February 10, 2008 Parkview Campus Western Michigan University Dear Students of Math 3740, Congratulations on your hire as our new mathematical consultants! As you may know, EcoSystems, Inc. has an immensely successful series of fish farms that provide a needed and ecologically sound food source for thousands of satisfied customers near our headquarters in Petkyos. As our hallmark has been the freshness of our fish, we have not been to expand our distribution to include other parts of the country, such as Kalamazoo. We have recently been offered the opportunity to take over a large lake not far from Kalamazoo which would permit the establishment of a fish farm in that location. Needless to say, it is essential that we obtain absolute assurance of our success before we undertake such a venture. It is our experience that the reproduction rate of the fish is proportional to the size of the fish population and limited by the number of fish that the farm can support. Additionally, we expect predation will be significant and it will produce a measurable effect on fish population whenever a significant number of fish are present. Several years ago, we hired an outside consultant to analyze our situation. She proposed the model 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 We understand that N is the number of fish, R,K,P, and A are constants, and epsilon is a parameter much less than one. We recall that the consultant said something about "substituting t=alpha*tau and N=beta*u to change variables in the equation, choosing the right values for alpha and beta, and changing the differential equation to the form 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where r and q are constants." She said, "If you write the second differential equation in terms of N and t, you will see what r, q, alpha, and beta need to be." We expect epsilon is much smaller than one, q is close to one, and, by appropriate care of our fish, r can be controlled between one and thirty. print(); # input placeholder Our consultant's contract, however, did not require analysis of the model due to the extreme shortsightedness of our company's board of directors. We do not understand how the fish population will change. Our new board has charged us with finding new consultants for the analysis of these equations to make them useful to our company. As our new consultants, you must analyze these equations, covering the following issues. (A) Do we expect a stable fish population from which harvesting could take place? (B) How large does our initial population of fish need to be to obtain this stable population? (C) How long will it take to obtain a stable population? We hope for a Maple document containing your report by February 22 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 February 20 at 5pm. We look forward to your response. Sincerely, S. L. "Ope" Fields Ecosystems, Inc. JSFH