The logistic map was originally used to approximate an animal population over time. Mathematically, it is written as:

A n+1 = rA n (1 - A n )

where:

A n represents the population at year n, with 0 <= A n <= 1. Of course, in reality a population is a whole number, but this makes the maths simpler. A value of 1 means the maximum population i.e. the capacity of the physical environment.

represents the population at year n, with 0 <= A <= 1. Of course, in reality a population is a whole number, but this makes the maths simpler. A value of 1 means the maximum population i.e. the capacity of the physical environment. r represents a combined rate for reproduction and starvation, with r > 0.

All initial populations A 0 eventually settle into one of three behaviours:

A fixed value. Periodic oscillation between values. Chaotic i.e. unpredictable values.

A bifurcation diagram shows these three behaviours. The x-axis shows r and the y-axis shows the population. The population eventually stabilises to a fixed value for r = 1 to 3, and you can see the periodic behavior starting after this, eventually becoming chaotic at around r = 3.5699457...

The Monte Carlo simulation on the left uses repeated random sampling to produce the bifurcation diagram. The algorithm is roughly:

For each value of r, generate a multiple points with a random initial population and random maximum age (remaining number of iterations). Set the population of each point to A n+1 = rA n (1-A n ) and update its remaining number of iterations. If the remaining number of iterations for any point reaches zero, then we pick a new random initial population and age for that point. Clear the plotting area and plot each point at (r, A n ). Go to 2.

The plot is updated for every iteration, so you can see how the populations are changing over time.