Deterministic Chaos

The Logistic Map

We could use the equation,

Pn+1 = (1 + B - D)⋅Pn - C*(Pn - 1)⋅Pn/2.

but our later calculations will be much simpler if we spend a bit of time now recasting the model.

Instead of measuring the actual population number Pn, suppose instead we use the related variable xn = Pn⋅((C/2)/(1 + B - D + C/2)), roughly the fraction of the maximum population supported by the environment.

In terms of xn, the population equation becomes

xn+1 = (1 + B - D + C/2)⋅xn⋅(1 - xn)

This is still too long, so we give the coefficient (1 + B - D + C/2) the name s. Finally, we have the Logistic Map

xn+1 = s⋅xn⋅(1 - xn)

the idealized model of a single-species wth limited resources. Robert May's pioneering study of this model, "Simple mathematical models with very complicated dynamics," is one of the main catalysts for the current interest in chaos.

Return to the Logistic Map.