Lab 5

In this lab we investigate fish population growth and the effects of various types of harvesting on the fish population over the long term.

P.F. Verhulst, a 19$^{th}$ centry Belgian Mathematician and Biologist, is credited as having suggested the best population growth pattern equation, the logistic model. It combines the concepts of exponential growth, $\frac{dN}{dt}=aN,$ and limited growth MATH into the following equation MATH Which can be readly modified the reflect harvesting into MATH where $H(t)$ is the harvesting function. We will explore the following 4 situations:

1. No harvesting (no fishing): $H(t)=0$

2. Constant effort harvesting: $H(t)=EN$ in 2,3,& 4 $E$ is a constant

3. Constant harvesting: $H(t)=E$

4. Perodic harvesting: MATH

For all of the above situations use $k=100$ and $r=4$ & $0\leq x\leq 30$ and $0\leq y\leq 150$

Situation 1. The purpose of this is to determine the biomass of fish the lake will hold. Examine the long term effect of various initial stockings of fish by the Fish and Game people. $N(0)=10$, $40,$ $70,$ $100,$ $130.$ Use algebra to find the equilibrium value and compare this answer to the graphs . What is the max. capicity of the lake?

Situation 2. Here the haversting effort is directly proportional to the fish population in the lake. For this problem consider the constant of proportionality, $E=3.$ Run graphs for this model for $N(0)=10,$ $20,$ $30,$ $40.$ Compare these graphs with the graphs in situation 1. Again use algebra to find the equilibrium value and compare this answer to the graphs and to situation 1.

Situation 3. Here we have a constant harvesting rate that is not connected to anything (except maybe the lake owners greed). We will look at 2 different harvesting rates $H=90$ and $H=110$ using 3 initial populations MATH (2 printouts, 3 graphs/printout). What do these graphs say about the long term outlook for fish in the lake?

(Special note: if the population is killed out ODE Ar. will give a note to the effect that it cannot continue and give the desired accuracy. Say ok and go on.)

Situation 4. Now we get fancy and look at perodic harvesting. Again we will look at 2 different harvesting situations: first where $E=51$ and second when $E=101$ . In each look at the initial conditions $N(0)=10,$ $N(0)=40,$ $N(0)=70.$ $N(0)=100$ (2 printouts, 4 graphs/printout). What do these graphs say about the long term outlook for fish in the lake?

Now use the questions with each section as a basis for a letter to the owners of the lake recommending a fishing scheme. Consider that you are the authority and they are paying for your recommendation so make it good.

Thanks to Tom Wangler, Illinois Benedictine College, and Paul Willams, ACC Physics, for the ideas upon which this lab is based.

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