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Generic Structures: First-Order Positive Feedback D-4474-1 1 Generic Structures: First-Order Positive Feedback Produced for the System Dynamics in Education Project MIT System Dynamics Group Under the Supervision of Dr. Jay W. Forrester Sloan School of Management Massachusetts Institute of Technology by Stephanie Albin Mark Choudhari March 8, 1996 Vensim Examples added October 2001 Copyright © 2001 by the Massachusetts Institute of Technology Permission granted to copy for non-commercial educational purposes 4 D-4474-2 Table of Contents 1. INTRODUCTION 5 2. EXPONENTIAL GROWTH 6 2.1. EXAMPLE 1: POPULATION-BIRTH SYSTEM 7 2.2. EXAMPLE 2: BANK BALANCE-INTEREST SYSTEM 7 2.3. EXAMPLE 3: KNOWLEDGE-LEARNING SYSTEM 8 3. THE GENERIC STRUCTURE 8 3.1. MODEL DIAGRAM 9 3.2. MODEL EQUATIONS 9 3.3. MODEL BEHAVIOR 11 4. BEHAVIORS PRODUCED BY THE GENERIC STRUCTURE 12 5. SUMMARY OF IMPORTANT CHARACTERISTICS 14 6. USING INSIGHTS GAINED FROM THE GENERIC STRUCTURE 15 6.1. EXERCISE 1: SOFTWARE SALES 16 6.2. EXERCISE 2: MAKING FRIENDS 17 6.3. EXERCISE 3: ACCOUNT BALANCE 18 7. SOLUTIONS TO EXERCISES 18 8. APPENDIX - MODEL DOCUMENTATION 20 9. VENSIM EXAMPLES 22 D-4474-2 5 1. Introduction Generic structures are relatively simple structures that recur in many diverse situations. In this paper, for example, the models of a bank account and a deer population are shown to share the same basic structure! Transferability of structure between systems gives the study of generic structures its importance in system dynamics. Road Maps contains a series of papers on generic structures. In these papers, we will study generic structures to develop our understanding of the relationship between the structure and behavior of a system. Such an understanding should help us refine our intuition about the systems that surround us and allow us to improve our ability to model the behaviors of systems. We can transfer knowledge about a generic structure in one system to understand the behavior of other systems that contain the same structure. Our knowledge of generic structures and the behaviors they produce is transferable to systems we have never studied before! It is often the case that the behavior of a system is more obvious than its underlying structure. Systems are then referred to by the common behaviors they produce. However, it is incorrect to assume such systems are capable of exhibiting only their most popular behaviors, and we need to look more closely at the other behaviors possible. In effect, our study of generic structures examines the range of behaviors possible from particular structures. In each case, we seek to understand what in the structure is responsible for the behavior produced. This paper introduces a simple generic structure of first-order linear positive feedback. We illustrate our study of the positive feedback structure with many examples of systems containing the structure. You will soon begin to recognize the structure in many of the models you see and build. In the exercises at the end of the paper, we provide you with an opportunity to see how you can transfer your knowledge between different systems. 6 D-4474-2 2. Exponential Growth Exponential growth is produced by a positive feedback loop between the components of a system. The characteristic behavior of exponential or compound growth is shown in Figure 1. Many systems in the world exhibit the exponential behavior of a process feeding upon itself. For example, in an ecological system, the birth of deer increases the deer population, which further increases the number of deer that are born. At your bank, your account balance is increased by the interest you earn on it, and the larger your balance gets, the more interest you earn on it! Another system which can be said to exhibit exponential growth is the knowledge-learning system. Simply stated, the more you know, the faster you learn, and then gain even more knowledge. 1: STOCK 2000.00 1050.00 1 1 1 100.00 1 0.00 3.00 6.00 9.00 12.00 Time Figure 1 Exponential Growth Curve These very different systems exhibit the same behavior pattern because the relationship between their components is fundamentally the same. They all contain the first-order linear positive feedback generic structure. Population is related to births in the same way your bank balance is related to the interest it earns and knowledge is related to learning. Let us begin to explore the nature of this relationship by looking at the structure of our three example systems. D-4474-2 7 2.1. Example 1: Population-Birth system Our first example shown in figure 2 is taken from the ecology of a deer population. The deer population is the