A discrete version of the logistic map is given by the recurrence relation

(Rewrite the continuous logistic equation in terms of N/K to see how this discrete version is arrived at.) By restricting r as above, the sequence remains positive.

This simple iteration leads to surprisingly complex behaviour. If r < 1, then the sequence x_n tends to 0, signifying extinction. If 1 < r < 2, then the sequence monotonically converges to 1−1/r. If 2 < r < 3, then the sequence converges to 1−1/r in an oscillatory fashion. But at r = 3 this equilibrium becomes unstable and bifurcates; for $3<r<1+\sqrt{6}$ the stable behaviour is an oscillation between two values of r. Then at $r=1+\sqrt{6}$ we get a further bifurcation, and the stable behaviour is oscillations between four values of r. Further period-doubling bifurcations occur until the behaviour becomes chaotic at around r = 3.57, but then oscillations still occur at some greater values of r.

By cobwebbing with the graphs of y = rx(1−x) and y = x for different values of r, a qualitive appreciation of how these behaviours arise is possible.