Chapter 2 - Systems

1. Introduction   A system is usually defined as a collection of interrelated parts. This definition includes such complex entities as a pond, a social club, or an electrical power grid. But it also includes such simple things as a pair of pliers, a compass, or needle-and-thread. We will sacrifice the logical security of the conventional definition in order to focus on more interesting systems:

A system is set of interrelated parts that give rise to emergent properties.

The relations among the parts should be so numerous or varied that the properties of the set cannot be readily understood by a casual inspection or by reading a parts list.  Chess and programming languages are examples of systems; tic-tac-toe and pig Latin are not.  Here are some other systems --

       Two systems might have very different parts or contents and so appear to be very different. Yet they could be very similar in terms of behavior, structure, or properties.  Government bureaucracies and army bureaucracies are good examples.
       In 1980 the independent scientist Dr.James Lovelock pioneered the concept of viewing large systems as organic units. His Gaia Hypothesis explains Earth as a living system that has evolved because of a "dialogue" among its constituents, all of whom are trying to change the environment to help them thrive. Several billion years ago oxygen made up a negligible part of the atmosphere. Then cyanobacteria (blue-green algae) appeared on the scene. These bacteria were the first living things to produce and tolerate a new and poisonous gas -- oxygen.  Atmospheric oxygen increased to 22% and this made it possible for life as we know it. Enormously complex interactions between the oxygen-breathers and everything else have maintained oxygen in this steady state.

       Radically different systems might be masked by the same name. For example, lumber companies (and even foresters) often claim that American forests have increased in size over the last several decades and that modern forestry has essentially replaced cut-over forests. A forest is a complex system and such systems once covered most of the southeast US. These forests, usually called Long Leaf Pine forests, consist of Long Leaf Pine, Turkey Oak and Wiregrass plus a panoply of animal life, fungi, shrubs and other trees, including half a dozen other pines. Replacements have been tightly-planted monocultures of Slash Pine or of Sand Pine. The result is a woodlot. Forest systems have been replaced by low-diversity tree farms. In the last 400 years there have been a few changes in the American forest.

2. Closed and Open Systems   A closed system is any system that does not exchange energy with its environment. We will use the term energy in a far more general way than you did in chemistry or physics. It includes the energy associated with material, as well as the energy "embodied" in money, information, or structures.  A truly closed system would have to be completely insulated or completely isolated and thus does not exist in nature. However, a well-insulated cooler or an unplugged refrigerator approximates a closed system, at least over a short period of time. Our solar system can also be considered an essentially closed system.

	Closed system		Tank w/o flows

Fig. 1  No energy exchange

Most systems are open -- they interact with their environment.  Energy exchange allows these interactions take place. A candle imports oxygen and it exports carbon dioxide, carbon particles and heat. The more complicated example of a pond takes in air, water and sediment, while yielding insects, fish and water plants.  Most human-designed organizations are open -- they have inputs and outputs.

            Fig. 2  Energy exchange

Much of the input for these bureaucracies is information and money – taxes for the public bureaucracies and investment for the private ones. The output is transformed information and various activities.

Exercise 2.1   Give an example of an open system.

Exercise 2.2   Give an example of a closed system.

3. Taxonomy   Classification, or taxonomy, provides a road map for thought. It can color our perceptions or predispose us to think along a certain path. For example, consider the conventional view of the biological kingdoms -- Plant and Animal.  Where do mushrooms fit . . . ?  They are not at all like animals, but they don't have chlorophyll like green plants, either. Still, fungi are more like plants than animals, so they can be shoe-horned into the Plant kingdom.
       Bacteria are even harder to fit in the Plant/Animal scheme than fungi. They are vanishingly small, one-celled entities, while everything else in the two kingdoms is multicellular. Moreover, all cells except bacteria have nuclei. Any taxonomy of living things will have some problems, but the two-kingdom scheme is clearly inadequate.  It has no clear place for bacteria, the most basic form of life!  
       The Five-kingdom scheme is much more satisfactory.  Bacteria are assigned their own kingdom. Animals, fungi, and plants are also each assigned in their own kingdom.  A fifth kingdom houses everything else -- the multicellular but usually very small organisms.

