## Exponential Population Growth

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)
```