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Jan 19, 1995 08:47 AM

by Eldon Tucker

JRC: I'd like to common on exponential population growth. It reminds me of something that I read in a book on chaos. If you have a population (p) that is continually growing at a growth rate of (g), with each generation (or iternation) you have the population increasing by: P = P * (1 + G) But nothing exists in a vacuum. There are always outside forces, external factors that compete for the available resources. There is also a limiting factor, giving us: P = P * (1 + G) * (1 - P) The limiting factor is small when the population is tiny; it grows in its ability to limit the population as it gets bigger and bigger. This equation shows the competition of a growth factor with an external resistance. When we normalize the equation, with "1" standing for the maximum possible population, and other values between 1 and 0 indicating what percentage of the largest size the population is at any point of time, we get some interesting mathematics. When we pick certain growth rates, G, we find that over a period of time, as we iterate the equation, as we see the changing population levels for that rate, a pattern may emerge. The pattern depends upon the rate. Values from 0 to 2 can be picked for the growth rate. For each possible value, when we iterate the equation over and over, we see a particular pattern emerge. For low values, the population drops to zero; a low growth rate leads to eventual extinction. For slightly higher values, the population stabilized to a single level, in stable adjustment with its external environment. For still higher growth rates, we get a cyclic change in the population levels. A population may cycle between, say, seven different levels, and continue to go through those levels over and over again. The plot of stable values that a population attains at different growth rates is called the Bifurcation Curve, and it is a graphic illustration of the theosophical law of cycles. A living system, at a certain rate of growth or self-feedback, ends up in death, a stable state, or an cycle of states. Depending upon the particular growth rate, the cycle may be unstable, with a slight change in the growth rate causing an entirely different type of cycle to arise. An interesting speculation could be regarding the sevenfold cycles that we have in Theosophy. It is said that the knowledge of the Masters, which we have fragments of in our literature, extends only as far as the Solar System. Perhaps *it*, the Solar System, is subject to seven-fold cycles, but elsewhere other cycles apply, like five-fold, fifteen-fold, etc., depending upon the growth rate or the self-iternation in other places. ---- It's been a while since I've looked at the Bifurcation Curve, and I'm writing from memory. I'm not sure that I've explained it well enough for anyone without a previous background in chaos. It's an interesting subject, though, and I thought I'd give a try at writing about it... -- Eldon Tucker (eldon@netcom.com)

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