Chaos in Nature
The applet's first equation x = cx(1 − x) is known as the Standard Logistics Difference Equation. It models, among other things, the way in which populations of reproductive life-forms wax and wane. In this context, the constant c represents the fecundity of the life-form concerned - that is, the ferocity with which it reproduces. This is moderated by a factor representing the degree of ease or difficulty with which the species is able to acquire its needs of life from its environment. The population is thus held in balance by a mechanism analogous to an amplifier with negative feed-back, or an engine with a torque-sensitive speed regulator, or a market economy regulated by interest rates.
Human and animal populations follow a smooth sigmoid curve like that produced by the applet for c < 2·1 in the equation x = cx(1 − x). Try c = 1·85 for a good example of a population sigmoid. Currently, human world population is thought to be at about 0·4 on the Time Graph's vertical scale. This corresponds to about 6,000 million. But this doesn't mean that the Earth cannot support more than 6750 million (0·45 on the graph, the point at which it levels off). It simply means that it cannot do so under the present level of technology and under the currently dominant socio-economic régime.
In complex dynamical systems especially, there is a delay between cause and effect. For human population this delay is very short compared with the reproduction cycle. That is why the curve it follows is a smooth sigmoid. But with insects, due to their high fecundity, the delay is significant. Effects of changes thus trail well behind their causes. This makes some insect populations wax and wane chaotically, corresponding to the behaviour of the equation for c = 3·6 or thereabouts. This has a future significance for humanity in a way which is more immediate and less obvious than one may suppose.
© November 1997 Robert John Morton