The logistic model:
n_t+1 = (birthrate - deathrate) [n_t - (n_t^2 / max_population)]
The logistic map is a famous algorithm in chaos theory:
R = birthrate - deathrate
K = max population (carrying capacity)
X_t =n_t / k
x_t+1 = R (x_t - x_t^2)
Robert May and Mitchell Feigenbaum studied the logistic map.
x is always between 0 and 1.
An example:
R = 2
x_0 = .2 (20% of the carrying capacity)
x_1 = 2 (.2 - .2^2) = .32 (32% of the carrying capacity)
x_2 = 2(.32 -.32^2) = .4352
...
slowly approaches .5 and reaches it at x_7 and it stays at .5 for all the following generations. .5 is called an attractor. In this case it is a fixed point attractor.
A graph of x_t compared to x_t+1 for the logistic map forms a parabola.
The logistic map is a model, a simplified representation of reality.