The Five Kingdoms of Life

Fig. 3

        One of the most impressive classification systems of all time is the Periodic Table of Chemistry.  Elements with similar properties such as copper Cu, silver Ag, and gold Au, end up in the same family column. The Periodic Table replaced a hodgepodge collection of elements by an orderly taxonomy.  It also predicted the discovery of elements.
       Even in physics, taxonomy plays an important role. The concept of energy is greatly clarified by dividing it into two classes -- kinetic, or active energy and potential, or passive energy.  Examples of kinetic energy are Niagara Falls, an arrow in flight, or a volcano in action.  Examples of potential (stored) energy are fat, a coiled spring, or a car battery.

Exercise 2.3   Produce another example from any field where taxonomy plays an important role.  It could come from a science such as botany, geology, zoology etc., or from commodities such as food, hardware, pharmaceuticals, etc.

4. Local, Global  Many fields have language that distinguishes between the relatively small, short-range or local and the larger, long-range or global. This Local-Global way of looking at situations provides a handy classification that could depend on time or on space or on some other concept. The table shows how four different fields express this concept.

       Geography is distinguished by having a handy intermediate term.  If the system of interest is the USA, then the region might be New England, while the local geographical unit might be Rhode Island. Or the system might be North America, the region Yucatan, and the local geographic unit the resort city of Cancun.

Exercise 2.4   Come up with two more examples of the local-global concept. Begin with a field such as history, photography or sociology and then try to find paired concepts in that field.

5. Stability   A stable system is one that can "Roll with the punch."  A system is said to be stable when small changes in its context or initial conditions cause small changes in its behavior.  Animals are examples of very stable systems.  Small injuries, weak toxicity, or drinking slightly less water than normal will not have much of an effect. The American Jeep or the British Land Rover is a mechanical example of a stable system.
       On the other hand, a tiny change in a computer program -- a mistake in one character -- can cause a program with 5000 lines of code to crash. This is the basis for "Don't touch that key!" cartoons. A system in which small changes in the context or initial conditions cause large changes in behavior is said to be unstable. Many engineers and scientists refer to an unstable system as sensitive.  Figure 4 shows a simple object that is stable in one position but not another.

       A more interesting example of the stability concept is provided by an ordinary box, as in Fig. 5.  The challenge is to spin the box around one of its three major axes and then send it spinning into the air, as shown in Fig. 6.  With just a little practice the box can be launched into a high, smooth orbit if it is spun around the shortest axis. The same is true for the longest axis. Try it. Now try the same thing be spinning around the middle-sized axis. No matter how keen your eye, how great your motor skills, or how good you are with a basketball, the box will wobble after a few spins.  In all three cases, the launch starts off OK, but motion about the "moderate" axis quickly becomes unstable.  Even if the box were launched with machine precision, it would quickly go into an exasperating wobble.

Fig. 6  The spinning box

Exercise 2.5   The change in the behavior of the box that is spinning around the middle axis is apparent. What is the small change in the environment or original condition that makes the box so resistant to control?  

       The journalist J. Gleick gave the most striking example of an unstable system in his book Chaos.  He argued that the flutter of a butterfly's wings in China could, via a series of events, lead to a tornado in the Midwest.  His book title comes for a term used for complex, unstable systems -- chaotic systems.

      Many seemingly new concepts such as chaos have been anticipated by myth, proverbs, or folklore. Let's take a look at the old saying, For the want of a nail the battle was lost.  The battle was lost because a critical message was delivered too late.  The message was delivered late because the horse carrying the messenger became lame. The horse became lame because he threw a shoe.  The horse threw a shoe because a small nail was wanting.  So a battle was lost because of a small nail that had not been correctly placed!
       Big, sophisticated, modern systems are not immune to such catastrophes due to small failures -- quite the contrary.  In the 1970's a tiny switch failed in the mammoth northeast electrical power grid.  In four seconds the entire system was brought to its knees. Cities on the power grid had a first-class brownout.  

Exercise 2.6   Give an example of a stable process or system.

Exercise 2.7   Give an example of an unstable or sensitive process or system.

Exercise 2.8   Can you think of case where a big, modern, presumably "fail-safe" system crashed for want of a nail . . . ?

6. Feedback Loops   The term feedback means more than just a response - it is a cycle of responses and co-responses. These are called feedback cycles, or feedback loops.  Feedback cycles often lead to a spiraling up, or a spiraling down of some behavior or characteristic.  The vintage Laurel & Hardy comedy film, "Tit for Tat" provides an example of a feedback loop in which action spirals up.  Hardy taps Laurel, who responds with a somewhat stronger tap. Hardy returns with a still stronger tap, and soon they are doing serious escalation. The responses keep getting stronger, and the violence spirals up.
      Suppose a bone-crunching spotted hyena breaks a tooth.  This will make it harder to crush bones, an essential source of calcium.  The hyena's tooth structure will tend to weaken.  This will make it still harder to crush bones, so the tooth structure will weaken still more.  The hyena's bone-crunching performance spirals down, leading eventually to death.

