D-4432 1 GENERIC STRUCTURES: S-SHAPED GROWTH I Produced for the MIT System Dynamics in Education Project Under the Supervision of Dr. Jay W. Forrester Sloan School of Management Massachusetts Institute of Technology BY Marc Glick Terri Duhon August 24, 1994 Vensim Examples added October 2001 Copyright © 2001 by MIT Permission granted for non-commercial educational purposesD-4432-2 3 TABLE OF CONTENTS ABSTRACT ............................................................................................................... 4 INTRODUCTION ...................................................................................................... 4 S-SHAPED GROWTH ............................................................................................... 5 S-SHAPED GROWTH STRUCTURE 1..................................................................... 7 EXERCISE 1: S-SHAPED GROWTH STRUCTURE 1............................................. 9 S-SHAPED GROWTH STRUCTURE 2..................................................................... 11 EXERCISE 2: S-SHAPED GROWTH STRUCTURE 2............................................. 14 STRUCTURE SIMILARITIES................................................................................... 17 CONCLUSION........................................................................................................... 19 APPENDIX A - Documentation.................................................................................. 20 S-SHAPED GROWTH STRUCTURE 1 (From Figure 5a)......................................................20 S-SHAPED GROWTH STRUCTURE 2 (From Figure 7b)......................................................21 APPENDIX B - Solutions to Exercises........................................................................ 24 EXERCISE 1..........................................................................................................................24 EXERCISE 2..........................................................................................................................25 APPENDIX C - Technical Documentation.................................................................. 29 S-Shaped Growth Structure 1..................................................................................................29 S-Shaped Growth Structure 2..................................................................................................29 VENSIM EXAMPLES………………………………………………………………… .31D-4432 4 ABSTRACT Generic Structures: S-Shaped Growth I is an introduction to the concept of generic structures, along with specific examples of such a structure. This paper assumes knowledge of STELLA1 software, and simple system dynamics structures such as positive and negative feedback loops, exponential growth, and S-shaped growth. Generic Structures: S-Shaped Growth I will examine two different structures that generate S-shaped growth. We will look at the multiple behaviors that each structure generates, and the simulations of each structure that produce the same behaviors. Then causal loop diagrams will be utilized to explain similarities between the structures. INTRODUCTION Generic structures, also known as transferable structures are structures that can be found in different situations. Generic structures facilitate learning by allowing one to transfer knowledge of one system to another. Figure 1 shows how a simple generic structure can be used to model different systems, simply by altering the names and parameter values depending on the specific system being modeled. Stock Generic Structure: Inflow OutflowBathtub Example 1: Water from Faucet Water through Drain Population Example 2: Births Deaths 1 STELLA is a registered trademark of High Performance Systems, Inc..D-4432-2 5 Figure 1 - Model of a generic structure followed by two specific models based on that structure. Once a person understands a generic structure, that person’s knowledge of other similar systems is expanded. “A generic structure that many people are familiar with is S-shaped growth.” The previous sentence is a popular misconception. S-shaped growth is commonly classified as a generic structure. However, it is not a generic structure, but rather a behavior. A structure generates a behavior, but a behavior is not a structure. They are not synonymous. Several different structures can exhibit the same behavior. Also, several different behaviors can be generated by the same structure. This can be achieved by changing either the initial values of the stock, or the parameter values. Despite the lack of direct relation between behavior and structure, the different behaviors a structure can generate are related. Additionally, different structures that exhibit similar behaviors must also be related. This is the focus of Generic Structures: S-Shaped Growth I. S-SHAPED GROWTH S-shaped growth is also called sigmoidal growth. Figure 2 shows a general example of S-shaped growth. The curve begins at less than the equilibrium amount. Initially, exponential growth is the dominant behavior of the curve. Then, at the inflection point2, the curve begins an asymptotic approach to an equilibrium. Examples of systems that exhibit this behavior are the growth of rumors, sale of new products, epidemics, and populations with limited resources. 2 The Inflection Point is merely the point where the transfer from exponential to asymptotic behavior occurs.D-4432 6 Stock Equilibrium Level Asymptotic Growth Inflection Point Exponential Growth TimeFigure 2 - STELLA generated graph of generic S-shaped growth. The examples listed above each exhibit S-shaped growth under certain conditions, yet they cannot all be modeled by the same structure. This paper examines two distinctly different structures that produce S-shaped growth. These structures are pictured in Figures 3 and 4. Population Births Deaths Population DensityBirths Normal ~ Deaths Multiplier Average Lifetime Area Figure 3 - S-Shaped Growth Structure 1D-4432-2 7 Non-Owners Owners Buyers Probability of Contact With Owners Probability of Buying Product Break Product Lifetime Population Interactions Rate Figure 4 - S-Shaped Growth Structure 2 S-SHAPED GROWTH STRUCTURE 1 The first structure can be used to model growth with a limiting factor. Two examples of systems governed by this structure are the growth of rabbits in a limited area (Figure 5a) and the construction of buildings