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