 Spiral-down feedback                                                   Spiral-up feedback

      Fig. 7  Feedback loops

Exercise 2.9   Give an example of a situation in which there is spiral down feedback. Be sure to make clear what action, behavior or characteristic is getting weaker.

Exercise 2.10   Give an example of a situation in which there is spiral up feedback.  Be sure to make clear what action, behavior or characteristic is getting stronger.

7. Self-organization and Self-maintenance   It is a common observation that living systems are capable of self-organization. They grow and reproduce, as well as repair and maintain themselves.  Much of this depends on feedback processes between the system and its environment.  Borderline systems such as viruses share many of the same properties of living things. Even prions a lowly form of protein discernable DNA or RNA, can replicate itself and cause Mad Cow disease.  In fact, many complex non-living systems can organize and maintain themselves. Think of tornadoes and other weather systems. Something as simple as a candle exhibits quite a bit of adaption to a (moderately) changing environment.  Even something as abstract as a computer program can exhibit adaptive behavior. Chess-playing programs are good examples of this.

8. Unintended Consequences   Meddling with a complex natural system is almost guaranteed to return long-range environmental costs that far exceed short-term benefits.  Natural systems have a seemingly endless number of feedback loops and other interactions. They are not serviced or repaired as easily as a car or a computer. The three examples below show how the Law of Unintended Consequences deals with excessive or unwarranted intervention.

Exercise 2.11   List an unintended consequence of some large, intrusive activity such as clear-cutting trees, turning a stream into a concrete channel, or cattle-grazing in the tropics.

Exercise 2.12   List two potential (or actual) environmental unintended consequences of genetic engineering.

Exercise 2.13   Unisex public toilets are becoming the mode, so that urinals are disappearing from men's rest rooms.  List an environmental unintended consequence of this.

9. A Question of Scale   Walden Pond, Lake Tahoe, and Lake Superior are examples of systems with many properties in common. They are self-maintaining, open storage systems that support fish and plant life. The enormous difference in physical size - or scale - warns us that there will be significant qualitative differences among them.  For example, Henry Thoreau's pond has large seasonal changes in temperature relative to the massive Lake Superior.
       In biology, there is a huge range of subsystems from bacteria to the biosphere (Gaia).  Many properties are affected by scale. Just by virtue of size, an elephant can wallow in gooey mud, yet be threatened by a three-foot drop.  A spiderling can be trapped by a speck of clear water, yet safely pass over a canyon.
      A simpler example is that of a physical model.  An airplane or a boat can be carefully scaled down by some factor such as 20:1 but the model will not exhibit the same properties as the original. This can be seen even in the static case.  If an accurate one-foot model is made of a fifty-foot building, a viewer of the model will tend to lose small items and details, such as doorknobs, bronze plaques or decorations.
      Very large numbers, such as the total estimated U.S. debt in the year 2000 of 14 trillion dollars ($14x10¹²), are hard to grasp. Such figures can be tamed by changing units or some similar device.  For example the U.S. debt can be reduced to average debt for each American of approximately $50,000. Very small numbers, such as the size of bacteria are also hard to grasp.  A typical bacterium will measure five microns, where a micron is a millionth of a meter.  It would take about 100 bacteria side-by-side to make up the width of the period at the end of this sentence.

Exercise 2.14   The killer whale Keiko (star of the 1993 film Free Willy) weighs about 8000 pounds. Express this weight in terms of a number of humans.

Exercise 2.15    ". . . the garden snail and its prospective mate will circle around each other at a pace of about 17 miles per year . . ." (C. Manes, "Slugfest," Wild Earth, Winter 1994-95).  Express this pace in a more reasonable scale.

Exercise 2.16   The current USA national debt is about 5.5 trillion dollars ($5.5x10¹² ).  Bring this gigantic figure down to earth by expressing it in terms of dollars per family, dollars per wage-earner, or dollars per capita.

