Unit 2: Modeling Population

Upon completing this module, students are expected to be able to:

• Create models of population growth using STELLA.
• Differentiate between balancing (negative) and reinforcing (positive) feedbacks on population growth.
• Use a predator-prey model to explore trophic relationships and population dynamics.
• Explain how carrying capacity leads to a stabilization of population.
• Experiment with an Easter Island population and resources model to explore the conditions that allow sustainable use of resources versus collapse of civilization.

This unit is intended to be used in a three- to four-hour class period that meets once a week. It can be used as part of this modeling course or it can be adapted as a lab exercise for courses in environmental science. For this module, students should come to class prepared to take a short quiz on the assigned reading. Thereafter they will be led through a series of prompts designed to help them create and experiment with a number of simple models using the iconographic box modeling software STELLA (see https://www.iseesystems.com/store/products/ for different options for purchasing student or computer lab licenses of STELLA or for downloading a trial version).

For those learning to use STELLA, we suggest the online "play-along" tutorials from isee systems. You can find them here: isee Systems Tutorials.

In preparation for the exercise, students should read the following: Unit 2 Student Reading.

Students should take the following quiz prior to coming to class to ensure they have done the assigned reading: Population modeling reading quiz (Microsoft Word 2007 (.docx) 47kB Aug11 16). An answer key for the reading quiz can be found here:

Population modeling reading quiz answer key -- private instructor-only file

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In class, students should be provided with the exercise found here: Population modeling student exercise (Microsoft Word 2007 (.docx) 2.2MB Nov15 16)

An answer key for the exercise can be found here:

Population modeling answer key -- private instructor-only file

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Instructors can download a version of the lynx and hares predator-prey model by clicking here: Lynx & hare predator-prey model (Stella Model (v10 .stmx) 15kB Aug11 16), the logistic growth model by clicking here: Logistic growth STELLA model (Stella Model (v10 .stmx) 7kB Aug11 16), and the Easter Island model by clicking here: Easter Island collapse model (Stella Model (v10 .stmx) 13kB Aug11 16). All models were created using STELLA Professional and should open on any subsequent version of STELLA. If you are using an earlier version of STELLA, the complete model graphics and equations can be found in the answer key so that you can reconstruct the models yourself.

We generally post the readings and assignments for students to an LMS site (e.g. Moodle, Blackboard, Canvas). Students can open the assignment in Microsoft Word on the same computer they are using to construct each STELLA model and then answer the questions by typing directly into the document. Students can then either print a paper copy to hand in to the instructor or email their modified file to the instructor. It is straightforward to copy graphs and model graphics from STELLA and to paste them into Word. Simply select the items to be copied, hit copy in STELLA, and then paste into Word. There is no need to export graphics to jpg.

We teach the course in a three- to four-hour block once a week because we have found that models require a lot of uninterrupted time to construct. If students have a 50- or 75-minute class period several times a week, they spend at least 20 minutes of subsequent class periods trying to figure out where they were in the exercise at the beginning of the week. This is not a good use of time, hence the recommended three- to four-hour class session once per week. However, we also know that sustaining attention for this length of time can be difficult. We therefore recommend allowing students the freedom to take breaks throughout the modeling session to get snacks or coffee.

A typical four-hour class session might be broken up into the following sections:

• 20-minute discussion of the reading to ensure all the students are familiar with the mathematics behind the model
• 1.5 to 2 hours to build the model
• 1.5 hours to conduct experiments

For instructors who have more limited contact hours with their students, we suggest that the model construction parts of this exercise be assigned as a pre-lab to be handed in a day or two before class along with the completed STELLA model itself. This would allow the instructor to determine whether students' models are working correctly and to provide feedback to address errors in construction, omissions in documentation, problems with unit conversions, and inappropriately sized time steps that might lead to spurious model behavior. Class time could then be devoted to running experiments and analyzing the results. If access to STELLA outside of class time is impossible due to computer lab scheduling or to financial constraints that prevent students from purchasing their own STELLA licenses, students could be asked to create a pencil and paper sketch of what their model should look like, annotated with equations and then sent to the instructor in advance of class for feedback. This should facilitate a faster model construction time during the limited class hours.

Barnosky, A.D., Matzke, N., Tomiya, S., Wogan, G.O.U., Swartz, B., Quental, T.B., Marshall, C., et al., 2011, "Has the Earth's Sixth Mass Extinction Already Arrived?" Nature, v. 471, p. 51–57.

Bologna, M. and Flores, J.C., 2008, "A simple mathematical model of society collapse applied to Easter Island," EPL, v. 81, doi: 10.1209/0295-5075/81/48006.

Cohen, J.E., 1995, How Many People Can the Earth Support? New York: W.W. Norton & Company, 532 p.

Cook, B.I., Anchukaitis, K.J., Kaplan, J.O., Puma, M.J.,, Kelley, M., and Gueyffier, D., 2012, "Pre-Columbian deforestation as an amplifier of drought in Mesoamerica," Geophysical Research Letters, v. 39, n. 16, article n. L16706.

Diamond, J., 2005, Collapse: How Societies Choose to Fail or Succeed, New York: Viking Press, 575 p.

Hunt, T., "Rethinking the fall of Easter Island," American Scientist, September-October 2006.

Kaufmann, R.K., and Cleveland, C.J., 2008, Environmental Science. Boston: McGraw Hill Higher Education, Chp. 5.

Kolbert, E., 2014, The Sixth Extinction: An Unnatural History, New York: Henry Holt & Company publishers.

Merritts, D.J., Menking, K.M., and DeWet, A.P., 2014, Environmental Geology: An Earth System Science Approach, 2nd Edition. New York: W.H. Freeman, Chp. 7.

Mieth, A., and Bork, H.-R., 2005, "History, origin and extent of soil erosion on Easter Island (Rapa Nui)," Catena, v. 63, p. 244-260.

Ricklefs, R.E., 2008, The Economy of Nature, 6th Edition. New York: W.H. Freeman, Chp. 15, 18.

Turner, B.L., II, and Sabloff, J.A., 2012, "Classic Period collapse of the Central Maya Lowlands: Insights about human–environment relationships for sustainability," Proceedings of the National Academy of Sciences, v. 109, p. 13908-13914.

Additional web resources include:

• World Population Clock, from Worldometers
  
• What Happened On Easter Island — A New (Even Scarier) Scenario, an NPR blog entry by Robert Krulwich.
  
• Researchers challenge old theory of Easter Island population collapse, a Washington Post article by Justin Wm. Moyer.
  
• Rethinking the Fall of Easter Island New evidence points to an alternative explanation for a civilization's collapse, an American Scientist Online article by Terry Hunt.