Deterministic Chaos

The Logistic Map

We could use the equation,

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

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

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

In terms of x_n, the population equation becomes

x_n+1 = (1 + B - D + C/2)*x_n*(1 - x_n)

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

x_n+1 = s*x_n*(1 - x_n)

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.