At r = 2, the population quickly reaches a steady state.

At r = 3.1, population now oscillates between two steady states.

At r = 3.5, the population oscillates among four steady states.

As r approaches 4, the steady states vanish and the population jumps around in complete disorder.

The purpose of May's bifurcation diagram is to be able to see the successive period doublings that constitute the descent into chaos.

Series of period-doubling plots.

Stack the data from all parameters into a single column and plot the column.

A color-coded bifurcation diagram is more informative.

Here, the time series is collapsed on itself, and we see each parameter's i th iteration at the same place on the x-axis.

A radial plot.

A bifurcation diagram created with Matlab.