1. Our Heritage and the Environment   Three and a half centuries  ago we moved from an earth-centered to a sun-centered perspective. Science, or the study of nature and the laws of nature, promised an entirely different world view from that of the humanistic Renaissance. The scientific revolution prepared to dethrone man from the center of the solar system and, by extension, from the center of the universe. This promise was never fulfilled. Instead, we had heavy emphasis on the man-made or design world, coupled with an increased exploitation of the natural world. The first Earth Day in April 1970 focused our attention on the damaged environment. Mathematics has joined the response to the Earth Day challenge. It is fitting that mathematics – one the most humanistic of all disciplines – should play a role in helping us to move from the current homocentric view of the world to a more biocentric view.

2. New Subject, New Approach   A new subject requires a new approach to learning the subject. We encourage you to experiment, to ruminate, and to engage in cooperative activity.  Unlike high school algebra, a problem or scenario might not have an exact, or even a right answer. Even when there is a precise answer, it might not be accessible and will have to be approximated or estimated.  For example, how many plant species were driven to extinction in the last decade of the 20th century?  In other cases there might be several "right" answers, or no satisfactory answer. For example, does the economic value from building a road through a National Forest offset the intrinsic value lost by destroying plants and wildlife . . . ?

3. A Broader View of Mathematics and Modeling   The conventional view of mathematics emphasizes numeracy. Just as literacy suggests being comfortable and competent with words and literature, numeracy suggests being comfortable and competent with numbers and mathematics. The quantitative aspect is important, but it might be more accurate to describe mathematics as the study of patterns. In our approach, visual representations and qualitative arguments will play an important role alongside quantitative and algebraic techniques. Mathematicians usually think of a model - a representation of some real object - as some sort of equation or other algebraic expression. Other representations are considered lightweight or deficient for modeling. We will take a broader view of what constitutes a model. Diagrammatic and other visual models typically offer an expeditious way to connect with the natural world. They often make it easier to see and to appreciate such concepts as open systems, closed systems and feedback. We will make extensive use of visual representations, and the first stage of our modeling process will almost always be a diagram.

4. The Visual   The visual approach is known and loved but it doesn't get much respect [1, 3].  Marston Morse, an outstanding mathematician who credited carpentry for greatly contributing to his geometric sense, noted that "The battle between algebra and geometry has been waged from antiquity to the present" [2].  To move to a more basic level, the paleontologist Stephen J. Gould observed that ". . . primates are visual animals" [8].  Visual representations can capture and deliver a great deal of information.

Fig. 1   Threats to ozone in Costa Rica

Look at the example in Figure 1.  The meaning of the pie chart is fast and direct.  The meaning of the storage tank diagram in Figure 2 (below, left) is less immediate.  But doesn't the arrow on the path line suggest an outflow of the contents Q . . . ?

Exercise 1.1   Suppose the tank in Fig. 2 is filled with 200 gallons of oil, and the outflow, in gal/min, is proportional to the amount of oil in the tank. That is, the outflow is only half as fast when 100 gallons are left in the tank, and the outflow is only one quarter as fast when 50 gals are left in the tank, and so forth.  Which of the below scenarios describes the outflow best?

                  Fig. 2  Storage tank with outflow                                                                                                Fig. 3   Three scenarios

The example in Fig. 4 shows that electrical kilowatt-hour consumption (vertical bars) and temperature (curve) go through cycles. It fairly quickly suggests cause-and-effect, but identifying the causal relation takes a little thought.

Fig. 4   Temperature & electric energy consumption over a four-year period

Exercise 1.2   Do the graphs in Fig. 4 represent energy usage in states like Maine and Michigan, or in states like Arizona and Florida?

