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MIT 15 988 - Use of Generic Structures and Reality of Stocks and Flows

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D-4646-2 1 1Mistakes and Misunderstandings: Use of Generic Structures and Reality of Stocks and Flows Prepared for the MIT System Dynamics in Education Project Under the Supervision of Prof. Jay W. Forrester by Lucia Breierova December 18, 1996 Vensim Examples added October 2001 Copyright © 2001 by the Massachusetts Institute of Technology. Permission granted to distribute for non-commercial educational purposes.2 D-4646-2 Table of Contents 1. INTRODUCTION 3 2. A FIRST ATTEMPT TO MODEL THE SYSTEM 3. MISTAKES AND MISUNDERSTANDINGS 4. OVERCOMING OUR MISTAKES AND MISUNDERSTANDINGS 5. KEY LESSONS 6. APPENDIX: MODEL DOCUMENTATION 9. VENSIM EXAMPLES 3 5 6 9 10 13D-4646-2 3 31. INTRODUCTION This paper studies two related mistakes and misunderstandings. First, it warns against the incorrect use of a generic structure. Secondly, it explains that stocks represent real-world accumulations and that flows represent changes over time in the stocks. This paper will examine two mistakes made in modeling a simple population of rabbits. 2. A FIRST ATTEMPT TO MODEL THE SYSTEM A student decides to build a simple model in order to study how an area with limited resources affects the growth of a population. Initially, ample resources do not limit the exponential growth of the population. As the population grows, however, resources become more scarce, and the rate of growth slows. The population asymptotically approaches an equilibrium value.1 The student expects the behavior of such a system to be S-shaped growth. Figure 1 shows a generic structure known to produce S-shaped growth.2 1 The equilibrium value is known as the carrying capacity of the environment. 2 For more information on generic structures producing S-shaped growth, please refer to: Terri Duhon and Marc Glick, 1994. Generic Structures: S-Shaped Growth I (D-4432), System Dynamics in Education Project, System Dynamics Group, Sloan School of Management, Massachusetts Institute of Technology, August 24, 30 pp.D-4646-2 4 Stock 1 Stock 2 flow 1 flow 2 FRACTION 1 fraction 2 Figure 1: Generic structure producing S-shaped growth The student applies the generic structure to model a rabbit population as shown in Figure 2. The model equations are in section 6.1 of the Appendix. Additional Rabbits the Area Can Support Rabbits births deaths death fraction BIRTH FRACTION Figure 2: First model of the rabbit population The model does indeed generate S-shaped growth of the “RABBITS” stock, as shown in Figure 3.D-4646-2 5 51: Rabbits 2: Additional Rabbits the Area Can Support 1000.00 500.00 0.00 2 2 1 1 2 1 1 2 0.00 12.00 24.00 36.00 48.00 Months Figure 3: Stock behavior of the first model 3. MISTAKES AND MISUNDERSTANDINGS Even though the model shown in Figure 2 generates S-shaped growth, it is not a realistic model. By simply fitting the system to a generic structure, the student did not realize that the structure does not represent the system. One of the stocks, “Additional Rabbits the Area Can Support,” is not an accumulation. The model implies that “Rabbits” are born from the stock of “Additional Rabbits the Area Can Support” through the flow of “births.” Thus, hypothetically, a rabbit, before being born, exists as an additional rabbit. Likewise, “Rabbits” that die become “Additional Rabbits the Area Can Support” through the flow of “deaths.” The model in Figure 2 consequently suggests that a rabbit can exist in two states: either as a rabbit, in the stock of “Rabbits,” or as an additional rabbit, in the stock of “Additional Rabbits the Area Can Support.” In the real-world system, however, dead rabbits do not accumulate as additional rabbits, and additional rabbits are not reincarnated into rabbits. Such a formulation of the model is incorrect and does not represent the real system. The student must rebuild the model.D-4646-2 6 In addition, the formulation of the “death fraction” equation is implausible. With the initial values of the two stocks, the “death fraction” initially has the value of 1/2000 rabbits per rabbit per month, implying an average lifetime of a rabbit close to 167 years. 4. OVERCOMING OUR MISTAKES AND MISUNDERSTANDINGS Figure 4 shows a corrected model of the rabbit population. Documented equations are in section 6.2 of the Appendix. Rabbits births deaths BIRTH FRACTION death fraction ~ effect of crowding NORMAL DEATH FRACTIONNORMAL RABBIT POPULATION Figure 4: A corrected model of the rabbit population In Figure 4, the total number of rabbits that the area can support is the converter “NORMAL RABBIT POPULATION.” The model only contains one stock, “Rabbits,” with an inflow of “births” and an outflow of “deaths.” The “death fraction” is the product of a “NORMAL DEATH FRACTION” (set to be one third of the “BIRTH FRACTION”) and the “effect of crowding.” The “effect of crowding,” shown in Figure 5, is a nonlinear function of the ratio of “Rabbits” to “NORMAL RABBIT POPULATION.”3 3 Further papers in the Mistakes and Misunderstandings series will explain how graph functions such as the “effect of crowding” should be constructed.D-4646-2 7 715.00 0.00 effect of crowding0.00 Rabbits / NORMAL 2.00 RABBIT POPULATION Figure 5: The “effect of crowding” graph function The “effect of crowding” has a value of 1 when the population of “Rabbits” is low. The “effect of crowding” then increases to the value of 3 when the number of “Rabbits” is equal to the “NORMAL RABBIT POPULATION” (the ratio of “Rabbits” to the “NORMAL RABBIT POPULATION” is equal to 1).4 Furthermore, the “effect of crowding” continues to increase if the ratio of “Rabbits” to “NORMAL RABBIT POPULATION” becomes greater than 1. Thus, when the ratio of “Rabbits” to “NORMAL RABBIT POPULATION” is 1, all resources are being used. The “effect of crowding” table function outputs a value of 3, and the “death fraction” is equal to the “BIRTH FRACTION.” (Remember that “BIRTH FRACTION” is equal to three times “NORMAL DEATH FRACTION.”) “Births” are then equal to “deaths,” and the system stabilizes at equilibrium with 1000 “Rabbits.” 4 The value of 3 was determined by dividing the “BIRTH FRACTION” by the “NORMAL DEATH FRACTION.”D-4646-2 8 The corrected model produces S-shaped growth, as shown in Figure 6. 1: Rabbits 1000.00 500.00 0.00 1 1 1 1 0.00 12.00 24.00 36.00


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