PREDICTIONS – 10 Years Later - Santa Fe Institute

10. IF I CAN, I WANT
look like an S-curve but rather like a straight line with some inevitable
fluctuations superimposed.
Statistical fluctuations—random in nature—can be found in both
rates of tourism and natural growth processes. For the former this randomness
is purer than for the latter, where fluctuations have to “arrange
themselves” so as to average on the overall trend of the S-curve. Individual
events may appear to be random, but groups of events must obey
the fundamental law governing the natural-growth process.
Yet there is another phenomenon that bridges the gap between the
randomness that characterizes tourism and the S-curves that characterize
natural-growth processes: the phenomenon of chaos.
FROM S-CURVES TO CHAOS
Oscillatory behavior is one of the mathematical solutions of the
Volterra-Lotka equations for a predator-prey system. 2 Such a system
can be described as follows, with the lynx as predator and hare as prey.
The hare population grows at a constant rate (exponentially) in the absence
of lynxes, while the lynx population declines at a constant rate—
through starvation—in the absence of hares. When they coexist, the
number of hares consumed is proportional to the number of lynxes,
while the growth in the lynx population is proportional to the number of
hares. This is a verbal description of the Volterra-Lotka system of differential
equations.
Without having to go into the mathematical solution, we can see that
starting from a situation where both species are scarce, the population of
hares will grow rapidly by reproduction. Then the lynx population also
increases until there are so many lynxes that the hare population declines.
This in turn causes the lynx population to decline, and with both
populations low, the cycle starts again.
The example used here is realistic. In the northern forests the populations
of these two species have been observed to fluctuate with a cycle
of ten years, and the pattern observed for the lynx population lags the
pattern for hares. 3 Differences from the real case occur, however, because
other species interfere making these oscillations less regular but
also increasing the stability of the ecological system.
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