x.
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, ...
by iterating
x.
We can even find an explicit formula for pn, namely:
n
p0.
The name exponential growth model comes from the fact that n is in the exponent.
We have derived a model for population growth,
- 1)
pn (1 -
).
that depends on the carrying capacity 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
x
(1 - x).
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 'normalised'
population. Since x is equal to
times a constant, it no longer stands for the size of the squirrel
population, but for the ratio of the squirrel population and the
carrying capacity. We say that the model is in non-dimensionalised
form.