Non Linear Models
As described in class, the next step towards realism is limiting population growth by letting the growth rate vary. We choose a function which is positive for small populations, zero when the population equals the carrying capacity (c) of the environment, and negative for populations greater than the carrying capacity. The difference equation becomes:
This nonlinear equation, called the logistic equation, cannot be solved "in closed form"; there is no simple formula for Pn in terms only of P0, r, and c. But the computer can easily calculate as many generations as we want, although on the applet given here (next page) we will limit the maximum number of generations to 100000.
As in class, we will use graphical displays to explore the predictions (behavior) of this equation for different values of r and P0. We normalize and set c = 1. In this formulation r is limited to lie between 0 and 3.