:

:

:

:

*A _{n}*

0
4

Combined reproduction and starvation rate *r*

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. - 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.

- Inspired and largely based on Sean Whalen's Java applet in Monte Carlo visualization of bifurcations in the logistic map.
- Logistic Map Simulation A Java applet simulating the Logistic Map by Yuval Baror.