PREDICTIONS – 10 Years Later - Santa Fe Institute

APPENDIX A
the two varieties will be multiplied by 1/(1– k). In n generations then we
will have the ratio of populations as
N 2 R o N 2
—— = ——— where R o = —— at t=0 (4)
N 1 (1–k) n N 1
Whenever k is small—in biology it is typically around 0.001—
Equation (4) can be approximated as
N 2
—— = R o e kn (5)
N 1
which is the same as Equation (3) since N 2 = M – N 1 . The only difference
between this case and the one in the previous section is that here
we have an initial condition (R o ) while there we had a final condition
(M). In a typical application the single occupancy implies determination
of the final ceiling M from early data, a procedure subject to large uncertainties
addressed in detail in Appendix B. The two-competitor
typical case deals with evolution in relative terms. The final ceiling of
the process is by definition equal to 100 percent. The determination of
the trajectory is not sensitive to the size of the niche, which may be
varying throughout the substitution process. This permits a reliable forecast
of the competitive “mix” even in situations where the overall
“market” may be going through unpredictable convulsions.
Multiple Competition: Many Competitors at the Same Time
One approach dealing with the many-competitor case developed by
Nakicenovic 7 is based on successive one-to-one substitutions. The ratio
of population i divided by the sum of all other populations follows
Equation (3) for all competitors except one, typically the leader, whose
share is calculated as what remains from 100 percent after all other
shares have been subtracted. Thus, the trajectory of every competitor
will in general be an S-curve—either a growing one or a declining
277