determine rlim NU]. Does this agree with your answer in Part >c-r_u a