The logistic difference equation (or logistic map) is defined as follows:
This equation, and more complicated variations of it, have been studied by ecologists since the 1950s. It can be used as a simple model of the population dynamics of many animals like fish and insects. Given a population xn at the nth year and a value for the growth parameter r, this equation returns xn+1, the population for the next year.
What is especially interesting about the logistic difference equation though is the qualitative change in the behaviour of the system for different values of the parameter r. Depending on what we set r to be, our modelled population may go extinct. Or it might settle at a particular size and stay there forever. Or it could bounce between two sizes, or between four sizes, or even bounce all over the place chaotically!
So as a first post for the new year (and the new blog!), we'll use some of the gorgeous Racket plotting facilities to explore this logistic difference equation and its chaotic behaviour.