Exercise 2.17   The U.S. maintains 45,000 miles of roads in the national forests (mostly to benefit commercial loggers).  It has been claimed that the length of these roads is ten times the length of the roads in the federal Interstate highway system.  Does this sound reasonable . . . ?  Check the claim by estimating the mileage of the highway system (or consult an appropriate source).

      There is another, closely-related meaning of scale.  It goes beyond contrasting order-of-magnitude differences in such properties as size, number or weight.  In the earlier example of a one-foot high physical model of a 50-foot high building, we quickly recognize what the model represents.  Still, our perception of the building is not quite right because of the loss of detail -- such items as door handles would tend to disappear.  This difference goes far beyond perception –  many of the physical characteristics of the small model and of the building are very different.  These differences occur between two objects of the same shape even if they are fairly close in size.
        These differences can be shown by looking at two simple objects with the same shape but fairly close in size.  Let's compare a jumbo egg three inches long with a mini-egg that is two inches long.  The large egg is 50% bigger than the small egg.  If you were going to color the eggs, would the large egg need just 50% more dye or paint than the small egg . . . ?  Suppose that you were going to buy a dozen of the large eggs to scramble for a club brunch but noticed that the small eggs were on sale. You could buy two dozen of the small eggs for the same price as one dozen large eggs. Would you go for the sale . . . ?  These questions are fairly easy to answer once we see how areas and volumes vary -- or scale -- with length. Since the geometry for the eggs is fairly complicated, we will work with round eggs or balls. (Round eggs are not far-fetched.  Alligators and turtles lay round eggs!)
      We will look at a ball with diameter D=2" and one that is 50% bigger, a ball with diameter D=3".  The formulas for the surface and volume of a ball are given below and the calculations are alongside.

So the large ball will have 7.07 square inches of surface area, almost twice the small ball's 3.14 square inches. There is an even bigger discrepancy in the volume. The large ball will have 14.14 cubic inches of volume, more than three times the small ball's 4.19 cubic inches!      The phrase "The large egg is 50% bigger than the small egg." was a bit misleading. We should have written something like "The large egg is 50% wider (or longer) than the small egg."  Likewise, with the ball, we should have written something like "The large ball has a diameter 50% bigger than the small ball."  These length-surface-volume relations apply to chicken eggs, people any other objects with similar shapes.

     Take a look at the first column of formulas – the key is there. The radius, a linear measure, is squared in the surface formula and cubed in the volume formula. The surface of an object is said to scale like the square of a linear dimension, while volume scales like the cube of a linear dimension. The scaling principle for two objects of the same shape is -- 

      Now back to the scrambled eggs and the club brunch.  Should you go for the sale . . . ?  That is, would you buy two dozen small eggs for the same price as one dozen large eggs?  The above calculations tell us that a large egg has over three times the volume of a small egg. This means they will weigh over three times as much and will make over three times as much scrambled eggs.  So that two-for-one sale was no bargain!  Even if the big-hearted store manager offers you three dozen small eggs for the same price as a dozen large eggs, say "No, thanks."
      Since we know the volume will increase with the cube of a linear dimension, we can find the ratio of volumes by cubing the ratio of the (linear!) size of the eggs. So the three-inch jumbo eggs would have  (3/2)³ = 27/8 = 3.37  times more volume than the two-inch mini-eggs.

Exercise 2.18   You know from experience that you need a bag of a dozen jumbo oranges to provide enough juice for a luncheon. Jumbo oranges are almost uniformly four inches in diameter, and a bag costs $3.00. The store turns out to have smaller, three-inch oranges on sale at a bargain price of ten cents. Would you buy 30 of the smaller oranges instead?  Explain.

Exercise 2.19   Verify that the areas in the three circles (below) are proportional to the square of their diameters. The three circles have diameters of 4, 6 and 8 units.  Find the areas by using algebra or by a graphical estimate.  Does it matter whether the units are centimeters, miles or light years . . . ?

Fig. 8  How areas scale with diameters

10. System Invariants   The Tokyo-Osaka Bullet Trains speed along the tracks at 120 mph.  Figure 9 pictures one of the wheels.  It would seem that every point on the wheel is moving forward at an enormous speed. A snapshot of the wheel should show a blur. But will all the points on the wheel show up as a blur . . . ?  That is, are all points on the wheel in motion?  There is a point on the wheel that is not moving, the one in contact with the track.  If you are not convinced, check the arrows. They indicate the approximate straightforward speed of points on the rim of the wheel.

