The
logistic map is a
polynomial mapping (equivalently,
recurrence relation) of
degree 2, often cited as an archetypal example of how complex,
chaotic behaviour can arise from very simple
non-linear dynamical equations. The map was popularized in a seminal 1976 paper by the biologist
Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by
Pierre François Verhulst. Mathematically, the logistic map is written
![](http://info.babylon.com/onlinebox.cgi?rt=GetFile&uri=!!ARV6FUJ2JP&type=0&index=1081)
where
![](http://info.babylon.com/onlinebox.cgi?rt=GetFile&uri=!!ARV6FUJ2JP&type=0&index=311)
is a number between zero and one that represents the ratio of existing population to the maximum possible population. The values of interest for the parameter
r are those in the interval (0, 4]. This nonlinear difference equation is intended to capture two effects:
- reproduction where the population will increase at a rate proportional to the current population when the population size is small.
- starvation (density-dependent mortality) where the growth rate will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment less the current population.