Human ecology
Discrete mathematics
Logistic map
Continuous vs. discrete

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