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S Curves Everywhere

If you have ever worked with predictive models and trend mapping you are no doubt familiar with the S curve. The ubiquitous S-curve (also known as the sigmoid function) has long been recognized by economists, technologists, and scientists as a strong tool for understanding patterns. Now professor Adrian Bejan at Duke University, with collaborator Sylvie Lorente from the University Toulouse, has developed an exiting new theory that explains the reason for the prevalence of this particular pattern throughout nature and the man-made world.

Their research shows that this phenomenon can be predicted entirely by recognizing in it a flow. The flow is not by diffusion alone, rather it is a combination of tree-shaped “invasion” by convection, followed by “consolidation” by diffusion perpendicular to the invasive lines. The S curve is not unique: its scales depend on the relative magnitude of the speed of the invading lines and the diffusivity perpendicular to the lines. Tree-shaped invasion covers the territory with diffusion much faster than line-shaped invasion. The predicted S-curve flow architecture unites the designs of spreading flows and collecting flows (e.g., mining, fossil fuel extraction, Hubbert peak) in all the realms of nature: animate, inanimate, and human-made.

Economic trends, population growth, the spread of cancer, or the adoption of new technology seem to follow certain patterns, says Bejan. A new technology, for example, begins with slow acceptance, followed by explosive growth, only to level off before “hitting the wall. Rebecca Henderson – one of my favorite MIT Sloan Professors who is now at Harvard Business School had some very interesting theories and practical models for working with S-Curves when developing a successful product and technology strategy. You should also check out her work as well.

Bejan’s theory, known as the constructal law, uses a large river basin as a visual description of flow systems, growing fast and far, with smaller branches growing laterally from the main channels. It is based on the principle that designs of flow systems develop over time by facilitating flow access — reducing and distributing friction or other forms of resistance.

  • A new technology, for example, after a slow initial acceptance can be imagined moving fast through established, though narrow, channels into the marketplace. This is the steep upslope of the “S.”
  • As this technology matures, and its penetration slows, any growth, or flow, moves outward from the initial penetration channels in a shorter and slower manner.

If your involved in predictive models, product development, technology development or analysis this is a very interesting study and I recommend you check it out. You can get the study below:


Ref.: A. Bejan and S. Lorente, The constructal law origin of the logistics S curve, Journal of Applied Physics, 110, 024901 (2011); [DOI:10.1063/1.3606555]


Written by David Frederick

July 21, 2011 at 9:49 AM

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