Easter Island Population Model

Easter Island Population Model Consider a problem that happened on Easter Island. Easter Island is a small island (about 150 square miles) in the Pacific Ocean about 2,000 miles from South America. It is quite isolated, making a study of the historical change in population a little easier. In about 400 AD there was a small population of settlers (say 24) 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 streams.1 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.2 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 fire source for cooking and heating.3 Since the society did not need to spend a lot of time in 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 hypothesized, 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, reaching around 7,000 by 1550.4 But by the 1600s there were almost no trees left on the island.5 By the 19th century there were only 110 people left on the island. So what happened? Why after so many years did the population collapse? What happened to all the trees? In your journal, draw a BOTG showing your hypothesis of what happened to both the population and the trees over time. (see the figure at the right.) If you were to build a model of this situation, list the variables you think might need to be included.

1. Pop 2. Trees high Draw your graphs in your journal. low 400

1950 year

1Clive

Ponting, A Green History of the World (Penguin Books, 1991), 3. 3. 3Ponting, 5. 4Ponting, 5. 5Ponting, 5. 2Ponting,

4.1 Easter Island Population Model Student Lessons Page 4-7 Modeling Dynamic Systems: Lessons for a First Course 2nd Edition © 2007 Diana M. Fisher

We will create a model to help us gain some insight, hopefully, into the turn of events. A time frame from 400 AD to 2000 AD might be sufficient to encompass the important dynamics that will help us understand what could have happened.

The Diagram A. The Population Structure 1. Set up a standard population structure (similar to the one set up in the Population Tutorial). Population

deaths

births

birth rate

death rate

2. The population will begin with 24 persons. We do not have actual data to support a birth rate, but we could calculate a reasonable birth rate from some assumptions. First, assume females comprised 50% of the population. Also, 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. Finally, assume that there were 192 births per 1000 reproducing females per year.6 Calculate a value for the birth rate per year. Show the calculation and final value (round answer to nearest thousandth). Be sure to include units. birth rate =

3.

Define births in the model.

Births =

4.

Define deaths in the model. Deaths =

6This value was calculated using the following logic. It seemed reasonable that each reproducing female (on average) would have produced 5 children that would have lived to reproducing age. Assuming a reproducing age span of 26 years, that means that there would be 5/26, or .192 child born per year per reproducing female, which translates into 192 births per 1000 reproducing females per year.


5. Define these components in your model. We will not set up 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.

B. The Tree Structure 6. Now let's determine how to add the tree resource to the model. Recall that there were a specific number of trees to begin with. An initial figure of 40000 trees would be reasonable. The island size is 150 square miles. For now we will consider trees a nonrenewable resource, since the natives did not replant the trees they destroyed. We want to keep track of the number of trees over time, so trees should be modeled by a (circle one of the choices) stock

or

flow

7. Since trees are only decreasing we need a flow that is (circle one of the choices) into

or

out of

the tree icon. Call this flow consumption. Draw these two new components under the population section of the diagram.

C. Defining the Death Rate 8. As trees became less abundant the quality of life decreased. The natives began living in caves and had fewer canoes to use to search for fish.7 Reduction in food and resources led to increased conflicts between the clans. So, in addition to the natural death rate, the eventual reduction of trees would have caused the "real" death rate to increase. 9. The "real" death rate is the product of two factors: the normal death rate (assume 67 per 1000 people per year) and the effect of tree supply on death rate.8 The death rate diagram segment should look like the one shown at the right. effect of tree 7Ponting,

supply on death rate

normal death rate

6. converter effect of tree supply on death rate is a special type of converter (called a multiplier) that is used to modify the value of another component over time. It is good style to have the "normal" converter contain a value that is applied when circumstances in the model are typical or not stressful, then have another converter alter the normal value when the system becomes stressed, as is the case here when trees become scarce. We will study more about multipliers later in this course.

