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The problem of modeling the recruitment, aging, mortality, and migration of
a shrimp population described by Richard Dudley is an example of the
general formulation issue of an aging chain. This formulation is treated
in detail in chapter 12 of Business Dynamics, including multiple cohorts
with the possibility of recruitment, mortality, and migration into/out of
each cohort. Several different ways to model the demographic structure of
a population are treated, and examples from world population growth to the
aging of the faculty at a university are discussed. For information see
http://www.mhhe.com/sterman.

John Sterman
jsterman@MIT.EDU

Part of the answer to this question depends on the purpose of the model (as
always). Creating a structure that accurately represents the distribution
of maturing shrimp over their three months in the lagoon may be a waste of
time if the pulse passes into a large stock of oceangoing shrimp with a
long residence time. On the other hand, if youre trying to set seasonal
catch limits or time activities in the lagoon so as to minimize disturbance
to the shrimp, the distribution could be very important.

>Alternative I tried using a delay whereby the stock "shrimp in lagoon"
>has a connector to a delay something like this: delay3(shrimp in lagoon,
>time in lagoon). This delay then determines the outflow from my lagoon,
>over an additional specified time period.

This approach may blow up when you check units, since shrimp in lagoon is a
stock and the delay is being used to create a flow. Fixing it by adding a
second time constant in the denominator doesnt really have a physical
justification. Also, its likely to fail extreme conditions tests - e.g. if
some other process (imagine a chemical spill) kills all the shrimp in the
lagoon, the outflow of maturation would still continue for some time due to
the delay3. As a general rule, outflows from physical stocks always need
1st order control, i.e. outflows must be connected to the stock, without
any integration or delay, so that when the stock is 0 the outflow is 0.

If the distribution matters, I prefer the explicit stocks approach you
mention. This makes the processes affecting the shrimp most clear to users.
The typical SD rule of thumb is to close your eyes and assume that 3 levels
are enough.

The trouble with explicit stocks is that if the shrimp are very regular -
as many animal gestation and maturation processes are - a large number of
stocks are required to get the right distribution. In the extreme, an
ithink conveyor or Vensim fixed or material delay will work as long as the
total lack of dispersion in the pulse is OK. The ithink conveyor will also
let you specify a "leakage" outflow, so as long as shrimp losses can be
specified as a fraction of the shrimp stock or inflow.

To get a realistic, high-order distribution, one shortcut would be to use
an Nth order delay function (like DELAY N in Vensim). You could also try
Vensims DELAY PROFILE, which lets you draw an arbitrary arrival
distribution. You can also use an Nth order smooth in place of an Nth order
delay as long as the delay time is constant (otherwise your physical
quantity wont be conserved). You would normally implement this as:
Shrimp in Lagoon = INTEG( inflow - outflow, initial shrimp)
inflow = ...
outflow = DELAYFUNCTIONOFSOMESORT(inflow, maturation time)
Because the various delays you might use here have a bunch of stocks
inside, you dont really need the stock of Shrimp in Lagoon, but its nice
to have for accounting purposes.

The trouble with this approach is that theres no correct way to represent
losses that occur while the shrimp are resident in the lagoon. There are
two ways around this. An easy way is to assume that all losses occur at
either the beginning or the end of the process, so that you can just
multiply the inflow or outflow by a constant survival fraction. This means
that the timing of losses reported by your model wont be right, but you
may not care as long as the distribution of maturing survivors is OK. A
better way would be to use arrays to build an explicit stock-flow structure
of the order you want, without having to write all the equations individually.

Apart from building an explicit 3rd order structure, none of the solutions
is very elegant. Using DELAY N-type functions creates a diagram with a
rate-to-rate connection and a redundant stock with no apparent connection
to its outflow both no-nos or at least confusing to users. Conveyors with
leakage look a little cleaner but are similarly confusing, as theres no
visual representation of the causality between stock and outflow. Arrays
make the feedback apparent, but hide the vintage structure. Theres
probably room for some creativity here.

Regards,

Tom

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Thomas Fiddaman, Ph.D.
Ventana Systems http://www.vensim.com
8105 SE Nelson Road Tel (253) 851-0124
Olalla, WA 98359 Fax (253) 851-0125
Tom@Vensim.com http://home.earthlink.net/~tomfid
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