Logistic Map

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

where:

is a number between zero and one, and represents the ratio of existing population to the maximum possible population at year n, and hence x₀ represents the initial ratio of population to max. population (at year 0)

r is a positive number, and represents a combined rate for reproduction and starvation.

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.

However, as a demographic model the logistic map has the pathological problem that some initial conditions and parameter values lead to negative population sizes. This problem does not appear in the older Ricker model, which also exhibits chaotic dynamics.

The r=4 case of the logistic map is a nonlinear transformation of both the bit shift map and the case of the tent map.