PREDICTIONS – 10 Years Later - Santa Fe Institute

APPENDIX A
The Predator-Prey Equations
Two species are locked together in a life-death relationship because one
serves as food for the other; for example, lynx-hare, or big fish-small
fish. The populations feature oscillations. The mathematical description
is obtained from the above equation:
dN 1
—— = a 1 N 1 – k 1 N 1 N 2 prey
dt
dN 2
—— = – a 1 N 2 + k 2 N 1 N 2 predator
dt
where N 1 and N 2 are the populations of prey and predator respectively.
The k constants represent the strength of the interaction between the two
species. More mathematical details can be found in Montroll and Goel’s
work. 3
The Malthusian Case: One Species Only
An illustrative example of this case is a population of bacteria growing
in a bowl of broth. The bacteria act as the agent that transforms the
chemicals present in the broth into bacteria. The rate of this transformation
is proportional to the number of bacteria present and the
concentration of transformable chemicals.
All transformable chemicals will eventually become bacteria. One
can therefore measure broth chemicals in terms of bacterial bodies. If
we call N(t) the number of bacteria at time t , and M the amount of
transformable chemicals at time 0 (before multiplication starts), the
Verhulst equation can be written as
dN ( M - N )
—— = aN ———— (1)
dt
M
275