The simple bulk biomass model (see definition 2.2 of tutorial 8) can be modified in various ways to suit real situations. Recruitment indices for deep-sea prawn are, for example, available and
should contain such information. It is known that cod feed to
a considerable extent on deep-sea prawn and this factor should
be entered straight into the model. Natural mortality is,
on the other hand, less well known and it is difficult to estimate the
increase in biomass between years.
\begin{defn}
\textbf{Prawn model}:

\[
B_{y + 1} = \alpha B_y - Y_y + \beta R_y - \delta D_y
\]

$ R_y $= recruitment index\\
$ D_y $=index of cod predation on prawn\\
$\alpha $= production excluding recruitment and mortality excluding predation by cod
\end{defn}

The system is described in such a way that biomass at the beginning of the year is multiplied by a
coefficient, catch is subtracted, recruitment is added and predation
is subtracted.\\
 
Unknown coefficients in the model are $q$, $\alpha $, $\beta $ ,
$\delta $ , q and $ B_0 $ . Given these coefficients, it is possible
to compute $ B_y $ for all the years. Estimation of the coefficients
is arrived at by establishing the values that match existing data on
catch per unit effort most closely.\\

This model can then be used to examine biomass changes from one year
to the next. For a given size of the cod stock, the effect of catches
on the stock can be examined.


 

\begin{xmpl}
The following data provides an example of the type of data needed for a prawn model:\\

http://tutor-web.net/[…]/prawnmodel.dat \\

The data can be entered into a spreadsheet and the calculations completed. When
using the spreadsheet, we begin by guessing the values of unknown
coefficients. These values are then used to update the stock and make
predictions as to catch per unit effort. Then the quadratic deviation
of the projection is computed. Finally, different values for the
coefficients are tested in order to examine which values give the
lowest quadratic sum.\\

http://tutor-web.net/[…]/prawnmodelexpanded.dat\\

As can be seen from the last column, an estimate is obtained of the
amount of prawn consumed by cod on an annual basis. This is denoted
as $\delta D $.\\


It is possible to plot measured and projected catch per unit effort. Although it is seen that the
predictions match the data quite closely, it is also clear that the
number of parameters is high compared to the number of data points.\\

This model was introduced in (Stefánsson et.al. 1994)). The results
of stomach content investigations in Icelandic waters have been
presented in various articles by Pálsson et.al.\\
\end{xmpl}

\begin{xmpl}
Herring models\\
Acoustic surveys provide an estimation of the size of the
herring stock at a certain point in time. \\
 
Measurements of stock size do, however, not give any information on
yield potential. This requires additional information on how the
stock reacts to catches. If, for example, renewal within the stock is
slow, hardly anything can be caught from it without depleting it by an
amount which almost equals the catch. If, on the other hand, renewal
is fast (high natural mortality rates, good recruitment and fast
individual growth) it will be possible to catch a bigger proportion
from the stock each year.\\
 
Stock-production models like the ones described above have therefore
been designed for herring stocks. These models are then estimated in
such a way that is most consistent with counts or catch per unit
effort.
\end{xmpl}