Posted: July 26th, 2022

system dynamic modeling

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Easter Island Population Model – Exercise
Background Information
Easter Island is a small island (about 150 square miles in area) in the Pacific Ocean about
2,000 miles from South America. In 400 AD there was a small population of settlers on
the island. The island was heavily forested, but the land was not very useful for farming
due to extremes in temperature and lack of fresh water. The only staples the settlers had
were chickens and sweet potatoes that they brought with them from their previous
location. There were not many fish around the island. There were, however, plenty of trees to support
the various needs of the new population.
These trees served as a natural resource from which huts and canoes could be built and
a strong rope could be fashioned. Trees were also used as a fuel source for cooking
and heating. Since the society did not need to spend a lot of time on food production,
much time went into social activities. The population began to divide itself into clans and
created rituals that began to dominate their social structure.
Part of the rituals involved the creation of stone statues. A competition between clans
resulted in larger, more elaborate statues being sculpted. Soon trees were cut down
to facilitate moving the statues from one location to another. The rope that was crafted,
it is hypothesised, may have played a part in transporting some large stone statues
from a volcanic rock quarry (near the volcano Rano Raruku) to other parts of the island, where they
remain today.
The population flourished for a number of years, however by the 1600s there were almost no trees
left on the island. By the 19th century there were almost no people left on the island. So what
happened? Why after so many years did the population collapse? What happened to all the trees?
Your task is to create a system dynamic model to help you gain insight into what happened
on Easter Island between 400 AD to 2000 AD.
Utilise your knowledge from previous tutorials to help you with constructing your system
dynamic model.
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Part 1 – System Dynamics Model
Population Structure
Step 1 – Build the Base Model for the Population
– Set up a standard population model structure (please refer to earlier tutorial exercises).
The population will begin with 24 people.
This is how this model section should look:
Step 2 – Set the Equations
We do not have actual data to support a birth rate, but we could calculate a reasonable birth rate
based on the following assumptions.
– First, assume females comprised 50% of the population.
– Second, since people died at a much earlier age at that time, we will assume that 75% of the
females would have been of reproducing age.
– Third, assume that there were 200 births per 1,000 reproducing females per year.
How should we use these in an equation to calculate a value for the birth rate per year?
Q1. Write your answer in the box below, including the calculation and final value (and round your
answer to nearest thousandth). Make sure to include the units. (1 mark)
Q2. Write the equation for the ‘Births’ flow in the model in the box below. Make sure to include the
units. (1 mark)
Q3. Write the equation for the ‘Deaths’ flow in the model in the box below. Make sure to include
the units. (1 mark)
Add these equations to your model. We will not set the ‘Death Rate’ yet. A graphical function
will be used as part of the definition of death rate, and this graphical function will depend on some
factors not yet discussed.
Birth Rate (people/people/year) =
Births (people/year) =
Deaths (people/year) =
Birth Rate Death Rate
Births Deaths
Population
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Tree Structure
Now let’s add the tree structure component to the model.
Step 1 – Build the Base Model for the Trees
– Use an initial figure of 50,000 trees.
For now we will consider trees to be a non-renewable resource, since the natives did not replant the
trees they used. We want to keep track of the number of trees over time, so should trees be modelled
using a stock or a flow?
Q4. Write your answer in the box below. Explain your answer. (1 mark)
Given that trees are only decreasing, how would you model this, using an inflow or an outflow?
Q5. Write your answer in the box below. Explain the reasoning behind this model structure. (1 mark)
Call this flow ‘Consumption’.
Add these two new components underneath the population section of your model.
Define the Death Rate Converter
As trees became less abundant the quality of life decreased. The natives began living in caves and
had fewer canoes to use for fishing. Reduction in food and resources led to increased conflicts
between the clans. In addition to the natural death rate, the eventual reduction of trees would have
caused the “real” death rate to increase.
Step 1 – Define the Death Rate
The ‘Death Rate’ is the product of two factors: the ‘Normal Death Rate’ (assume 70 deaths per 1,000
people per year) and the ‘Effect of Tree Supply on Death Rate’.
