**By Tad Calkins**

The ‘S’ curve is a simple and compelling representation of a system’s growth under inherent limitations. First developed to explain Thomas Malthus’ demographic data, of Malthusian Catastrophe fame, it has been used to model everything from new technology adoption to COVID-19 spread — ‘think flatten the curve’.

One of the most common representations of evolution in a complex system is the ‘S’ curve, also known as the logistic function. It is a visually compelling symbol of cumulative growth with inherent limitation. It is usually shown as a plot of system status versus time, for example population numbers, complexity, market capture, or revenues. The S curve describes a system changing slowly at first, followed by rapid increase, and finally leveling off and changing slowing again. The logistics curve has been found to be broadly applicable to many fields including technology forecasting, epidemiology, population demographics, and others, helping to describe and predict changes in complex systems.

At the heart of a logistics curve’s ability to predict a system’s future state is whatever limitations are incorporated in the system. Since built-in limitations are a feature of many natural world systems, this helps explain the model’s broad applicability.

The origin of the logistic function is in the population growth modeling by Belgian mathematician Pierre-Francois Verhulst (1804-1849). He based his model on the work of the English demographer and economist Thomas Malthus (1766-1834). This equation described the self-limiting growth of a population. Later generalization of Verhulst’s work led to the Volterra-Lotka model of predator-prey interaction, which has been extremely successful in modeling the evolutionary dynamics of systems with competition in biology, ecology, technology, and business. In more recent times the logistic curve has been used extensively to elegantly capture complex system behavior. This has led to many different names for the curve in literature: Logistics curve, Verhulst-Pearl equation, Pearl curve, Richard’s curve (Generalized Logistic), Growth curve, Gompertz curve, S-curve, S-shaped pattern, Saturation curve, Sigmoidal curve, Foster’s curve, Bass model, among others.

The limitations may be explained more rigorously by saying the rate of growth is proportional to both the amount of growth already completed and the amount of growth remaining. Mathematically, this is the time-based rate dependence of the system parameter, N (dN/dt). We want to relate this time dependence to the limitation (limiting value), M, which is also known as the carrying capacity. We relate these through k, the intrinsic growth rate.

dN/dt = kN(1-N/M)

The magnitude of N depends on M, while the dynamics and stability depend on k. The last term (1-N/M) can be thought of as a regulatory mechanism which causes disturbances to monotonically, gradually, fade away. In fact, N approaches the carrying capacity M at a rate dependent on k. At large t (longer time), the globally stable equilibrium is N asymptotically approaching M.

A solution to the equation can be found through separation of variables to get

N=M/[1+Be^{(-kt)}] where B = (M-No)/No

Additional understanding of the logistics equation can be found by plotting equation 2 as shown in this Figure:

Example application?

Perhaps the Singularity is not an exponential curve, but a sigmoidal one. Tim O’Reilly thinks so in “It’s not exponential, it’s sigmoidal” Tim opines that this discrepancy between curve types is why Ray Kurtzweil is wrong in this book, The Singularity. We don’t think that it will matter much over the next three decades, which are what concern us. Over these coming decades, we expect the curve to look vertical.