calculus

[20 pts] 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 = KN (1 - N) where N(t) is the population at time t, k is the growth rate, and & is a constant. For the purpose of this problem, we will take _ = 20 and k = 1/4. I. [3 pts] The direction field for the differential equation is shown below. A. Sketch the solution to the initial value problem It = KN (1 - 2 ) , N(0) = 2 for t 2 0. B. From the direction field, what do you expect the maximum number of cases to be? That is, make a conjecture about lim N(4) based on the direction field. Write your answer in the box. 1-+00 lim N(t) = 1-+00 Il. [3 pts] Substitute the values for k and [ into the equation and show that the differential equation can be brought 80 into the form N(20 - N) - dN = di.80 Ill. [12 pts] Solve the initial value problem N(20 - N) -dN = dt, N(Q) = 2 by integrating the left and righthand sides. Explicitly solve for NV in your final answer. IV. [2 pts] Use your answer to Part III to calculate lim N(t)