Discrete mathematics

Logistic map

The logistic map (and its continuous counterpart, the logistic equation) are well studied models in population biology. Its formula is very simple:

**x(n+1) = mu * x(n) * (1 - x(n))** {Eqn. 1)

where x is the relative size of the population (ie. it varies between 0 and 1) and mu is the growth coefficient.

To find stable solutions to this equation, an analytical approach might be taken:

If **x(n+1) = x(n)** then x is stable

Let **x(s) = x(n+1) = x(n)** be a stable solution of this equation {Eqn. 2}

Therefore, subtituting eqn. 2 into eqn. 1, we have:

**x(s) = mu * x(s) * (1 - x(s))** {Eqn. 2.1}

Clearly the two solutions of this equation are either:

**x(s) = 0** {Eqn. 2.2a}

or

**1 = mu * (1 - x(s))** {Eqn. 2.2b}

From eqn. 2.2b, we get

**1 / mu = 1 - x(s)**

**x(s) = 1 - 1 / mu** {Eqn. 2.3}

Graph of eqn. 2.3

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Created 10/12/99