\documentclass[12pt,twoside]{article} 
\usepackage{epsf}
\pagestyle{empty}
\renewcommand{\today}{February 17, 1997}

\special{src: 6 LAB2C.TEX} %Inserted by TeXtelmExtel

% \ilf is inline formula in displaystyle
\newcommand{\ilf}[1]{$\displaystyle #1 $}

\special{src: 11 LAB2C.TEX} %Inserted by TeXtelmExtel


\special{src: 14 LAB2C.TEX} %Inserted by TeXtelmExtel

\begin{document}
\noindent{\bf Math 2326
\hfill
Laboratory  2C}\\
\begin{center}
{\bf Aquaculture\footnote{This laboratory is based on a group project in {\em ``Fundamentals of Differential Equations''} by R. Kent Nagle and Edward B. Saff.}\\(Prerequisite: Laboratory 1A or 1B)}
\bigskip\end{center}

\special{src: 24 LAB2C.TEX} %Inserted by TeXtelmExtel

\noindent{\em  Aquaculture is the art of cultivating the plants and fish indigenous to water. In the example considered here, it is assumed that a batch of catfish are raised in a pond. We are interested in determining the best time  for harvesting the fish so that the cost per pound for raising the fish is minimized.

\special{src: 28 LAB2C.TEX} %Inserted by TeXtelmExtel

A differential equation describing the growth of fish may be expressed as 
\[\frac{dW}{dt}=k W^\alpha,\]
where $W(t)$ is the weight of the fish at time $t$, and $k$ and $\alpha$ are empirically determined growth constants. The functional form of this relationship is similar to growth models for other species. Modeling the growth rate or metabolic rate by a term like $W^\alpha$ is a common assumption. Biologists often refer to the equation above as the {\bf allometric equation}. It can be supported by plausibility arguments such as growth rate depending on the surface area of the gut (which varies like \ilf{W^{2/3}}) or depending on the volume of the animal (which varies like $W$). }

\special{src: 34 LAB2C.TEX} %Inserted by TeXtelmExtel

\begin{enumerate}
\renewcommand{\theenumi}{{\bf \arabic{enumi}}}
\item Solve the allometric equation for $\alpha\not=1$.

\special{src: 40 LAB2C.TEX} %Inserted by TeXtelmExtel

\item The solution in {\bf 1.} grows large without bound, but in practice there is some limiting maximum weight $M$ for the fish. This limiting weight may be included in the allometric equation by inserting a dimensionless variable $S$ that can range between 0 and 1 and involves an empirically determined parameter $\mu$.

\special{src: 44 LAB2C.TEX} %Inserted by TeXtelmExtel

Namely we now assume that 
\[\frac{dW}{dt}=k W^\alpha S,\]
where $S$ has the form 
\[S=1-\left(\frac{W}{M}\right)^{\mu}.\]

\special{src: 51 LAB2C.TEX} %Inserted by TeXtelmExtel

When $\mu=1-\alpha$, this equation becomes a {\bf Bernoulli equation}, and has a closed form solution. Solve the equation, when $k=12$, $\alpha=0.75$, $\mu=0.25$, $M=81$ ounces and $W(0)=1$~ounce. The constants are given for $t$ measured in months.

\special{src: 55 LAB2C.TEX} %Inserted by TeXtelmExtel

\item The differential equation describing the cost $C(t)$ of raising fish for $t$~months has 2 parameters: $K_1$ specifies the cost per month (due to costs such as interest, depreciation and labor); $K_2$, the cost for food,  multiplies with the growth rate (because the amount of food consumed by the fish is approximately proportional to the growth rate).

\special{src: 59 LAB2C.TEX} %Inserted by TeXtelmExtel

That is 
\[\frac{dC}{dt}=K_1+K_2 \frac{dW}{dt}.\]
Solve this equation when $K_1=0.4$, $K_2=0.1$, $C(0)=\$1.10$, and $\frac{dW}{dt}$ as determined in {\bf 2.}

\special{src: 65 LAB2C.TEX} %Inserted by TeXtelmExtel

\item Explain why it is optimal to harvest the fish at the time when the ratio \ilf{\frac{C(t)}{W(t)}} is at minimum. Estimate the optimal harvesting time to the nearest month.

\special{src: 69 LAB2C.TEX} %Inserted by TeXtelmExtel


\special{src: 72 LAB2C.TEX} %Inserted by TeXtelmExtel

\end{enumerate}
\vspace*{\fill}

\special{src: 77 LAB2C.TEX} %Inserted by TeXtelmExtel

\end{document}