How should we use these in an equation to calculate the death rate per year?
Q6. Write your answer in the box below. Make sure to include the units. (1 mark)
This is how this model section should look:
Death Rate (people/people/year) =
Effect of Tree Supply
on Death Rate
Normal
Death Rate
Death Rate
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Step 2 – Define the Dimensionless Multiplier
The ‘Effect of Tree Supply on Death Rate’ is a dimensionless multiplier. It depends upon
a ratio comparing the number of trees that are actually available per person per year to the desired
number of trees per person per year.
– Add two new converters to represent this. These two new converters are:
o ‘Actual Number of Trees Available per Person’.
o ‘Desired Number of Trees per Person’. Set this value to 0.5.
– Connect these two converters to the ‘Effect of Tree Supply on Death Rate’.
Q7. Write the equation for the ‘Effect of Tree Supply on Death Rate’ in the box below. Explain the
units. (1 mark)
This is how this model section should look:
Step 3 – Set up the Graphical Function
The ‘Effect of Tree Supply on Death Rate’ converter is defined as a graphical function hence the
output value will change based on the ratio of the number of trees available per person and the
desired number of trees per person.
– Set the data points to 11. (Hint: Graphical Function Editor – Points).
– Set the lower value of the horizontal axis to 0 and the upper value to 2.
– Set the lower value of the vertical axis to 0.9 and the upper value to 1.4.
When there are lots of trees available, we expect the normal death rate to be in effect. This will be
the case as long as the actual number of trees per person is equal to or greater than the desired
number of trees per person.
– Set the ‘Effect of Tree Supply on Death Rate’ output value to 1 for all points where the ratio
(input) value is 1 or larger.
When the actual number of trees decreases, it begins to affect the death rate, slowly at first.
– When the (input) ratio is 0.8, the ‘Effect of Tree Supply on Death Rate’ will increase the death
rate by 1.4% above normal – to reflect this, set the output value to 1.014.
– At a ratio of 0.6, the death rate increases 4.5% above normal.
– At a ratio of 0.4, the death rate increases 8.5% above normal.
– At a ratio of 0.2, the death rate increases 15% above normal.
– At 0, the death rate is 25% above normal.
Effect of Tree Supply on Death Rate (dimensionless) =
Actual Number of Trees
Available per Person
Desired Number of Trees
per Person
Effect of Tree Supply
on Death Rate
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Hint: set up these data points using the table that can be found in the Points tab in the
Graphical Function Editor.
Q8. Paste a copy of your graphical function into the box below. (2 marks)
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Connect Population and Trees
As the population increases, what do you think will happen to the consumption of trees?
Would it increase, decrease or remain the same? Why?
Q9. Write your answer in the box below. Explain your answer. (2 marks)
Now we will connect the population segment to the trees segment. (It is good practice not to have
connection arrows crossing other model components for a clear representation).
Step 1 – Add the Connecting Components
Given that the population is determining the consumption of trees, ‘Population’ is required to be part
of the ‘Consumption’ flow definition. In addition, the amount of trees each person consumes each
year should also be accounted for in this definition. (Note: we will set up a structure similar to the
death rate structure).
– Add three new converters to represent this. These new converters are:
o ‘Normal Tree Consumption per Person per Year’. Set this value to 0.025 trees per
person per year.
o ‘Actual Tree Consumption per Person per Year’.
o ‘Effect of Tree Supply on Tree Consumption per Year’.
– Connect ‘Normal Tree Consumption per Person per Year’ and ‘Effect of Tree Supply on
Tree Consumption per Year’ to the ‘Actual Tree Consumption per Person per Year’.
– Connect ‘Actual Tree Consumption per Person per Year’ to ‘Consumption’.
We need to determine the number of trees that are available per person.
– Add a converter called ‘Available Trees per Person’. Determine the inputs to this
converter, given that it is dependent upon the number of people and the number of trees.