5. The Qualitative   The term qualitative is not meant to exclude numbers but it rarely is number-intensive. A "wise-guy" rule for qualitative mathematics is -- Know the answer to a problem before trying to solve it.  One should have an estimate, a pictorial solution or at least a good idea of the form of a solution before trying to solve a problem analytically.  This can be done via pictures, "what-if" questions, or thought (gedanken) experiments. The guiding principle for qualitative mathematics is:

 Find (a reasonable) answer to a problem before trying to solve it analytically.

The qualitative approach to problem-solving or modeling can be characterized as a "no holds barred" mathematics. It is often little more than an application of common sense and includes --

Order-of-magnitude arguments or "ball park" estimate   A request to estimate the product of 75.36 and 84.21 suggests a reach for the calculator, but a quick approximation is 80x80 or 6400.

Best and worst case scenarios   You want to estimate how long it will take to drive to a destination five signal lights away.  Suppose the best time, seven minutes, is when you can make all five lights, and that the worst time, eleven minutes, is when you hit five red lights.   So a reasonable estimate is nine minutes.

Drawing a diagram or picture and looking at special cases   Suppose we would like to know how many two-way interactions there are for seven species. (There is an algebraic formula for doing this, but you might be unfamiliar with it or have forgotten it.)  We begin by drawing simple diagrams and count the number of interactions for two species, three species and four species, as in Figure 5.  This might suggest how to solve it for a larger number of species.

Exercise 1.3   Draw a figure to find out how many two-way interactions are possible among five species.

Exercise 1.4   How many two-way interactions are possible among seven species?

Fig. 6   The Aransas National Wildlife Refuge, near Corpus Christi, Texas

Exercise 1.5   Make a rough estimate of the square mileage in the Aransas Refuge. Then choose one of the following ways to improve this rough estimate:  a) Count all the squares with areas inside the boundary and then approximate what proportion of the area of the boundary squares are inside;  b) Count the number of squares whose areas are inside the boundary and the number of squares whose areas are inside or on the boundary and average these numbers.

6. The Computational   Numerical solutions, or numerical verification of special cases, are quickly becoming an accepted part of mathematics courses.  Spreadsheets, once the province of business, are recognized as an important aid for solving many problems. It is natural to turn to computers for solutions of almost any problem that involves intense computation.  However, computers are still somewhat large and costly while scientific calculators are small and cheap yet powerful.  Such tedious operations as finding the square root of 327 or the cube of 23.7 are carried out by touching a couple of buttons. There is an even more powerful calculator, the graphing calculator or grapher. Graphers are slightly larger than scientific calculators and are somewhat more expensive, but they have many more operations and are much more flexible.  A major advantage of a grapher is its screen, which allows a variety of visual representations. Many of the equations that come up in this course can be solved with a grapher.

7. "Decay" and the Tank Model   In Exercise #1 we looked at how the oil level in the tank fell due to an outflow. The third scenario (correctly) tells us that the loss is rapid at the beginning and slows down as the oil level drops. This transfer of stored material, energy or other asset often follows the same scenario – the rate of transfer is proportional to the amount in storage. In the case of radioactive material, the loss of mass is referred to as decay.  A more abstract example is provided by the discounting of prices.
      Compute hardware prices drop an average of 3% each month.  So an $1100 laptop in April will drop $33 to $1067 in May. This $1067 laptop will then drop $32 to $1035 in June. Therefore, the dollar "loss," the discount, is proportional to the amount in "storage," the current price. The next example will be worked out in detail.
      Lithium batteries are used for powering laptop computers.  When not in use, a lithium battery (like all batteries) loses a small amount of its energy charge. At 90°F it will lose 3% of its charge every day.  How much of its original charge will it have left after a week at that temperature?  You might not know how to carry out an accurate computation, but it is easy to come up with an estimate.  Each day 3% is lost, so in seven days, approximately 21% of the original charge is lost. Therefore, the battery has about 79% of its original charge.
      The actual answer is 80.8% or about 81%. This is slightly larger than our estimate because the 3 percent loss is relative to a diminishing basis. Here are the calculations --

