Another type of exponential growth is the Malthusian growth model. Sometimes called the simple exponential growth model, it is essentially exponential growth based on a constant rate of compound interest. Named after the Reverend Thomas Robert Malthus. At best, it can be described as an approximate physical law as it is generally acknowledged that nothing can grow at a constant rate indefinitely (Cassell's "Laws Of Nature," Professor James Trefil, 2002). Professor of Populations Joel E. Cohen has that the simplicity of the model makes it useful for very short-term predictions and of not much use for predictions beyond 10 or 20 years ("How Many People Can The Earth Support," 1995).

The Malthusian growth model is the direct ancestor of the logistic function. Pierre Francois Verhulst first published his logistic growth function in 1838 after he had read Malthus' "An Essay on the Principle of Population."

The sigmoid function

A logistic function or logistic curve models the S-curve of growth of some set P. The initial stage of growth is approximately exponential; then, as competition arises, the growth slows, and at maturity, growth stops. The untrammeled growth can be modelled as a rate term +rKP (a percentage of P). But then, as the population grows, some members of P (modelled as − rP2) interfere with each other in competition for some critical resource (which can be called the bottleneck, modelled by K). This competition diminishes the growth rate, until the set P ceases to grow (this is called maturity).

A logistic function is defined by the mathematical formula:

for real parameters a, m, n, and τ. These functions find applications in a range of fields, from biology to economics.

For example, in the development of an embryo, a fertilized ovum splits, and the cell count grows: 1, 2, 4, 8, 16, 32, 64, etc. This is exponential growth. But the fetus can grow only as large as the uterus can hold; thus other factors start slowing down the increase in the cell count, and the rate of growth slows (but the baby is still growing, of course). After a suitable time, the child is born and keeps growing. Ultimately, the cell count is stable; the person's height is constant; the growth has stopped, at maturity.

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