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\noindent{\bf Math 3226
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Laboratory  2C}\\
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{\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)}
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\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.

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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 $W$). }

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\item Solve the allometric equation for $\alpha\not=1$.

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\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 $\mu$.

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

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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.

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\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).

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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.}

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\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.

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\end{enumerate}
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\flushright\tiny\copyright\  H. Knaust. \today .
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