James Gleick
Chaos: Making a New Science
Chapter 3. Life's Ups and Downs
Physics vs. Ecology
- Ecological phenomena more complex than physical phenomena.
- Ecologists know their models are approximations.
- Ecology equations simpler.
- Physical equations of double pendulum linked by spring are set up a
priori. Ecological equations are set up to fit empirical data.
Discrete maps instead of differential equations.
- Markov property: next value is a function of the current value.
- To compute the next year's population, apply the map to this year's
population.
- Discrete feedback
- Microphone: out of control.
- Thermostat: in control.
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.
- Unusual mathematician interested in biology.
- Knew Smale's work and conjecture.
- Fluid dynamicist in institute gave Yorke a copy of Lorenz's paper.
- Yorke realized Lorenz's equation was an applied counterexample
to Smale's conjecture of the instability of chaos.
- Yorke sent it to Smale. Every copy of Lorenz's paper circulating at
Berkeley had Yorke's address label on it.
- York felt physicists learned not to see chaos, since it is everywhere
in nature (history, baseball).
- Physics is writing and solving differential equations.
- Textbooks focus on solutions.
- Sampling fallacy: Most equations don't have solutions. The ones that
do involve conserved quantities that don't exist in chaotic ones.
- So when people saw chaos, they explained it away. Investors see cycles
in markets. Experimentors blame the experimental setup or treat disorder
as noise.
- Stanislaw Ulam: "Calling chaos nonlinear dynamics is like calling
zoology the study of non-elpephant animals."
- Paper in American Mathematical Monthly: "Period Three Implies
Chaos".
- This named the study of Chaos.
Robert May, theoretical physicist turned mathematical biologist.
- 1971: Talked to biologists at Princeton when at Institute for Advanced
Study.
- Biologists usually not very mathematical.
- Started studying predators and prey. Simplified back to looking at
logistic equation for a single population (repeated theme in development
of chaos theory: recall Kuhn on simplification and isolation of an anomaly).
- Studied equation as r increases.
- Steady state extinction, 2 cycles, 4 cycles, 8 cycles, ....with increasing
equilibrium values.
- May studied what happens after "accumulation point". Chaotic
behavior, never visiting the same population twice. Windows of order occasionally
appear with odd periods (3, 7, etc.) , followed by a faster regimen of
bifurcations and chaos.
Yorke's theorem: Any one-dimensional system that has a 3-cycle also
has cycles of every other length and chaos.
A. N. Sarkovskii:
- Stopped Yorke on sightseeing cruise in East Berlin.
- Russian mathematicianwho had the result earlier: "Coexistence
of Cycles of a Continuous Map of a Line into Itself".
- Kolmogorov began strong school of Chaos research in Soviet Union in
the 1950s.
- In Soviet union, chaos revolution wasn't new.
- Smale and Yorke caught on faster in Soviet Union than here.
Frank Hoppensteadt, NYU Courant Institute of Mathematical Sciences.
- Mathematician, with interest in biology.
- Did iterative plot of bifurcation graph of logistic equation on Control
Data 6600.
May saw pictures and collected similar equations from genetics, economics,
fluid dynamics.
Culture war in biology:
- Vague ecology.
- Clean molectular biology.
Instability in ecology over population biology
- Steady state with some exceptions.
- Erratic fluctuation with exceptions. Fluctuation due to noice obliterating
deterministic signal.
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).
- New York City measles epidemics
- Canadian lynx population over 200 yrs.
- Molecular biologists do systems studies of proteins.
- Physiologists looked at chaotic features of heartbeat.
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."