PREDICTIONS – 10 Years Later - Santa Fe Institute

APPENDIX A
Mathematical Formulation of S-Curves and the
Procedure for Fitting Them on Data
The behavior of populations growing under Darwinian competition is
described in general by the celebrated Volterra-Lotka equations. These
equations—one for every competing species—specify the rate of
growth which is made up of several terms.
dN i
n
—— = a i N i – ∑b ij N i N j
dt j=1
The positive term is proportional to the actual size. The parameter a i
reflects the capacity of the species to reproduce, or, in other words, the
rate of growth of population i in the absence of predators. The negative
terms represent growth reduction due to loss of life caused by the predator.
The parameter b ij can be interpreted as the rate at which predator j
will grow assuming there is an infinite amount of prey i. The properties
of the solutions to these equations have been described by Elliot Montroll
and N. S. Goel 1 and more recently by M. Peschel and W. Mendel. 2
We can distinguish the following special cases.
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