PREDICTIONS – 10 Years Later - Santa Fe Institute

APPENDIX A
and its solution is
M
N(t) = ———— (2)
1 + e –(at+b)
with b a constant locating the process in time. 4
We can manipulate mathematically Equation (2) in order to put it in
the form
N(t)
———— = e (at+b) (3)
M – N(t)
Taking the logarithm of both sides, we obtain a relationship linear with
time, thus transforming the S-shaped curve of Equation (2) into the
straight line of equation: at + b. The numerator of the left side of Equation
(3) is the new population while the denominator indicates the space
still remaining empty in the niche. In the case of one-to-one substitutions
(see below), the numerator is the size of the new while the
denominator is the size of the old at any given time. If on the vertical
logarithmic scale we indicate the fractional market share instead of the
ratio new / old, we obtain the so-called logistic scale in which 100 percent
goes to plus infinity and 0 percent to minus infinity.
M is often referred to as the niche capacity, the ceiling of the population
N(t) at the end of growth. In order to solve Equation (1), M must
be constant throughout the growth process, but this requirement can be
relaxed for one-to-one substitutions.
One-to-One Substitutions: Two Competitors in a Niche
This case originally treated by J. B. S. Haldane 5 and later reported by
A. Lotka 6 deals with the situation where a mutant, that is a new variety
that we can call N 2 , has a small genetic advantage k over the old variety
N 1 . This means that at every generation the ratio of individuals in
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