Fig. 9  A turning wheel

       In many processes or transformations it seems that everything is changing, everything is varying. This seemed be the case for the wheel of a speeding auto or train. Yet a snapshot or an analysis reveals a nonmoving or unvarying point.  Just as there are unvarying points, so there can be steady or unvarying characteristics or properties. These are all referred to as invariants. When the invariant is a point, as in the case of the train wheel, it is usually called a fixed point.
       In physics, engineering and mathematics there are many invariants, often referred to as constants.  In the spinning wheel, for example, the ratio of its circumference C to its diameter D is  C/D = 3.14159 . . . , the constant p.  Big wheel or little wheel, fast moving or slow moving, the ratio is always the same. An important invariant from physics is the speed of light (in a vacuum or in space), c = 3x1010 cm/sec = 186,000 miles/sec. Another important one is the Planck quantum constant h.  
       Are there invariants in the natural environment . . . ?   Yes, but they will not be as precise or as "tight" as c, h, p or any other invariants from geometry or physics. This is inevitable when dealing with natural, and especially living systems things are less uniform and harder to measure. So be prepared for some "abouts" and "approximates".   First we will look at mammal heartbeats and stream meanders. Then we will look at a system invariant from physics that deals with energy.

Heartbeats   Relatively large mammals such as camels and humans tend to live longer than smaller mammals such as cats or mice. It would seem that larger mammals have more extensive experiences than smaller ones.  However, smaller mammals have faster heart rates.  For example, humans have a normal heart rate of 60 beats per minute while cats have a heart rate of 150 beats per minute. Biologists noticed that the product of normal heart rate and expected longevity was approximately the same for animals of different sizes. In other words, the expected lifetime number of heartbeats B for an animal is a biological invariant –

B =  2 billion  =  2x10⁹  =  2,000,000,000  or  B =  2 billion heartbeats in a lifetime.

Let us do the calculation for Homo sap.  First to find the number of minutes in a year -- 60 min/hr x 24 hrs/day x 365 days/yr = 525,600 min/yr.  If we use 70 years for longevity, we get 70 x 525,600 = 36,792,000 minutes in a lifetime. A heart rate of 60 beats/min yields 60 x 36,792,000 = 2,207,520,000, or about 2.2 billion beats.  If a heartbeat is a basic unit of experience then elephants, humans and shrews are in the same boat.  

Exercise 2.20   Calculate the expected number of lifetime heartbeats for a relatively large mammal other than Homo sap.  Is your answer approximately equal to B?

Exercise 2.21   Calculate the expected number of lifetime heartbeats for a relatively small mammal. Is your answer approximately equal to B?  Flowing water provides another good example of an invariant in nature.

Meanders   Irrigation ditches and concrete channels look and behave (as well as smell) very differently from a natural stream.  Streams meander as they run through soil.  A meander looks like this --

Fig. 10  Stream meander

       The undulating path of a stream can often be approximated by a sine wave.  If the run of a stream is 190 yards, the width will be about 19 yards. If the width is 8 ft, the run will be about 80 ft.  It turns out that the ratio of the run to the width is (approximately) 10.  In other words the Run/Width ratio is a stream invariant--

Run / Width  =  190yd / 19yd  =  80ft / 10ft  = 10  .

Notice that the units -- yards and feet in these two examples -- cancel out.

11. Energy   Suppose a closed system starts out today with a given budget of energy Q(0).  Recall that there is no energy (or material) exchange across the borders of a closed system. The quantity of energy Q(0 )in the system might be 30,000 Calories (or 8000 BTU or 500 kw-hr).  Now let Q(t) represent the quantity of energy in the system at any later time t.  Here t could be tomorrow, next week, or next year -- it does not matter -- the quantity of energy remains constant. That is, Q(t) = Q(0), the same quantity of energy that was there initially.   In other words, the quantity of energy in a closed system is an invariant -- energy can neither be created nor destroyed.  In physics this fundamental law of nature is referred to as Conservation of Energy.

Fig. 11  Quantity of energy in a closed system

       Suppose That a fish tank, or the building you are now in, is sealed off.  All living things, even bacteria, would eventually die, viruses would go into a crystal-like state, clocks and other devices would run down. Throughout this process of degradation, the quantity of energy remains constant.  As we shall see later (Chapter 7, Energy Principles), there is a way to measure the quality energy.  Unlike the quantity of energy, the quality of energy will tend to run down hill in a closed system.