8The


10. The effect of tree supply on death rate 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 effect of year. Create two new converters: first, actual tree supply on death rate number of trees available per person and second, desired number of trees per person (which is actual number of 0.5). Connect these two new converters to the trees available effect of tree supply on death rate. Double-click per person desired number of on the effect of tree supply on death rate. Define trees per person the effect ... equation as actual number of trees.../desired number of trees.... (Notice that the units cancel.) Click Become Graphical Function. We want to define this effect... converter as a graphical function so the output value will change based on the ratio of the number of trees available over time. In your journal answer these questions (in complete Death fraction sentences). How do you think the number of trees high affects the death rate? For example, what do you Draw your think happens to the death rate when there are plenty of trees normal graph in your journal. (relative to what people need to survive)? What happens to the low death rate if there are very few trees (relative to what people need few plenty Trees available to survive)? What if there are just enough trees? After writing per person your description, sketch a graph with Trees available per person on the horizontal axis and set the horizontal scale from few (on the left), to plenty (on the right). Write Death fraction on the vertical axis and set the vertical scale from low, to normal, to high. (See the figure above.) Then sketch a graph that matches the description you wrote earlier. 11. Set the number of data points to 11. Set the lower value of the horizontal axis to 0 and the upper value to 2. Set the vertical scale lower limit to .9 and the upper limit 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 is equal to or greater than the desired number of trees. So set the effect… 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 maybe the effect of tree supply will increase the death rate by 1.5% above normal (so set the effect… output value to 1.015). A (input) ratio of 0.6 causes the death rate to increase 5% above normal. At a ratio of 0.4 the death rate increases 9% above normal. At a ratio of 0.2 the death rate increases 14% above normal. And at 0% the rate is 20% above normal.


On the grid below sketch the graph that represents the effect of tree supply on death rate from the given information.

D. Connecting Population and Trees 12. As the population increases the consumption of trees will increase

or

decrease

Draw a connection from population to consumption. (Make this connection a wide arc that swings around the outside of the two structures. It is better style not to have connection wires crossing each other.) 13. It makes sense that the consumption should be in trees per year, since it is a flow and all flows must be in “stock units” per “time.” Because the population is generating the consumption of trees, we will need to have population as part of the consumption flow definition. The other part of the definition involves how many trees each person is consuming each year. This is the component we still need to add. Actually we are going to define consumption per person per year using three converters. We will set up a structure similar to the death rate structure. Add three converters just above the consumption flow, one called normal tree consumption per person per year, another effect of tree supply on tree consumption per year, and the third actual tree consumption per person per year.


Connect each converter normal tree consumption per person per year and effect of tree supply on tree consumption per year to actual tree consumption per person per year. Define actual tree consumption per person per year as the product of the two converters upon which it depends. This structure (when it is completed) will allow the actual yearly consumption of trees per person to change over time based upon how many trees are left on the island. (Are you remembering to put in units as you define converters?) Connect actual tree consumption per person per year to consumption. Define consumption in your model. (Recall: Consumption is how many trees all the people will consume each year.) Consumption =

14. The normal tree consumption will be set to 0.025 tree per person per year. The effect of tree supply on tree consumption per year will be defined as a graphical function and will depend upon the ratio of the actual number of trees available per person and the desired number of trees per person. Connect both of these converters to the effect of tree supply on consumption per year. Double-click on the effect... converter. Set up the ratio and then click Become Graphical Function. Set the number of data points to 11. The minimum ratio value will be set to 0 and the maximum value will be set to 2. View the following table to determine convenient values for the vertical scale. Then define the effect of tree supply... output values as shown in the table.9 actual number of trees available.../desired number of trees consumed... 0 .2 .4 .6 .8 1 1.2 1.4 1.6 1.8 2

effect of tree supply on consumption per year 0 0.2 0.3 0.4 0.5 1 2 3 5 10 20

9Normally

a multiplier component should have an output value of 1 when the system is not stressed (i.e., when the ratio is one). Notice that that is the situation described here. 4.1 Easter Island Population Model Student Lessons Page 4-12 Modeling Dynamic Systems: Lessons for a First Course 2nd Edition © 2007 Diana M. Fisher

15. There is one converter left to create. We need to determine the number of trees that are available per person. Create a converter called available trees per person. This converter depends upon the number of people and the number of trees on the island. How would you define this converter? available trees per person =

16. We have yet to 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 do this 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. We will 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) {trees/person/year}.

