This article discusses the dynamics of the S-curve phenomenon in nature. The S-curve phenomenon and its physics principle unite the spreading flows with the collecting flows, and the animate flows with the inanimate flows. The history of the volume of heated soil versus time follows an S-shaped curve that is entirely deterministic. It is also predicted that when the invading channels are tree shaped as opposed to single pipes, the entire flow from point to volume occurs faster, more easily, along a steeper S-curve. The S-curves of nature are history records of tree-shaped spreading on areas and volumes that are eventually filled during consolidation by transversal diffusion. The prevalence of S-curve phenomena in nature rivals that of tree-shaped flows, which also unite the animate, inanimate, and human realms. This phenomenon is so common that it has generated entire fields of research that seem unrelated: the spreading of biological populations, cancer tumors, chemical reactions, contaminants, languages, news, information, innovations, technologies, infrastructure, and economic activity, and the evolution of technology performance versus cumulative R&D spending.
The life of every successful innovation— whether it is an idea or a technology— has a remarkably similar trajectory. In the beginning, when familiarity is confined to a few, acceptance spreads slowly to a wider population. At some later point, the spread of the innovation reaches critical mass and begins a sharp rise in the rate of new adopters. Finally, there is a saturation point, and the rate of spreading tails off when the total number of adopters appears to have hit a ceiling.
We can see this flow, from its point source to the big area, and we see it morphing in time, invading and wetting an area. This flow is like the Okavango delta, which grows every year and then “hits the wall” in the Botswana desert, months after the rainy season upstream in Angola. There is no wall in the desert, just empty space, lots of it. Yet, the spreading of the delta stops.
This slow-fast-slow spreading history lies at the heart of the design of nature at the scale of the landscape. The flow of the Okavango delta is part of the inanimate world, unlike the spreading of humanity, with its ideas and technology, on the globe.
When it is graphed, the history of the covered area or volume follows an S-curve. Here is how to predict it:
When a heat pump cools a home during the hot and humid season, it must dump heat into the ambient. Where the human settlement is sparse, the dumping of the heat is not a critical design feature. The atmosphere—the big sewer in the sky—does the job. The same environment serves the heat pump during the cold season, when the heat pump must suck heat from the ambient and inject it (multiplied) into the home. What was sewer in summer is manna from heaven in winter.
It is not nearly as easy when the human settlement is dense. No one wants to live in somebody else's exhaust. In this evolutionary direction, which, by the way, is the future of all humanity, the “environment” is as dear as the plot of land on which the home is built. The heat pumps of the future must dump heat to the ground and suck heat from it.
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How to spread heat from one river mouth (the heat pump) to a finite-size delta (the soil around the home) is a design problem that we solved. First, the heat must be spread by fluid flow through pipes, throughout the territory. During this initial “invasion” phase, the volume of the heated soil around the pipes is small, but it increases at a growing rate. Second, after the hot fluid has invaded all the channels on the territory, the heat is transmitted from the channels perpendicularly to the neighboring soil. This is the consolidation phase: the root “solid” in “consolidation” suggests how heat is filling the soil interstices held between neighboring channels.
We found that the history of the volume of heated soil versus time follows an S-shaped curve that is entirely deterministic, i.e. predictable. Everything about this S-curve is known because both phases, the invasion and the consolidation, are known. We also predicted that when the invading channels are tree-shaped as opposed to single pipes, the entire flow from point to volume occurs faster, more easily, along a steeper S-curve.
The S-curves of nature are history records of tree-shaped spreading on areas and volumes that are eventually filled during consolidation by transversal diffusion.
When anything spreads on a territory, the curve of territory size versus time is S-shaped: slow initial growth is followed by much faster growth, and finally by slow growth again. The corresponding curve of the rate of spreadings versus time is bell-shaped. This phenomenon is so common that it has generated entire fields of research that seem unrelated: the spreading of biological populations, cancer tumors, chemical reactions, contaminants, languages, news, information, innovations, technologies, infrastructure, and economic activity, and the evolution of technology performance versus cumulative R&D spending (as in Mechanical Engineering, December 2009). The S-curve phenomenon is also visible in the history of citations received by every publication, and is responsible for the increase, as time passes, in the h-index of every author.
The S-curve is a natural phenomenon, not the mathematical expression of a particular S-shaped curve. In fact, by analyzing the invasion-consolidation flow we showed that the S-shaped curves are not unique. The natural phenomenon is the observation that in many and highly diverse flow systems the covered territory increases in time according to a curve that resembles an S.
The prevalence of S-curve phenomena in nature rivals that of tree-shaped flows, which also unite the animate, inanimate, and human realms. This is no coincidence. Both phenomena are manifestations of the natural tendency to generate evolving designs that flow more easily. This tendency is the constructal law.
These predictions apply equally to the behavior of collecting flows, which draw streams from areas or volumes and carry them to discrete points. For collecting flows, the scales of the S inflexion point indicate the all-important regime of peak production rate, known as the “Hubbert peak” in oil extraction. It is not a coincidence that oil extraction technology has evolved from single-line invasion (the single well) to tree invasion.
The S-curve phenomenon and its physics principle unite the spreading flows with the collecting flows, and the animate flows with the inanimate flows. In the human realm, they unite the designs for urban infrastructure with the underground architectures for mining (coal, metals, etc.), and teach the physics basis of “limits to growth” and when a spreading population and technology can be expected to “hit the wall.” They also cover the periodic phenomena of spreading and collecting, such as respiration (inhaling, exhaling), drug delivery, excretion, rainwater (from river basin to delta), and blood circulation.
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There is a lot of doomsday talk today about the world being in an “explosion” phase, or on an “exponential growth” curve. In view of the constructal-law origin of the S-curve, all such talk is about the first part of an S-curve phenomenon. What looks like explosion today will look like hitting the wall tomorrow.
This is how a new technology spreads. The invasion-consolidation scenario happens naturally, not because industry and government leaders dictate it. Decades ago, only a few countries made autos. Now it seems that autos are made everywhere, but the advanced countries make them better, with more modern designs and methods. Each such design is the start of its own S-curve of how it spreads on the globe.
All these phenomena are described in their own language by the constructal-law prediction of the S-curve. If translated correctly, the S-curve reveals when “exponential growth” must end and be replaced by “hitting the wall.”
This research was supported by a grant from the National Renewable Energy Laboratory, Golden, Colo.