stock, and the births of deer is the net inflow to the stock. The amount of deer births is equal to the amount of female deer that reproduce and is calculated as a compounding fraction (called birth fraction) of the total deer population. The births = deer population * birth fraction. deer population births birth fraction Figure 2 Model of a population-birth system 2.2. Example 2: Bank balance-interest system Our second example in figure 3 shows the relationship between a bank balance and the interest it earns. The bank balance is the stock, and the interest earned is the inflow to the stock. The amount of interest earned every year is equal to a compounding fraction (interest rate) of the bank balance. The interest earned = bank balance * interest rate. bank balance interest earned interest rate Figure 3 Model of a bank balance system 8 D-4474-2 2.3. Example 3: Knowledge-learning system This third example is of a more abstract system. Figure 4 shows that the stock of knowledge is increased by the net inflow learning. The rate of learning is the knowledge spread out over the time to learn. Therefore, learning is equal to the knowledge divided by the time to learn. The time to learn is known as the time constant of the system. Basically, the more you know, the faster you learn. The learning = knowledge/time to learn. knowledge learning time to learn Figure 4 Model of a knowledge-learning system Note that in the equation of the rate, we divide the stock (knowledge) by time to learn. This is analogous to multiplying by a fraction as seen in examples 1 and 2. The units for time to learn are the units of time like weeks or months. The units of the compounding fraction would be unit/unit/time. As is evident from Figures 2, 3, and 4, all three of these systems have essentially the same structure. 3. The Generic Structure We will now study the generic structure and then explore the possible behavior it can produce. D-4474-2 9 3.1. Model Diagram stock flow compounding fraction or time constant Figure 5 Model of the underlying generic structure The model diagram of the first-order positive generic structure is shown in figure 5. In the equation of the rate, we multiply the stock by the compounding fraction or divide the stock by the time constant. The time constant is simply the reciprocal of the compounding fraction. 3.2. Model Equations The equations for the generic structure are stock(t) = stock(t - dt) + (flow) * dt DOCUMENT: This is the stock of the system. It corresponds to the deer population, the bank balance, and the stock of knowledge in the examples above. UNIT: units INFLOWS: flow = stock*compounding_fraction DOCUMENT: The flow is the fraction of the stock that flows into the system per unit time. It corresponds to the births, the interest earned, and the learning in the examples above. UNIT: units/time compounding_fraction = a constant 10 D-4474-2 DOCUMENT: This is the compounding fraction or growth factor. It determines the inflow to the stock. The compounding fraction corresponds to the birth fraction and the interest rate in the examples above. It is the amount of units added to the stock for every unit already in the stock, every time. UNIT: units/unit/time Note: If we had a time constant instead of a compounding fraction the equation for the flow and the time constant would be INFLOWS: flow = stock/time constant UNIT: units/time D-4474-2 11 time_constant = a constant DOCUMENT: This is the time constant. It is the adjustment time for the stock. It corresponds to the time to learn in the above example. This is the time for each initial unit to compound into a new unit. UNIT: time From the comparison of the two possible equations for the rate, we notice that the multiplier in the rate equation is given by 1 multiplier (for the stock) in the rate equation = compounding fraction = time constant 3.3. Model Behavior The characteristic feature of exponential growth is its constant doubling time, i.e. the time it takes for the stock to double remains constant. For example in Figure 6, it takes the stock 7 years to double from 100 to 200 and also the same time to double from 800 to 1600! 1: STOCK 2: FLOW 3: COMPOUNDING FRACTION 1: 1600.00 2: 3: 1.10 1: 800.00 1 2: 3 3 3 3 3: 0.10 1 1 1: 1 2 2: 0.00 2 2 3: -0.90 2 0.00 7.00 14.00 21.00 28.00 Time Figure 6 Results of a simulation of the positive feedback generic structure. To find the doubling time of the stock, we need the time constant of the system. The time constant may either be given to you directly (as the time to learn in Example 3 above), or 12 D-4474-2 if you have a compounding fraction, the time constant is simply it’s reciprocal.
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