James Gleick

Chaos: Making a New Science


Chapter 3. Life's Ups and Downs

Physics vs. Ecology

Discrete maps instead of differential equations.

Example: x[n+1]= rx[n]. Malthusian exponential explosion: x[5] = rrrrrx[0].

Logistic difference equation: large populations exhaust resources, so new population smaller. x[n+1] = rx[n](1-x[n]).

The logistic equation generates chaos at high values of r (biotic potential). 1950s ecologists must have seen oscillations, but assumed it was oscillation around an asymptotic equilibrium.

James Yorke: Institute for Physical Science and Technology, U. of Maryland.

Robert May, theoretical physicist turned mathematical biologist.

Yorke's theorem: Any one-dimensional system that has a 3-cycle also has cycles of every other length and chaos.

A. N. Sarkovskii:

Frank Hoppensteadt, NYU Courant Institute of Mathematical Sciences.

May saw pictures and collected similar equations from genetics, economics, fluid dynamics.

Culture war in biology:

Instability in ecology over population biology

False dichotomy: Either steady determinism or random noise. Bifurcation diagram of logistic equation showed both orderly and disorderly regimens. Highly structured across r values, but indistinguishable from noise at fixed r values.

May studied response to interventions in evolution of system to model effects of innoculation programs in epidemiology. Perturbations downward can occasion sharp upward spikes. This was observed in Britain's rubella vaccination program. Spikes lead health officials to conclude that the program was a failure.

New successes led to more cooperation between biologists and physicists (Kuhn: nothing succeeds like success).

May saw that there was nothing intrinsically biological about chaos.

Messianic article in Nature (1976): Every science student should be given a calculator and told to play with the logistic equation to overcome the misleading textbook impressions of a standard science education.

"The mathematical intuition ... developed [by a standard scientific education] ill equips the student to confront the bizarre behavior exhibited by the simplest of discrete nonlinear systems. Not only in research, but also in the everyday world of politics and economics, we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties."