Environmental Sustainability

(Professor Womersley)
Final “Modeling” Homework
(Due last day of finals week, ten percentage points)

I. Instructions:
This homework assignment, worth ten percentage points of your grade (a whole letter grade!), requires you
to demonstrate your understanding of exponential mathematics and its application in human ecology.
Specifically, you will build and employ an ecological “model” that simulates an exponential phenomenon
in human ecology. There are several possibilities of phenomenon to model, and, in fact, we have already
demonstrated some in class. There are also several means by which the model may be calculated, detailed
below. You may choose any phenomenon to model, as long as it contains the “feedback” or intrinsic
change that typically leads to exponential effects, and as long as you clearly demonstrate an
understanding of feedback in dynamic systems. You may also choose any means to calculate the model,
as long as the work and results are clearly explained. (Hint: approach this like a lab assignment, make an
informal lab notebook as you model away, then clean up your notes a little and hand them in as a lab
report).

A word or two about ecological models: At heart, models are simply mathematical equations to provide
predictions of change in a system. We have used them for calculating population change, calculating oil
depletion, and even, as a demonstrator, calculating the time taken to repay a loan. Any of these examples
can be used for this homework, or you may choose an entirely different system not discussed in class, as
long as it is mathematically valid. In any case, most of you will likely need to consult with me and even get
some coaching.

You can calculate models using 1) difference equations, which can be automated in Microsoft Excel, by 2)
using an intrinsic growth equation (an application of calculus), or 3) by using proprietary ecological
modeling software such as the Stella ® program demonstrated in class, which is available to you as a free
30 day trail download from ISEE Systems Inc., its creators, at
http://www.iseesystems.com/community/downloads/STELLA/STELLADemo.aspx

If you use any automated method, you must perform at least 50 iterations of the model.

You can also use pencil and paper to calculate a difference table. If you use a hand-calculated difference
table, you may do only 10 iterations.

Below is a list of models, followed by a list of techniques with brief instructions. More detailed instruction
will be given in class.

II. Models to choose from:

As stated above, you may choose any phenomenon to model, but if you choose something different or new,
you should talk it over with me first. The examples below are those that we discussed in class and are set
up for you to some extent. Make sure you complete the entire problem and show all your work.

Loan payback: Calculate the payback period of a student loan. (Not your own, but hypothetical. I don’t
need to know how much money you owe!) Use a specific interest rate. Think about how different kinds of
jobs and living situations make it easier or more difficult to pay back such a loan, and briefly detail the
results of your thinking in your report. What causes the exponential effect in this case?

Population growth: Pick a country and predict its population for a year at least ten years from now. The
CIA maintains an excellent database called The World Factbook that you can easily find on the Internet at
http://www.cia.gov/cia/publications/factbook/index.html
Using the other data available from the CIA and a brief web search, detail any problems your country may
experience supporting this population. Be careful. Some countries have shrinking populations. What causes
the exponential effect in this case?
Oil depletion: Using reasonable assumptions about the levels of known reserves, population and economic
growth, and the price-conservation effect, calculate the depletion curve of world oil over the next century
or more. You will need to do some research to come up with reasonable assumptions. You may limit your
calculation to conventional oil, or include unconventional oils shales and sands. One good place to start is
www.hubbertpeak.com
What causes the exponential effect in this case?

Crude climate model: It is possible and even somewhat accurate (although not that scientifically
defensible for reasons you should understand by now) to predict global average annual temperature change
based on carbon dioxide levels in the atmosphere. Make this prediction for the next 100 years and compare
it with the IPCC predictions. You will need to calculate a rate of change for the rise of atmospheric
concentration of carbon dioxide over time using regression analysis to complete this assignment. What
causes the exponential effect in this case, if any?

III. Types of calculation technique

A. Calculating models using difference equations and tables:

By calculating the change in a system from one time period to another using a table, using the output from
one line in the table as the input to the next, changes can be predicted over short time spans with relative
ease. The equation is

Pt = P0 + (P0 x R)
where R is the discrete rate of change in a given time period. You must re-iterate for every row/time period
in the table.

The example below is a population growth example.

Year US population in Growth rate Growth rate X New population

that year Population
2005 298 million 0.009 (0.9%) 2.68 million 300.68 million
2006 300.68 million 0.009 (0.9%) 2.71 million 303.39 million
2007 303.39 million 0.009 (0.9%) 2.73 million 306.12 million

….and so on. You will have to make a longer table to achieve the time span required for our examples.

Below is an oil depletion example. In this case,

oilt = oilo – consumption + discovery

Year Known oil Estimate of Consumption Discovery Known oil

reserves undiscovered oil (annual) (annual) reserves
(at time zero) (at next time
period)
2005 950 170 35 5 920
2006 920 165 36 5 889
2007 889 160 36 5 853

Difference tables can be automated in Microsoft Excel. Most of you know how to do this already using the
“drag and fill” feature in Excel. If you use Excel, hand in your table as well as your notes.
B. Using intrinsic growth equations:
This technique employs the branch of calculus called “instantaneous” change. For any population, the
instantaneous equation for population growth is given by

Pf = P0 ert
where,

P0 is the population now or at the beginning

Pf is the population in the future
e is the base of the natural logarithms
r is the instantaneous rate of change, found by the method of derivation (finding the derivative)
t is the number of time periods

Use years for t and the instantaneous rate of change as r. Multiply these together right away to get rt. Most
scientific calculators have an ex key. You will need to know how to calculate the instantaneous rate of
change.

C. Using Stella ecological modeling software.

Stella is a program that allows you to automate difference equations. Stella uses a graphics interface to help
you construct box models of flow charts. The computer derives the equation from the way you piece the
boxes and arrows together. You can use Stella for any of these examples. I will teach you how to use it.
Learning the program takes a little time, but you can more quickly complete the assignment. The program
is available at
http://www.iseesystems.com/community/downloads/STELLA/STELLADemo.aspx

Below are examples of Stella graphics showing models for population change and oil depletion like those
used in class:

population

population change

undiscovered known oil

discovery consumption

price

As I’ve mentioned over and over, it’s the red arrow “feedback” loops that lead to the intrinsic change that
makes ecological phenomenon exponential. Exponential phenomena are surprising and even scary at times,
one reason why every educated person should have some knowledge of ecology.

When you’ve decided on a model, if you think you would prefer to use Stella, come by my office or make
an appointment for some help. I’ll coach you in the use of the program. This takes twenty minutes to half
an hour.

Make sure you have your data when you show up!