The Simulation Define the time specification for the model to start at year 400 and end at year 2000. Set the DT to 0.25. 17(a). We are interested in looking at the graphs of population and trees. Define a graph pad to include each of these components. Set up appropriate vertical scale values for each. To do this consider: Trees will have a maximum value of

Population could have a maximum value of

(If you don't know the maximum value for population, run the simulation once and then check the MAX box for the definition of population in the graph pad.) On no more than 2 sheets of paper, print out the graph containing Population and Trees, print the diagram, and print the equations. 17(b). Set up a second graph pad to include population, birth rate, death rate, and actual number of trees available per person. Set the minimum y-scale for birth rate, 4.1 Easter Island Population Model Student Lessons Page 4-13 Modeling Dynamic Systems: Lessons for a First Course 2nd Edition © 2007 Diana M. Fisher

death rate and actual number of trees available per person to 0 and the maximum to 0.5. Print out the new graphs. (Note: If the scale for birth rate shows up as 0 or 1 for upper limit double click on the birth rate name on the graph pad and select format 0.0. You will have to do the same for death rate and actual number of trees available per person before the change takes effect.) Looking at the second graph printout, why do you think the population started to decline?

18. Define a table to include trees and population components. How many years does it take the population to double?

Is the doubling time the same for different time periods? Why or why not?

What was the largest value for the population? When did the population start to decline? How many people were left in 1900? How many trees were left in 1900? 4.1 Easter Island Population Model Student Lessons Page 4-14 Modeling Dynamic Systems: Lessons for a First Course 2nd Edition © 2007 Diana M. Fisher

19. In about 1877 a Peruvian slave ship landed on the island and took all but 110 natives on board to be used as slaves.10 Many of these natives died. Add the slave ship scenario to the model. Think about how to represent the slave ship factor in the diagram. Should it be included as part of a flow already in the model or should it be a new flow?11 Print the diagram, graph (Population and Trees), and only the population stock/flow equations (with slave ship) for this new model, on one sheet of paper. (Hint: At this point you need to learn about some special functions called STEP and PULSE.12 Ask your teacher about these functions or look them up on the STELLA overview sheet.) 20. (Remove the slave ship component.) What would happen if trees were being replanted at a rate of 50 per year starting in year 1600? (Actually this is somewhat unrealistic since new trees could not replace other trees immediately. It would take years of maturing before they would be harvestable. This is a problem that we could actually figure into our model, but we won’t right now). Add this new factor to the tree section of the model. Print out the new graph (Population and Trees). Change the year the replanting starts to 1700. Rerun the simulation. Draw a feedback loop that relates the population with the number of trees. Feedback loop diagram:

10Ponting,

1. A good modeler will design the diagram so it tells a significant part of the story just by its appearance. There is an art to creating a visually meaningful, easy-to-read model diagram. 12The two functions have the following format: STEP(amount, time to change), PULSE(amount, first time, interval between times). 11


Explain what is happening to the population with the replanting policy. Explain why the results make sense using the feedback loop you drew (above) as part of your explanation.

21. (Extra Credit) 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 not only to grow to a useful size but also, because these were coconut palm trees, allow time for them to bear fruit. Add to the replanting model segment a factor that indicates the time trees take to mature (say 20 years) and that only mature trees are consumed. To do this we need to disaggregate the tree structure (i.e., have more than one tree stock) into, say, saplings and mature trees. Have the replanting start in the year 1600. Print out the model diagram, graph (Population and Trees), and the equation involving the tree stock and flow segments (that show your change). Explain how this delay affects the results you noticed in problem 20.
