\magnification=\magstep1
\centerline{MONT 105N -- Analyzing Environmental Data}
\centerline{A modeling example}
\centerline{Class, January 28, 2019}
\bigskip
\noindent
Consider the following situation:
\bigskip
\item{$\bullet$} A park service wants to maintain a stock of rainbow trout
in a (relatively small) lake as a resource for recreational fishermen.
\item{$\bullet$} The lake manager knows that for a long period, the
number of trout caught every week has been 50 and that the total population
of the lake has been 600 adult trout.  
\item{$\bullet$} The trout population in the lake has been maintained
by buying adult trout from a commercial hatchery and introducing them
in the lake on a regular basis (restocking).  
\item{$\bullet$} Orders of new trout for restocking take exactly {\it five weeks} to arrive
from the hatchery to the lake (a {\it delivery delay}).
\item{$\bullet$} Now suppose that at some point the demand for fishing increases and 
there will be 55 fish caught every week.
\item{$\bullet$}  In order to plan for the restocking under the new situation (and provide
some flexibility in case the demand changes again in other ways), the
manager decides to institute the following system:
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\itemitem{1.}  Each week, she will compute the {\it average} of the numbers of fish
caught in the {\it five previous weeks}.  She calls this the {\it perceived need for 
restocking}.  
\itemitem{2.}  Having too many fish in the lake makes the fishing too easy (the fishermen
want a challenge, after all!).  But having too few fish
makes the fishing too difficult.  The ideal number is for the total number of fish
in the lake to be $12 \times $ the number of fish that will be taken.  This the 
{\it desired number} of fish she aims to maintain.
\itemitem{3.}  Each week the manager decides to do a count of the actual number of the
fish in the lake and finds the difference between the desired number from point 2 above
and the actual number; she calls the difference the {\it discrepancy}.
\itemitem{4.}  While it would certainly be possible to try to order enough new fish
to make up the demand plus the discrepancy each week, she decides it's probably 
better (partly because of the delivery delay) to try to ``spread out'' making up the
discrepancy over several weeks--say {\it three weeks}.  This might also change
in the future, so she thinks of this as an adjustable parameter.  In other words,
the order of fish from the hatchery will be 
$${\rm (fish\ caught\ in\ previous\ week)} + ({\rm discrepancy})/3$$
to start.  Note that not trying to make up the full discrepancy immediately means
that the manager is building in a {\it response delay}.  Also, note that 
in some extreme situations it could be true that the discrepancy was very negative
and the formula above could yield a negative number.  How should the manager handle
that??
\bigskip 
\noindent
We want to try to model this situation to understand what will happen over a period
of about 2 years -- i.e. 100 weeks.  
\bye 



B.  Download a personal copy of the shared 
Google spreadsheet {\tt Fish Stock Management} from the course homepage.  
(Please don't edit the shared one!)
This incorporates the difference equation model for the fish stocking
situation we discussed in class.  
\item{$\bullet$} Column C gives number of fish in the lake as a function
of time in weeks.
\item{$\bullet$} Column D gives the number of fish removed by fishermen, with the 
$10\%$ increase from 50 fish to 55 fish starting in Week 6.
\item{$\bullet$} Column G gives the ``perceived need for restocking'' -- 
the number of fish taken, {\it averaged over the 5 most recent weeks}.  
\item{$\bullet$} Column H gives the Desired Number of fish (always
$12 \cdot $Perceived Need)
\item{$\bullet$}  Column I (the ``Discrepancy'') 
is the difference between the Desired Number of fish and the actual
number of fish present in the lake from Column C.  
\item{$\bullet$}  Column E gives the orders made by the lake manager
to a commercial fish hatchery for fish to restock the lake.  The amount
of each order is computed by adding the perceived need and a 
{\it fraction of the Discrepancy}.  The spreadsheet is set up so the fraction 
is $1/$(the number in cell {\tt J2}). 
I called this the {\it make-up period}.  This make-up period is set 
to 3 weeks as we discussed in class, but you can change the value in 
that cell to experiment and answer questions (4) and (5) below. 
We can think of this as a {\it response delay} where the orders are planned to 
make up the discrepancy, but not all at once.
We are trying to make up the discrepancy over a period of 
some number of weeks.  
\item{$\bullet$}  Column F gives the deliveries from the hatchery.  Notice
that there is a {\it 5-week delivery delay} between the time the order is sent in and
the time the order is delivered.  
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\noindent
Using this information, and modifying the spreadsheet as needed,
Answer these questions:
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\item{(1)}  Describe in words what is happening to the number of fish
on hand in the lake over the 100 weeks shown.  Is this necessarily 
a problem?  Why or why not? 
\item{(2)}  Why is the Discrepancy value negative in some weeks?
What does that mean in real-world terms?
\item{(3)}  Suppose the perceived need for restocking is averaged
over the most recent {\it three weeks} instead of {\it five weeks}.
This will involve changing the entries in Column G starting in row
9,  but everything else remains unchanged.  Copy the new plot
into your document.  
Also describe how the number of fish on hand changes (relative
to the original plot)  the number of fish in the lake changes (relative
to the original plot shown in the spreadsheet when you downloaded your personal copy) 
\item{(4)}  Now suppose the perceived need for restocking is 
again averaged over the most recent 5 weeks, but delivery delay is reduced to 3 
weeks instead of 5 weeks.  
(Note: You'll see how and where the delivery delay is included if you look 
at the entries in Column F, starting at Row 9.)
Again, copy the new plot
into your document.  
Also describe how the number of fish on hand changes (relative
to the original plot).   
\item{(5)} Now suppose the response delay described above
is decreased from 3 weeks to 1 week, but nothing else is changed.  (This means
you will want to set the delivery delay back to 5 days and the perceived need is 
computed by averaging the most recent 5 weeks again.)
Roughly speaking this means that the stocking manager is going to try to
making up discrepancies between the desired number of fish and the actual number
of fish in just one week rather than spreading it over three weeks. 
Again, copy the new plot into your document.  
Also describe how the number of fish on hand changes (relative
to the original plot). Is this what
you thought should happen?
\item{(6)}   Describe a strategy for reducing the severity
of the oscillations in the fish stocks (making
the amplitude smaller, and/or the period longer).   Discuss.  Is this what
you would have guessed intuitively.  
\bigskip
\noindent
Note:  You may want to look at the discussion of the car dealer's inventory
example in the excerpt from Meadows' {\it Thinking in Systems} on the course
homepage.
\bye
 
