One-Dimensional Dynamical Systems
Part 2: Logistic Growth Model
The squirrel model in the introduction is called the exponential
growth model. Suppose the current number of squirrels is
p0, and pn denotes
the number of squirrels after n years. Then we can construct
the orbit p0, p1,
p2, p3, ...
We can even find an explicit formula for pn, namely:
The name exponential growth model comes from the fact that n is in the exponent.
We have derived a model for population growth,
that depends on a 'saturation' number E and the growth rate parameter . If we substitute pn = E xn, and write xn+1 = f(x) and xn = x, then the logistic growth model reduces to the simple dynamical system
Notice that the parameter E has disappeared. In fact, the variable x = * does not stand for the total number of squirrels in a specific year. We can think of x as representing a 'normalized' population. Since x is equal to times a constant, it can be thought of as expressing the population of squirrels as a fraction of some adjusted saturation value.
Written by Hinke Osinga
Comments to: email@example.com
Created: Apr 1 1998 --- Last modified: Wed Apr 8 18:19:53 1998