Chapter 2
Where should we go when we get to the intersection of 191 and 491?
Suppose we started at Pierce College and our destination is the Albuquerque International Balloon
Fiesta
And now it is obvious which choice to make at the intersection of 191 and 491.
What are the important issues we should consider when we think about where we (humanity) has been
and where we are going?
QAW
Quantitative Assessment of the World
Page 21
Fishbank Debrief
Interpret the Fish Stock (Top Left) graph. Explain what happened and why.
Interpret the 2 Net Recruitment & Catch graphs in the bottom row. Explain what happened and why.
Interpret the Ship Market Value graph. Explain what happened and why.
Interpret the five graphs below simultaneously.
Is this just a game or is it real? See debrief ppt starting at page 10. Can show
other results first.
Recruitment means reaching a certain size or reproductive stage. With fisheries,
recruitment usually refers to the age a fish can be caught and counted in nets.
MEY – Maximum Economic Yield
MSY – Maximum Sustainable Yield
S/Smax Survival Rate/Maximum Survival Rate
The reason for these results is that individuals don’t often perceive themselves as
part of a system and that the system can behave in ways that are not obvious to
the individual.
What are systems?
Systems have three components – Elements, Interconnections and Purpose.
Example: Education System:
Elements: Students, Teachers, Books, Schools, Computers, Subject matter, etc
Interconnections: reading, talking, listening, interacting between elements
Purpose: Transfer knowledge to students along with the skills to develop new
knowledge and ways of thinking on their own.
Example: Transportation System
Elements: Roads, Airports, Water, railroads, Cars, Airplanes, boats, trains,
construction equipment, mechanics, etc
Interconnections:
Purpose: Allow people and freight to move from one place to another
Systems can be hierarchical.
In groups, go to the board and show a hierarchy of systems that hold relevance to
us.
The Country is a system
Within this system are natural and governmental systems
Within the government system are educational,
transportation, military, social services systems.
Within each of these are other systems, e.g. individual
offices, etc
How can we understand them mathematically? We use System Dynamics
Modeling
Google Images of System Dynamic Models
The mathematics of the long term behavior of a system
To understand the long run behavior of a system requires the use of an iterative process.
To understand this process, consider a house was built on 10 acres of land far from a city, so
that the water to the house will come from a well drilled into the local aquifer. The size of the aquifer is
generally not known, but for this example we will say it contains 1000 cubic meters of water. The only
way in which water leaves the aquifer is through the well when water is used by the homeowner. The
only way water enters the aquifer is when rain percolates through the ground. Suppose, for this
example, the home owner uses 45 cubic meters each month1. Suppose also, that precipitation produces
an average of 25 cubic meters each month. It is probably already obvious that this aquifer will
eventually run dry. Algebra could be used to produce a nice linear equation to show when this would
happen.
Algebra: Let y = amount of water remaining. Let x = months then y = 1000 + (25 - 45)x or y=1000 – 20x.
By setting y equal to zero, we can determine that the aquifer will be completely empty in 50 months.
Because systems are more complicated than this example however, we will look at this as an
iterative process. This will be shown in a table.
Month
0
1
2
3
4
Usage
0
45
45
45
45
Replenish
0
25
25
25
25
Remaining
1,000
980
960
940
920
This demonstration has been with a linear process which is why a linear equation could be used
to determine when the aquifer would be dry. Now, let’s make it more complicated.
1
Based on http://www.epa.gov/WaterSense/pubs/indoor.html March 26, 2014
Suppose the owner was aware of the size of the aquifer and managed water usage so that each
month he would only use 4.5% of the water in the aquifer. Since the homeowner cannot control the
precipitation, then that will remain constant.
Month
0
1
2
3
4
Usage
0
0.045(1000)=45
0.045(980)=44.1
0.045(960.9)=43.24
0.045(942.66)=42.42
Replenish
0
25
25
25
25
Remaining
1,000
980
960.9
942.66
925.24
The graph below shows the long run behavior of this system functioning under these conditions.
Notice how the volume of water will reach the point of not changing at about 556 cubic meters.
Ultimately, the homeowner will only be able to take out 25 cubic meters of water each month. The
graph below was made in Excel from this iterative process being completed for 160 months.
This graph resembles an exponential decay function although it is more complicated because
the increase is linear while the decrease is exponential.
Family Water
Usage
Precipitation
Aquifer
Percolation
Withdrawal from
Well
The graph below shows the amount of water remaining on the y axis. It reaches 0 in 50 months, as
expected.
