What is S-shaped growth?

S-shaped growth, also called logistic growth, is the pattern in which a quantity rises slowly at first, then quickly, and finally levels off as it approaches a ceiling. It describes how products are adopted, how populations settle, and how markets saturate. It matters because it shows that fast growth is not permanent and tells you when the ceiling is likely to bite.

How it works

Two feedback loops combine. A reinforcing loop drives exponential growth at the start. As the quantity approaches its limit, a balancing loop gains strength and slows growth. The usual rule is growth = rate × quantity × (1 − quantity / limit). The factor in brackets shrinks from nearly 1 to 0 as the limit is reached. The steepest point comes at half the limit.

The shape holds only if the limit stays fixed and the response to it is quick. If the signal of the limit is slow, the system may overshoot and then swing back. The carrying capacity is the name for the limit in ecology. In marketing, the Bass diffusion model gives an S-shaped adoption curve.

Example

Say a town has 10,000 households that may adopt a service. Each month, adopters grow by 20 percent of themselves times the share still unconverted. With 100 adopters, the gain is about 20. At 5,000 adopters the gain is about 500, the maximum. At 9,000 adopters the gain is about 180. The curve is slow, fast, then slow again, ending near 10,000. This is an illustration with round numbers.

How to model it in Machinations

Use a Pool for adopters and a Pool for the remaining potential, linked by a Converter that moves resources from one to the other. Make the Converter's rate depend on both Pools through state connections. Play the diagram with a chart of the adopters Pool, and use a Lever on the ceiling or the rate to see the curve change.