March 9, 2008 Parkview Campus Western Michigan University Dear Students of Math 3740, Thank you for your previous analysis. It is helpful for the our current discussion of our business plan. We are also considering an alternative location and we request further analysis. At this second location, there are no natural predators, and we believe the population of fish is modeled by a differential equation of the form 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 for constants R and K. Note that by changing the constants R and K, we can set the units of P and t to be any measure of population and time. For the sake of being clear, we will measure population in hundreds of fish and time in months. At this new location, we are considering harvesting at a rate which is proportional to the population of the fish. (A) We think that means our differential equation should be 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 for the harvesting rate h. Does this seem correct to you? If not, please advise us how to change our model. (B) We do not want to drive our fish population to extinction by our harvesting practices. Do you expect there is a nonzero harvesting rate which does not drive our fish population to extinction? How large does our initial population of fish need to be to avoid extinction? We expect that your answer to (B) will depend on the values of R and K. The values of these constants are classified, so we need an analysis from you for all values of R and K. You may start your analysis assuming R=.5 and K=354, but we expect that you will explain how your answer would change with different values of R and K. We expect a Maple document containing your report by March 28 at 5pm. If you should find in the course of your investigation you find that you have questions regarding the project, you are to contact your instructor who has agreed to spend her copious spare time to serve you as a mathematical adviser on this project. However, due to her other commitments, she will be unable to aid you after March 26 at 5pm. We look forward to your response. Sincerely, S. L. "Ope" Fields Ecosystems, Inc. JSFH