Math 224--Differential Equations: Project 2

Farming the great ranges of the West

by Gavin LaRose (glarose@umich.edu), Nebraska Wesleyan University, February 1996

Following your successful "debut" as mathematicians in the independent mathematical and scientific consulting community, further contracts are pouring into your company. Most recent of these is one from EcoSystems, Inc., a fish-farming company well-known in Leseatt, TG but less so in the Lonlinc area. Flush with your success at the previous contract, you have been assigned the EcoSystem contract.

plain TeX file with project -- uses gothic initial capitals from the gothic fonts
PostScript file of project

The letter...

EcoSystems, Inc.

29 February 1996

Rigorous Mathematical Contractors (RiMaC), Inc.
Suite 3, Strawmarket Business Plaza
Lonlinc, SK 04685

Dear RiMaC:

As you may know, EcoSystems, Inc., has an immensely successful series of fish farms that dot the scenic Tonwashing coast a short distance from Leseatt and provide a needed and ecologically sound food source for thousands of satisfied customers who are reached daily by our exclusively electric fleet of bright blue delivery vans. As our hallmark has always been the freshness (not to mention the natural and unsullied nature) of our fish, we have to date unfortunately not been able to expand our distribution to include other parts of the country, such, of course, as Lonlinc.

However, we have recently been offered the opportunity to take over a large lake not far from Lonlinc, which would permit the establishment of a fish farm in that location. Needless to say it is essential that if we approach such an undertaking it be from a position of absolute assurance that it will be able to succeed, and it is for the analysis of a model of the farm that we are approaching your firm.

It is our experience that the rate at which the fish in a farm reproduce is both proportional to the number of fish present, and limited by the number of fish that are able to be supported by the space in the farm. Additionally, especially in such a location as Skanebra, we expect predation to be significant. While through vigilant maintenance of the farm it should be possible to restrict this to reasonable level, predation will produce a measurable effect on the fish population whenever there are significant numbers of fish present.

To model this situation, an outside consultant proposed the model
dN/dt = R N (1 - N/K) - P (1 - e^(-N^2/ep A^2))
Her contract, however, was from several years ago, and required only that she come up with this (and a number of other) model(s)---which we have only recently come to require. Owing to a clerical error, much of the explanation that was associated with this particular model was misplaced, though we understand that N is the number of fish, R, K, P and A are constants, and ep is a parameter in magnitude very much less than one. A short distance further in the limited documentation that we have on the model, the consultant concludes that "by substituting t = a T and N = b u into this equation, it is possible to choose a and b to simplify it to the form
du/dT = r u (1 - u/q) - (1 - e^(-u^2/ep))
where r and q are, again, constants." In this equation, (i.) ep is small; (ii.) q is close to 1; and (iii.) r is related to the reproduction rate of the fish, which we can control through feeding policy---we expect that it may range from r=1 to perhaps r=30.

Your contract states that you are to justify and analyze the model proposed in the second equation, covering in particular the following issues:

• the derivation of the second equation from the first;
• an analysis of the validity of the second equation as a model for the fish population in a fish farm;
• an analysis of whether, based on the model equation, we may expect a stable fish population from which harvesting could take place; and, if so,
• an analysis of how large an initial population of fish will be required to obtain this stable population, and the length of time required for the stable population to be established.

As we must make a decision on the option to buy the Skanebra lake by the 18th of March, it is necessary that we receive your finished report by the 15th. If you should find in the course of your investigation that you have questions regarding this project, you are to contact, as a team, Dr. Gavin LaRose, who has in his copious free time agreed to serve as a mathematical consultant for this project. You should further contact him (also as a team) by the 7th of March to report on your progress.

We look forward to seeing your finished report, which should conform to the enclosed technical report requirements.

Yours sincerely,
``Chuck'' R.D. Arwin
President, EcoSystems, Inc.

Encl: technical report requirements

The technical report requirements...

EcoSystems, Inc.

Technical Report Requirements

It is of paramount importance that the reports submitted to EcoSystems, Inc. be written so as to be easily understood and readily applicable. All members of the engineering and design team to whom the report will be submitted boast strong academic records including substantial mathematics. However, they are not guaranteed to remember all that they have learned. Therefore they should be assumed to understand Calculus and perhaps the fundamentals of differential equations.

Reports should further:

• Be written in the first person plural.
• Include mathematical formulas and/or figures in the report as appropriate to describe the methods used and results obtained. While it is preferable that equations be typewritten (like the remainder of the report), it is permissible for them to be handwritten in the typewritten explanation---as a model, your researchers may wish to consider the format of the letter introducing the equations for this problem;
• Clearly explain how the mathematical formulas that are included bear on the problem being solved, and how the solution(s) obtained answer(s) the question(s) that the report is to answer.
• Consist of
  • An Introduction, indicating generally the problem(s) to be solved, and giving an explanation of the organization of the remainder of the report.
  • A Body, describing the mathematical problem(s) that were solved to answer the question(s) posed in the introduction, and the solution(s) to them.
  • A Conclusion, summarizing the results obtained from the solution described in the body and clearly stating their relevance to the original problem as described in the introduction.

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last modified on 29 Feb 1996