So at the end of week, .977 = .8080 = 80.8 % of the original charge is left.  
      Likewise, at the end of ten days, approximately 10·3% = 30% of the original charge is lost so that battery has about 70% of its charge left.  Calculation yields .9710 = 0.737, which rounds to 74%. This ten-day estimate is not as good as the seven-day estimate.  What do you think will happen if you go to two weeks?
     These situations (and many others) have a common characteristic.  In each case the rate of transfer of energy, money or other asset is proportional to the amount of material, energy, or asset still in storage.  In other words, all of these situations can be modeled by a tank.
      In Fig. 7 the tank represents the battery, and Q represents the energy charge. The graph shows the depletion of charge Q at the rate of 3% = .03 per day.  Note the path down to the "ground" or energy sink.  Any energy that takes this primrose path to this sink is irretrievably lost.

Fig. 7   The tank model

Exercise 1.6   Estimate how much energy remains in the Lithium battery at the end of two weeks. Compare with the calculated value. How good is your estimate?

      The US Department of Agriculture reported in 1970 that topsoil from Iowa, the main breadbasket of the USA, was being lost to erosion at the rate of one percent yearly.  Assume that this loss continues.  Use 1970 as the base year.  Use the lithium battery example to guide you with the following three exercises.

Exercise 1.7   Estimate how much Iowa topsoil was left in 1980.  Carry out the exact calculation and compare the two results.  How good is your estimate?  Do the same for 1990.

Exercise 1.8   Estimate how much Iowa topsoil was left in 2000. Carry out the exact calculation and compare the two results. How good is your estimate?   Do the same for 2010.

Exercise 1.9   Estimate how long it will take to lose half of the topsoil that Iowa had in 1970.  Use this estimate as the basis for a more accurate trial-and-error calculation.  The length of time it takes for 50% of an original stock to decay, erode or waste away is called its half-life.

8. Energy as a Central Concept   The romantic view holds that Love makes the world go around.  However, it takes energy to feed thoughts, support emotions, and implement the actions associated with love.  If we were somehow cut off from solar energy and from the internal energy of the earth's core, life as we know it would soon cease.  Earth would become cold and lifeless, more like Mars than our blue planet.  It is energy that makes the world go around. Our focus will be on two key aspects of energy –

      Also important is power or energy flow . The most common example of this is electrical power, measured in the familiar kilowatts (= 1000 watts).  It is worth noting that electrical energy is measured in kilowatt-hours. Your electric meter measures energy usage, and your electric bill is based on the number of kilowatt-hours (kw-hr) consumed.
     There is one more related concept, entropy.  It is defined as a measure of the unavailability of energy in a system. That is, an increase in entropy means a decrease in available energy.  In a tank, entropy is represented by the pathway to the sink.  We can associate entropy with the tendency of a system to degrade, degenerate or destruct. This can take the form of corrosion, disease, theft or almost any form of loss.

Exercise 1.10   Here is a "What's wrong with this picture?" question for Fig. 4.  Do you see any inconsistency between the labeling of the graph and the units on the vertical axis . . . ?

9. Overview of a Five-stage Modeling Process   Most of the modeling in this book will be done with a five-stage process. This process is described in detail in later chapters, so it will just be outlined here.

•   Construct an energy diagram of the target entity, process or system

•   Draw a freehand, qualitative Energy vs Time graph

•   Derive a flow equation

•   Solve the flow equation numerically (Graphing or Scientific calculator, spreadsheet or computer code.)

•   Plot a standard Energy vs Time graph with the numerical data

Fig. 8   Energy Diagram template

Figure 8 is a copy of a plastic template of H.T. Odum's energy diagrams.  We will use several of the shapes, but the two of them will play a major role. The first is the tank, which we have seen, and the second is the workgate.  Four workgates diagrams are on the right of the template, just below the middle.  The tank stores energy, and the workgate transforms energy.

---

Next  Previous  Contents

---

BAF/	MAR 87 - DEC 01