PREDICTIONS – 10 Years Later - Santa Fe Institute

10. IF I CAN, I WANT
Oscillations can also be observed in the absence of a formal predator—for
example, in a rat colony that is given a fixed daily food
supply. 4 The larger, more aggressive rats may start hoarding food in
order to attract females when food becomes scarce. The timid rats huddle
together, do not reproduce, and eventually die. The overall
population decreases, and at the end only aggressive rats survive. They
then propagate rapidly to reach a condition of an overpopulation of rats
“enriched” in aggressiveness. A new cycle of hoarding may start, and
eventually, extinction by degenerative belligerence may result.
But one can find oscillations in populations of much less aggressive
species. Sheep introduced in Tasmania in the early nineteenth century
grew in numbers to fill their ecological niche of about 1.5 million by the
1840s. 5 During the following one hundred years their population oscillated
erratically around that ceiling, with diminishing amplitude. These
fluctuations reflect changes in the birth and death rates, which in turn
reflect the economy, epidemics, climatic changes, and other phenomena
of a chaotic nature.
In scientific terms chaos is the name given to a set of ongoing variations
that never reproduce identically. It was first observed when
mathematical functions were put in a discrete form. Since populations
are made up of discrete entities, a continuous mathematical function
offers only an approximate model for the real situation. Discretization is
also dictated by the need to use computers, which treat information in
bits and pieces rather than as continuous variables.
There are many ways to put natural growth in a discrete form. In
their book The Beauty of Fractals, 6 H. O. Peitgen and P. H. Richter
devote a large section to chaos. The section, which comprises threefourths
of their book, starts with Verhulst 7 dynamics and ends with a
discrete Volterra-Lotka system. They produce chaos mathematically by
discretizing the natural law of growth described in Chapter One. They
cast Verhulst’s law into a difference, rather than a differential equation.
As a consequence the solution becomes a sequence of small straight
segments. The overall population rises similarly to the continuous case,
but now it does not reach the ceiling smoothly. It overshoots, falls
back, and goes through oscillations. For some parameter values these
oscillations do not subside; they either continue with a regular—simple
or complicated—pattern or simply break into random fluctuations,
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