D-4653-2 1 1Mistakes and Misunderstandings: Table Functions Prepared for the MIT System Dynamics in Education Project Under the Supervision of Dr. Jay W. Forrester by Leslie A. Martin July 15, 1997 Vensim Examples added October 2001 Copyright © 2001 by the Massachusetts Institute of Technology. Permission granted to distribute for non-commercial educational purposes.D-4653-2 2 Table of Contents 1. THE SYSTEM 3 2. A FIRST ATTEMPT TO MODEL THE SYSTEM 3. FIRST MISTAKE AND MISUNDERSTANDINGERROR! BOOKMARK NOT DEFINED. 4. A SECOND ATTEMPT TO MODEL THE SYSTEM 5. SECOND MISTAKE AND MISUNDERSTANDING 6. OVERCOMING OUR MISTAKES AND MISUNDERSTANDINGS 7. KEY LESSONS 8. APPENDIX: EQUATIONS FOR THE CORRECTED MODEL 14 9. VENSIM EXAMPLES 16 4 7 10 11 13D-4653-2 3 31. THE SYSTEM “A population model? Oh sure, I know how to make population models. They’re all over Road Maps...” I sat down at my desk and stared at the information I had gathered to build my model. Aardvarks. Stout, pig-like animals up to six feet long (hmm, longer than I am tall) with a long snout, rabbit-like ears, and short legs. Coat varies from glossy black and full to sandy yellow and scant. The aardvark is native to Africa. Today approximately one million aardvarks roam the savannah south of the Sahara. The name aardvark is Afrikaans for “earth pig.” Photo of aardvark removed due to copyright restrictions. Figure 1: An aardvark My population model needed to describe the spread of the aardvark population. A few aardvarks searching for food discover a lush savannah. They settle down and, as seasons pass, reproduce and grow in number. Typically, a young aardvark is born to each middle-aged female aardvark once a year. After several years, though, the savannah is no longer a paradise for aardvarks. Many aardvarks compete for the same food, the same shelters. Due to limited natural resources, aardvarks die at a faster rate. Eventually, the population in the savannah stabilizes. I realized that the key to my population model would lie in correctly representing the constraint of the environment on the aardvark system. The death fraction is aD-4653-2 4 nonlinear function of the aardvark population. A small increase in population for a population that is small (when compared to the maximum population the environment can support) only slightly changes the death fraction. A small increase in population for a population that is large (with respect to the maximum population), however, causes a large increase in the death fraction. Therefore the relationship between population and death fraction is nonlinear. Table functions are excellent tools for modeling nonlinear relations. Table functions are graphical relationships, usually nonlinear, between two variables. Table functions graph a table of data points. They allow a modeler to easily specify relationships that are not amenable to algebraic description. Table functions often aggregate several real processes into one relationship. In general, table functions are used for the representation of nonlinear variables. The effect of the constraint of the environment, which is negligible when relatively few aardvarks live in the savannah and overwhelming when many aardvarks roam around, is a nonlinear variable. 2. A FIRST ATTEMPT TO MODEL THE SYSTEM In my first attempt to model the aardvark system, I created the model in Figure 2. Aardvarks are the total number of aardvarks currently living in the savannah. The number of aardvarks is increased by births and decreased by deaths. In the model the birth fraction is constant while the death fraction changes with the current number of aardvarks. Aardvarks births deaths ~ BIRTH FRACTION death fraction Figure 2: My first aardvark population modelD-4653-2 5 5Female aardvarks give birth to one baby each year. Estimating that about half of the aardvark population is female, and more than half of those females are able to reproduce (not too young, not too old) then approximately one third of the aardvarks are able to give birth each year. So I entered an approximate “BIRTH FRACTION” of 0.3 aardvarks/aardvark/year (which is equivalent to 0.3 per year). I then tackled the “death fraction”. I plotted the “death fraction” as a function of the current number of “Aardvarks” in the table function shown in Figure 3. My first step in obtaining the curve was to identify the values of the “death fraction” for its extreme cases, the limits of the curve. I know that aardvarks have an average lifespan of 10 years. So initially the “death fraction” is 0.1 per year.1 Very large aardvark populations, however, are constrained by the environment. I referred to an encyclopedia to find out that, at most, 500 aardvarks live in a 1000 acre savannah. So when a 1000 acre savannah holds 500 aardvarks, the population should be at equilibrium... which means that the number of aardvark deaths must equal the number of births. Therefore, when the population of aardvarks reaches 500 animals, the “death fraction” should equal the “BIRTH FRACTION”, 0.3 per year. My next step in obtaining the curve was to estimate the overall shape of the curve connecting the extreme points. The curve between 0.1 and 0.3, between the two extremes, increases slowly for relatively small populations when natural resources abound and then increases sharply for large populations as the natural resources become scarce. 1 The death fraction is the reciprocal of the time constant. Here, the time constant is the average lifespan of an aardvark, the time for an aardvark to die, 10 years.D-4653-2 6 Figure 3: The table function for my first model With my table function complete, I ran the model and analyzed the graphs I obtained. Figure 4 depicts the behavior of the aardvark population model: S-shaped growth. Proud of my modeling skills, I submitted my work. 1: Aardvarks 1500.00 250.00 0.00 1 1 1 0.00 7.50 15.00 22.50 30.00 Figure 4: Behavior of the aardvark populationD-4653-2 7 73. FIRST MISTAKE AND MISUNDERSTANDING One week later the professor returned our papers. I eagerly flipped the paper over and my jaw dropped as I couldn’t even recognize my work, buried beneath a bloody trail of red ink. I quickly hid it in my notebook and did not look at it again until late that afternoon, when I had reached the privacy of my own room. Bold letters along the front page blared “inputs and outputs to a table function