D-4476-2 1 Exploring S-Shaped Growth Prepared for the MIT System Dynamics in Education Project Under the Supervision of Dr. Jay W. Forrester by Leslie A. Martin October 3, 1996 Vensim Examples added October 2001 Copyright © 2001 by the Massachusetts Institute of Technology Permission granted to distribute for non-commercial educational purposesD-4476-2 3 Table of Contents 1. INTRODUCTION 6 2. THE GENERIC STRUCTURE 2.1 SHIFTING LOOP DOMINANCE 8 2.2 GENERIC STRUCTURE FOR S-SHAPED GROWTH 9 2.2.1 APPLYING THE GENERIC STRUCTURE TO THE RABBIT POPULATION MODEL 13 2.2.2 RUNNING THE RABBIT POPULATION MODEL 15 3. EXAMPLES 20 3.1 CABBAGE PATCH KIDS 20 3.2 PROTESTANTISM 25 3.3 IMMUNE RESPONSE 33 3.3.1 A HEALTHY IMMUNE SYSTEM 33 3.3.2 AN IMMUNE SYSTEM AFFECTED BY HIV 38 4. CONCLUSION 42 5. APPENDIX: MODEL EQUATIONS 42 5.1 S-SHAPED GROWTH GENERIC STRUCTURE 42 5.2 RABBIT POPULATION MODEL 43 5.3 CABBAGE PATCH KIDS MODEL 45 5.4 PROTESTANTISM MODEL 45 5.5 IMMUNE RESPONSE MODEL 47 6. VESIM EXAMPLES 49 8D-4476-2 6 1. Introduction One spring a young couple and small child arrive in Provence, France, carrying bulging suitcases, ardent enthusiasm, and two abnormally large rabbits. They dream of settling down in the countryside and living off of the land. Jean, the husband, is unable to contain the excitement in his voice as he unveils his prized possession to his new neighbors: a carefully folded sheet of paper covered with ink-blotched calculations. Proud to have stumbled upon a get-rich-quick phenomenon known by the name of exponential growth, he explains how his two rabbits will become four then eight then sixteen and soon hundreds upon hundreds... and he is thought to be a madman. So that summer the rabbits become intimate and produce more rabbits which in turn produce even more rabbits. By mid-summer Jean and his wife have more rabbits than they can control. Rabbits are everywhere. Jean and his wife are ecstatic. They dream eagerly of all the profits they will soon make. But one week Jean realizes that some of his rabbits are dying. They are dying of dehydration. The faster, stronger rabbits quickly lap up the small pools of rainwater in the meadow and leave nothing for the others. Jean prays for a rain storm, but the skies in Provence burn with a throbbing sun. Jean and his wife have to travel several miles to a well to get more water. The young couple finds itself unable to bring back enough water to care for all the rabbits. More die. Jean panics. Now the death rate is so high that for each young rabbit that is born a weak, older rabbit dies. Jean’s population has stopped growing. A month later, a year later, he still has the same number of rabbits. The villagers mock the young couple for having bragged about such unrealistic dreams. A disillusioned Jean learns the hard way about limits to growth. True, sustained exponential growth cannot exist in the real world. Eventually all exponential, amplifying processes will uncover underlying stabilizing processes that act as limits to growth. The shift from exponential to asymptotic growth, or from positive to negative feedback, is known as sigmoidal, or S-shaped, growth.D-4476-2 7 Stock positive feedback negative feedback Time Figure 1: S-shaped growth Figure 1 displays a typical S-shaped growth curve. Positive feedback, which generates exponential growth, is tapered by negative feedback, which produces stabilizing growth. S-shaped growth can be observed in a wide variety of phenomena. The spread of fads, rumors, or even a religion is characterized by S-shaped growth. Attention span, concern, and interest also exhibit S-shaped growth. Market saturation and epidemics are classic examples of S-shaped behavior. The cellular growth of a plant and physical and intellectual development in small children, along with the body's immune response, are all subject to S-shaped growth. This paper will begin by exploring population dynamics, taking as an example Jean's population of rabbits. Exploring S-Shaped Growth will first study the shifting loop dominance that produces S-shaped growth. It will then present a generic structure of stocks and flows associated with S-shaped behavior. Finally this paper will examine several different examples of systems which generate S-shaped growth, emphasizing how a whole range of very distinct phenomena can produce similar behavior.D-4476-2 8 2. The Generic Structure 2.1 Shifting Loop Dominance Shifting loop dominance produces S-shaped growth. A system that will exhibit S-shaped growth starts out in a positive feedback loop. A large increase in the positive loop awakens a dormant negative loop. The negative loop does not just spontaneously appear. It is present the entire time, but its strength depends on the strength of a variable in the positive loop. When the positive loop begins to amplify all the variables involved in its cycle, the negative loop is also amplified until the dominance shifts and the negative loop takes over. Figure 2 represents shifting loop dominance. -critical variable+ stabilizing factorsamplifying factors Figure 2: Shifting loop dominance The critical variable in the behavior of Jean’s rabbit population is the number of rabbits that roam around his farm. As the population of rabbits increases, so does the number of rabbit births. More rabbits, more happy couples, more baby rabbits... Births reinforce a positive feedback loop. A negative loop, however, lies dormant. As the population increases the total supply of water stays fixed, so the amount of water available for each rabbit decreases. When the amount of water per rabbit drops low enough, the rabbits will no longer have enough water to sustain themselves, and the weaker rabbits will start to die. The negative loop reduces the population growth rate until the amount of water per rabbit is just large enough to support the rabbit population.D-4476-2 9 Systems that exhibit S-shaped growth behavior are characterized by constraints, or limits to growth. In the case of Jean’s rabbits, the system’s constraint is a fixed supply of water. The constraint fixes the maximum number of rabbits that Jean’s farm in Provence can support. Likewise, the constraint to the spread of a rumor is the number of people who could potentially be reached. That number could be the total student body on a college campus or a nation of TV viewers, depending on how juicy and pertinent the gossip.