Step 2 – Set the Equations
Q10. Write the equation for the ‘Actual Tree Consumption per Person per Year’ in the box below.
Make sure to include the units. (1 mark)
Q11. Write the equation for the ‘Consumption’ in the box below. Make sure to include the units.
(1 mark)
Actual Tree Consumption per Person per Year (trees/person/year) =
Consumption (trees/year) =
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Q12. Write the equation for the ‘Available Trees per Person’ in the box below. Make
sure to include the units. (1 mark)
Step 3 – Define and Set up the Graphical Function
The ‘Effect of Tree Supply on Consumption per Year’ converter is defined as a graphical function.
The equation for the ‘Effect of Tree Supply on Consumption per Year’ will depend upon the ratio of
the ‘Actual Number of Trees Available per Person’ and the ‘Desired Number of Trees per Person’.
– Connect the ‘Actual Number of Trees Available per Person’ and the ‘Desired Number of
Trees per Person’ to the ‘Effect of Tree Supply on Consumption per Year’.
Set up the ‘Effect of Tree Supply on Consumption per Year’:
– Set the data points to 11.
– Set the lower value of the horizontal axis to 0 and the upper value to 2.
Use the following table to define a graphical function for the ‘Effect of Tree Supply on Consumption
per Year’.
Actual Number of Trees Available per Person /
Desired Number of Trees per Person
Effect of Tree Supply on
Consumption per Year
0 0
0.2 0.063
0.4 0.125
0.6 0.25
0.8 0.5
1 1
1.2 2
1.4 4
1.6 8
1.8 16
2 32
Step 4 – Define the Actual Number of Trees Available per Person
If there are lots of trees available per person, we want to allow people to use the desired number of
trees per person. But if there are fewer trees available than the desired number, we must limit each
person to his/her apportioned share.
To illustrate, we will have the ‘Actual Number of Trees Available per Person’ depend upon the
‘Available Trees per Person’ and the ‘Desired Number of Trees per Person’.
Use a special command called MIN (for selecting the minimum value in a list of values). Define the
‘Actual Number of Trees Available per Person’ as MIN (available trees per person, desired
number of trees per person).
Available Trees per Person (trees/person) =
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Simulate the Model
– Set the simulation Start Time to the year 400 and Stop Time to the year 2000.
Pay attention to the time units.
– Set the DT to 0.25.
Step 1 – Produce a Graph
We are interested in the behaviour of population and trees over time.
1. Create a graph and add: ‘Population’ and ‘Trees’.
2. Create a second graph and add: ‘Population’, ‘Birth Rate’, ‘Death Rate’, and ‘Actual Number
of Trees Available per Person’.
o Set the scale for ‘Birth Rate’, ‘Death Rate’, ‘Actual Number of Trees Available per
Person’ from 0 to 0.5.
o Select Format [right click on variable name] and set the precision to 0.00 or Free Float
for these variables.
Answer the following questions.
Q13. Looking at the first graph, explain the observed trends in the population and trees over time.
Include a copy of your first graph in your answer. (2 marks)
Q14. Looking at the second graph, why do you think the population started to decline? Include a
copy of your second graph in your answer. (2 marks)
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Step 2 – Produce a Table
We are interested in the values of population and trees over time.
– Create a table and add: ‘Population’ and ‘Trees’.
Answer the following questions.
Q15. How many years does it take the population to double? (1 mark)
Q16. Is the doubling time the same for different time periods? Why, or why not? (2 marks)
Q17. What was the largest value for the population? (1 mark)
Q18. What year did the population start to decline? (1 mark)
Q19. How many people are left in 1900? (1 mark)
Q20. How many trees are left in 1900? (1 mark)
Tree Replanting Structure
At the moment there is no replanting of trees in the model. We will account for this now.
What would happen if trees were being replanted at a rate of 100 per year, starting in the year 1700?
Remember, there is a maturation delay as saplings take many years to mature and become
harvestable. At this stage, we will not take this into account, however, we will factor this in later on
(Step 3).
