Looking at the world population growth as a function of time we see that it increases exponentially. Population growth like this is due to a relationship where the next generation depends on the previous generation:
Pn+1 = bPn
where b is the birth rate. Things die and the population increase is modified by the death rate d.
Pn+1 = (b - d)Pn
If the death rate is greater than the birth rate then the population experiences exponential decay.
In many populations there are environmental limits. In the case of insects in a closed container, but having food introduced at a constant rate, the environmental limit will correspond to some maximum population. In the case of human populations the food supply in a region might limit the population. If the maximum population is called Mp then the available generation is (Mp - Pn) and the fraction possible at any given generation, n, is
(Mp - Pn)/Mp
The growth is therefore,
Pn+1 = (b - d)Pn(Mp - Pn)/Mp
This equation has the form
Xn+1 = aXn(1 - Xn)
and is called the Logistic equation.
This representation of population growth by a discrete equation contains more flexibility (more information) than its continuous, differential equation counter part. The differential equation does not exhibit fluctuations.
Probem: Set up the discrete population growth equation on a spread sheet with inputs xn for parameters a. Set up a graph of xn+1 vs. n = 1 to 50 and try different values of a from 0 to 4, etc.