Logistic equation

Q9: What is the logistic equation?
A9: 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" for more
information.