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\begin{document}

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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\Fishing - Make the most of it!}





\markboth{Fishing - Make the most of it!}{Fishing - Make the most of it!}

\author{William Mitchell\\
Macalaster College\\
St. Paul MN 55105 USA\\
wmitchel@macalester.edu
}

\makeStitlePDFLaTex


\begin{abstract}
We offer students a harvesting model for operating a fishery over a 25 year horizon and ask them to write a report on optimal harvesting policy with their analyses for fishing industry experts (not necessarily mathematicians).
\end{abstract}


\section*{SCENARIO DESCRIPTION}

You have been hired to manage a fishery for the next twenty-five years. This means you have to choose
the rate of harvesting of the fish. After carefully studying the previous behavior of the fishery, you write
down a logistic growth model (\ref{eq:fish1}) with for $y(t)$, the number of fish in the population at time $t$:

\begin{equation}\label{eq:fish1}
y'(t) = 0.2 y(t) \left(1 - \frac{y(t)}{10000}\right) - h(t)\, , \quad y(0) = 2500\, .
\end{equation}

Here $h(t)$ is the harvesting rate, with units of fish per year. The harvesting rate is chosen by you. It does
not have to be a constant. The fishery has been depleted before you arrive, so the initial population is
below the maximum value.

Your goal is to balance the desire to harvest lots of fish with the imperative to maintain a viable fish
population for the future. One way to combine and mathematize these goals is to try to maximize the
function $J$ given by (\ref{eq:Jval}):

\begin{equation}\label{eq:Jval}
J = 2000 \alpha + \int_0^{25} 14 e^{-0.1 t} h(t)\,dt\, .
\end{equation}

The term before the integral sign is a sustainability bonus which you can earn by leaving at least 1500 fish alive at the end
of your term. More precisely, take $\alpha  = 1$ if $y(25) \ge 1500$ and $\alpha = 0$ otherwise. The integral term will be large if lots of
fish are harvested during your term; we are assuming a price of \$14 per fish, and the decaying exponential
term refects the fact that most people would rather have a fish right now than next year!

If you immediately harvest all of the fish, you will  have $J =$ \$35,000, so try to do better than that. I have not
done much better than \$50,000 just by making guesses. To find the best possible answer, rather than just a good one, you should explore \emph{control theory}. 

To answer this question, please write a report with the following ingredients:

\begin{itemize}
\item Write an introduction explaining the meaning and units of the numbers $0.2$, $10000$, and $2500$ in the
initial value problem. 
\item Calculate this fishery's {\it maximum sustainable yield\/} (MSY), and explain your calculation. To do this,
suppose that the population level is constant, so the right hand side of the differential equation \eqref{eq:fish1} is zero, and the harvest rate is constant, $h(t) = H$. 
The MSY is the largest possible value of $H$ under these circumstances.
\item Propose a harvesting plan by defining a function $h(t)$. This will mostly likely be an explicit formula
like $h(t) = t+\sin(t)$, but in principle I would accept any construction of a function which is defined for  $t \in  [0, 25]$.
\item Report the value of $J$ that you achieve with your harvesting policy $h(t)$. Bigger is better! A small
part of your grade depends on making $J$ large.
\item Include two graphs in your report, showing $h(t)$ and $y(t)$ from $t = 0$ to $t = 25$. These graphs
should have captions of two or three sentences narrating what you see happening in the graphs.
\item Write a discussion of how you arrived at your function $h(t)$ and why you think it is effective. Give
an interpretation of your proposal targeted at fishing industry experts (not mathematicians). This
is a much larger part of the problem score than the value of $J$.
\end{itemize}

The report should include the two figures generated in your software package.
If you are not sure how to begin, you can start with the guess $h(t) = 100 + 10t$ and then compute $y$ and $J$, and then change $h$ to try to improve your score. You might also use other function types, like polynomials
or $A e^{b t}$ or $A e^{b(t-25)}$. 

An iconic text in the area of fishery management\cite{Clark1976} appears in the references.

\begin{figure}[t]
\centering

\vspace*{-.1 true cm}
\includegraphics[width=12.5cm]{ColinClarkTextFrontisPieceCartoon.jpg}


\parbox{11.7 true cm}{\caption{\label{ClarkFishery} From the frontispiece of \cite{Clark1976} as produced from {\it Punch\/}.}}

\end{figure}

\begin{thebibliography}{98}

\bibitem{Clark1976} Clark, Colin W. 1976. {\it Mathematical Bioecomonimcs: The Optimal Management of Renewable Resources\/}.  New York:  John Wiley \& Sons.



\end{thebibliography}

\teacher{
\section*{NOTES FOR TEACHERS}



I have used this as a homework assignment to be completed in teams of two students. 
I would suggest introducing the problem in class by demonstrating a guess for $h(t)$. 
For example, if you constantly harvest fish at the rate of 250 per year, what happens to the population and how much money do you make over the 25 years? 

I provided a MATLAB script (attached as a separate file in Supporting docs for this Modeling Scenario). This makes the illustrations, given a guess for $h(t)$. It requires the free Chebfun extension of MATLAB, which my students were using in that semester. In the future I'll switch away from MATLAB and will post a Python or Julia version then. 

One issue is that the population can blow up to $-\infty$ in finite time if there is too much harvesting. The differential equation should really be piecewise defined so that $y'(0) = 0$. My code throws an error if this happens, which leads to confusion for a few students.  

You can get a very good result just with a constant harvesting rate (possibly a weakness of this assignment). The choice $h(t) = 382.5$ leads to $J = \$51192.9$.  
The best student solution I've seen so far is the piecewise function   
\[h(t) = \begin{cases}
400,&0\le<t<1\\ 
375,&1\le t \le 24\\
1000,& 24<t\le 25
\end{cases}.\] This leads to $J = \$51278.87$. 

This assignment occurs several steps down the road of LaTeX proficiency in my class, but the problem could be done without a typesetting requirement. 

The summary ``for fishing industry experts, not mathematicians" provides an opportunity to give feedback on scientific writing. I tend to make critical comments but grade generously if they put in any paragraph of text here. 


}





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