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# 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
*p*_{0}, and *p*_{n} denotes
the number of squirrels after *n* years. Then we can construct
the orbit *p*_{0}, *p*_{1},
*p*_{2}, *p*_{3}, ...
by iterating

*f*(*x*) =
*x*.
We can even find an *explicit* formula for
*p*_{n}, namely:

*p*_{n} =
^{n}
*p*_{0}.
The name exponential growth model comes from the fact that *n* is
in the exponent.

### Nonlinear dynamics

We have derived a model for population growth,

*p*_{n+1} - *p*_{n} =
( - 1)
*p*_{n} (1 -
).
that depends on a 'saturation' number *E* and the growth rate
parameter . If we
substitute *p*_{n} = *E*
*x*_{n}, and write
*x*_{n+1} = *f*(*x*) and
*x*_{n} = *x*, then the logistic growth
model reduces to the simple dynamical system

*f*(*x*) =
*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 '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.

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Created: Apr 1 1998 ---
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