Suppose that a population grows according to a logistic model with carrying capacity 6000 and k = 0.0015 per year.
(a) Write the logistic differential equation for these data.
(b) Draw a direction field (either by hand or with a computer algebra system). What does it tell you about the solution curves?
(c) Use the direction field to sketch the solution curves for initial populations of 1000, 2000, 4000, and 8000. What can you say about the concavity of these curves? What is the significance of the inflection points?
(d) Program a calculator or computer to use Euler’s method with step size h = 1 to estimate the population after 50 years if the initial population is 1000.
(e) If the initial population is 1000, write a formula for the population after years. Use it to find the population after 50 years and compare with your estimate in part (d).
(f) Graph the solution in part (e) and compare with the solution curve you sketched in part (c).