# 27 What is the logistic equation?

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This article is from the sci.fractals
FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous
contributions by others.

# 27 What is the logistic equation?

It models animal populations. The equation is x -> c x (1 - x),

where x is the population (between 0 and 1) and c is a growth

constant. Iteration of this equation yields the period doubling route

to chaos. For c between 1 and 3, the population will settle to a fixed

value. At 3, the period doubles to 2; one year the population is very

high, causing a low population the next year, causing a high

population the following year. At 3.45, the period doubles again to 4,

meaning the population has a four year cycle. The period keeps

doubling, faster and faster, at 3.54, 3.564, 3.569, and so forth. At

3.57, chaos occurs; the population never settles to a fixed period.

For most c values between 3.57 and 4, the population is chaotic, but

there are also periodic regions. For any fixed period, there is some c

value that will yield that period. See "An Introduction to Chaotic

Dynamical Systems", by R. L. Devaney, for more information.

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