The basic equation of population dynamics is

X = aX (1-X )

This is iterated over the succeeding generations. Each X represents the number of individuals in each generation.

This is also the basic equation of chaos theory, or at least one of them. (This one is a parabola, but any equation whose graph has an upward hump will produce chaos.)

The parameter “a” is low for slow growth, high for ra-pid growth. For “a” values between 1 and 3, there is a sin-gle attractor: the population will settle to a steady con-stant value. For values of “a” between 3 and 3.43, there are two attractors: the population will oscillate between two values, a higher and a lower one, as each adjustment “over-shoots”. (Cf. my essay The Key.) This is the first bifur-cation. For “a” values between 3.43 and 3.57, there are further bifurcations, to 4, 8, 16…attractors. For “a” greater than 3.57 there is an “infinite” number of attractors; we enter a state of chaos. The population will fluctuate wildly, seemingly unpredictably, though the results are strictly deterministic if we carry out the iterations.

It is as if at low speeds we have orderly laminar flow as in a liquid, but at a high speed, when we force it, we get into turbulence. The liquid molecules have no time to adjust.

I tried various values on my computer. Some I did by manual computation. I want to mention one that shocked me deeply.

When I chose a = 4 (which is in the chaos range) and X = .05 (X can range between 0 and 1, i.e. .5 is a middle range), then X = 1.0 (because it’s 4 times .5 times .5), which is the highest possible value – population at its peak. The next iteration gives X = 4 times 1.0 times 0 = 0;
that is zero population, or extinction. Further iterations will then always yield zero. It seems that this crash from maximum flourishing to total extinction will happen whenever that maximum value of 1.0 is reached. The population crashes when it hits the ceiling of sustainability. The chaotic re-gime may not always lead to 1 and then 0, but it may.

One can hardly imagine anything more dramatic. I hesi-tate to draw any real-life conclusions. The model (equation) may be too simplistic.