Aquifer
1,000
Cubic Meters
750
500
250
0
0
5
10
15
20
25
30
Time (Month)
35
40
45
50
Aquifer : Linear
The stock-and-flow model below reflects the change that was made in the story when the family
used 4.5% of the water that remained in the aquifer each month. This is reflected by the extra
connector between the stock (Aquifer) and the variable (Family Water Usage). The result is shown in
the graph below the model. Notice it is similar to the graph that was produced from the iterative
process in Excel.
Usage Rate
Family Water
Usage
Precipitation
Aquifer
Percolation
Withdrawal from
Well
Percolation Rate
Aquifer
1,000
Cubic Meters
875
750
625
500
0
20
40
60
80
100 120
Time (Month)
140
160
180
200
Aquifer : Exponential
Other changes to the model were the result of the need for consistence in units. The
volume of water in the aquifer is measured in cubic meters. Since the change every month was chosen,
then the inflow and outflows must have units of cubic meter per month. Precipitation will have units of
cubic meters but the percolation rate will be the amount of precipitation that percolates each month,
therefore it will have units of 1/month or month-1. By multiplying these two values, the inflow will have
units of cubic meters per month. The usage rate by the family is 0.045 per month (units of 1/month or
month-1). By multiplying this usage rate times the volume in the aquifer, the Family Water Usage will
also be in cubic meters per month.
Explain Clouds in the model
Vantana Systems, Inc produces a free student version of their systems modeling software called Vensim.
This software can be downloaded from their website at http://vensim.com/vensim-software/. The free
version is called Vensim PLE (personal learning edition).
First Model
The first model will be an investment with a constant interest rate.
Select the New Model icon (or File, New Model)
Change the FINAL TIME to 40.
Change the Units for Time to years. Click OK.
The icons that will be used for this model.
1.
2.
3.
4.
5.
Use the Box Variable to create the stock. Click once on the Box Variable icon, click once on the
work space, then type the name Investment in the space that is provided.
Use the Rate icon to create the inflow by positioning the cursor about 4 or 5 inches to the left of
the stock, clicking once and then moving the cursor into the stock and clicking a second time.
Type the name Interest Earned into the space provided.
Use the Variable icon by clicking an inch or two above the inflow. Type Interest Rate into the
space provided.
Use the Arrow icon to make the connectors. The first connector is created by clicking once on
the Interest Rate, once between the Interest Rate and Interest Earned and finally once on the
valve for Interest Earned.
The second connector starts with a click in the stock, a click between the stock and the inflow
and a final click on the inflow valve.
Your stock and flow model should look like this.
Interest Rate
Investment
Interest Earned
Now that the model has been made, the initial values, formulas and units must be assigned. Select the
f(x) icon and click on Investment. Put a $ in the space for units and 1000 in the space for Initial Value.
That indicates that a $1000 will be put into the investment at the beginning.
Next, select the inflow and make the following changes in Units and Equation. Don’t type in the words
for the equation, just click on them in the space for Variables.
Select the Interest Rate. You will see a similar form as the two shown above. Change units to 1/year.
The initial value should be 0.01 which means a 1% annual interest rate.
To check if the units are correct, go to the drop down menu at the top called Models and select Unit
Check. You will be asked if you want to save the sketch to enable automated backup. Select Yes then
give the file a name (such as Chapter 2 Investment Model) and click save. You should now get a
message that says Units are A. O. K.. If not, you will have to find the error in the units you entered.
Type 1 percent for the Simulation results file name near the top of the screen then click the Simulate
button. You won’t see anything obvious happen.
Click the IO Object button and you will be shown a new dialogue box. Select Output Workbench Tool
then click on the Level button icon and select Investment. Finally select Graph from the list of options in
the last space.
Use the Hand icon to move the generic graph to an appropriate location then click on the lock icon and
you should see a graph such as this show up in the space.
At this point, I have not found a way to edit the y axis scale in this version of Vensim.
To compare the effect of different interest rates on the amount of money in the investments, we will
run a simulation.
In the space for Simulation Results file name, change the original name to Interest Rates Vary then click
on the SyntheSim icon. Doing so will put a slider in the interest Rate variable. Click on the arrow so that
the range of possible values can be modified. Set the Min to 0, the Max to 0.2 and the Increment to
0.005. Now click on the bar with the interest rate and slide it back and forth and notice the change in
the blue line on the graph. The y-axis scale will adjust. If you put the interest rate at 0.06 (6%), your
final model and graph should look like this. Notice that with 1% interest, the account will be worth only
a little more than $1,450 in 40 years. If your interest rate is 6%, the account will be worth about
$10,000 in 40 years.
It might be helpful to repeat the entire process for model 1 again, just to become more comfortable
with the icons and steps.