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Step 1 – Add Tree Replanting
– Add tree replanting to the tree section of your model.
Use the STEP function to set the rate of tree replanting. Define tree replanting as STEP (amount,
time of change).
– Rerun the simulation.
Step 2 – Produce a Graph
– Update your graph of ‘Population’ and ‘Trees’ with your new simulation results.
Answer the following questions.
Q21. Looking at your graph, explain the observed trends in population and trees over time. Include
a copy of your graph in your answer. (2 marks)
Q22. How many people are left in 1900? (1 mark)
Q23. How many trees are left in 1900? (1 mark)
Step 3 – Add Tree Maturation Delay
It is not reasonable to expect a tree that is planted one year to be of harvestable size by the next
year. It would be necessary to wait a reasonable amount of time to allow the tree to grow to a useful
size. Therefore, you have to add a tree maturation delay to your model. The delay indicates the time
trees take to mature (i.e. 30 years) and that only mature trees are consumed.
– Disaggregate the tree model structure.
To model this delay, disaggregate the ‘Trees’ stock into two stocks: ‘Saplings’ (which are young
trees) and ‘Mature Trees’ (which are mature trees ready to harvest).
Note: to model the tree maturation delay, use a normal stock (reservoir) and a converter, not a
conveyor.
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Step 4 – Add Sapling Death and Cyclone Components
– Add Natural Sapling Death Component
In reality, not all of the saplings planted will survive. Modify your model to account for sapling deaths,
which we assume to be 50% of the replanted saplings. Note that sapling deaths are 50% of what
is replanted, not 50% of the total number of saplings.
– Add Cyclone Component
Because Easter Island is in the tropics, it is prone to being hit by cyclones that can kill trees. Modify
your model again to account for mature tree and sapling loss due to the cyclone. In the year 1800 a
severe cyclone hits Easter Island and destroys 50% of mature trees and all saplings.
Use the PULSE function to model the loss of mature trees and saplings due to the cyclone. Define
this function as PULSE (amount, time of first pulse, time interval to repeat pulse).
– Rerun the simulation.
Step 5 – Produce a Graph and Table
– Update your graph of ‘Population’ and ‘Trees’ with your new simulation results.
Answer the following questions.
Q24. Looking at your graph, explain the observed trends in population and trees over time. Include
a copy of your graph in your answer. (2 marks)
Q25. How many people are now left in 1900? (1 mark)
Q26. How many trees are now left in 1900? (1 mark)
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Step 6 – Completed Model
Q27. Insert a picture of your completed model that shows: tree replanting, sapling
deaths, tree maturation and cyclone loss.
Insert two graph outputs:
– First graph: mature tree numbers and human population between 400-2000 AD without tree
replanting, sapling death, tree maturation and cyclone loss.
– Second graph: mature tree numbers and human population between 400-2000 AD with tree
replanting, sapling death, tree maturation and cyclone loss. (5 marks)
Note: to copy a model from Stella, drag a box around the model using your mouse and press
Ctrl + C on your keypad, then go to your document and press Ctrl + V. To copy a graph from Stella,
right click on the graph, select copy, then go to your document and press Ctrl + V on your keypad.
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Part 2 – Causal Loop Diagrams
Q28. Draw a feedback loop that relates the population with the number of trees. Make sure to include
all polarities and a loop label to your feedback loop. Explain your feedback loop. (2 marks)
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Q29. In the Easter Island Population Model, what common mode of system behaviour
does human population represent?
a) Draw the generic CLD that explains this behaviour and then
b) Draw a second CLD that replaces the variables in the generic CLD with those relevant to
Easter Island.
Make sure to include all polarities and loop labels in your CLDs. (5 marks)
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Q30. What system archetype best represents the Easter Island situation?
a) Draw the generic CLD that explains this system archetype and then
b) Draw a second CLD that replaces the variables in the generic CLD with those relevant to
Easter Island.
Make sure to include all polarities and loop labels in your CLDs. (5 marks)