This project is to go along with differential equations:

Learning Objectives addressed:

1. Understand sequences, limits, and continuity algebraically, numerically, visually, and verbally.

3.   Be able to model simple scenarios of change through either difference equations or differential equations.

5.   Recognize limits and derivatives in the practical and professional world, particularly in environmental and life science.

6.   Be able to use a computer algebra system and spreadsheet system to investigate or evaluate given problems.

7.   Work in groups to investigate problems and communicate solutions on an introductory level.

 

Project methods:

A small lake trust in Maine is undergoing an investigation that has both environmental and economic impacts.  The exact numbers presented here have been simplified for ease of analysis.  

To build up a population of trout in a small lake, 200 young trout are added each year. In addition, the population increases its own numbers by 20% each year. Let xn denote the size of the population after n years.

a) If x0 = 1200, determine the largest n such that when xn <2800.

b) Once xn =2800 the lake is no longer stocked and fishermen will catch 600 fish per year. What is the fate of the population?

 

Now extend this problem.

c) Suppose you have only have enough money to stock the lake for the 3 years it takes to reach 2800.  What level of fishing could be sustained when the population reaches 2800?

d)  Suppose you want to harvest 600, and that is your main goal.  How long should you stock the lake before switching to allowing permitted fishing?

e) Take the recommendation from part d.  What if you accidentally had overestimated your reproduction rate, and it is really only 15%?  What happens to your population long-term following the guideline recommended in part d?

 

Communicating the solution:

Consider the above, and write a 2-3 page, group report, outlining the management problem and your solution.  In so doing, take into consideration all of the mathematical work and discussion from above. Write the report as if you are writing a recommendation to the lake trust board, using summarizing graphs as necessary.  Submit through Canvas as a url (GoogleDocs).

 

Components of a “lite” management report for this lake (see rubric on Canvas):

1. Title describing the charge of the report

2. Background on the known information prior to the study.  Statement of intent for the report.

3. Investigation under financial constraints: (a-c)

4. Investigation under ecotourism constraints: (d)

5. Possible pitfalls, violation of assumptions (e)

6. Overall recommendation

7. Appendix which includes a summary of work (may be a separate file)  This part will not be graded, however, if there is a mistake in any of the calculations, this gives me the opportunity to go back and see how the mistake was made and determine appropriate partial credit.

Team work:  If you do not contribute in pre-defined, agreed upon ways, you will not receive credit.  Regardless of any individual’s participation, the entire team is responsible for handing in a full report. Please acknowledge all team members on the final report, and do the participation quiz after submitting the report.