**Up:** *Logistic Growth Model*

# One-Dimensional Dynamical Systems

## Part 2: Logistic Growth Model

#### Derivation of the model

The exponential growth model does not take into account the
possibility that there may be finite resources. For an overcrowded
population the growth rate should decrease, due to scarcity of food
and other essentials. In a more realistic model we assume that the
population only grows when the number of squirrels is less than a
certain 'saturated' or equilibrium population *E*. The growth
rate should decrease as the number of squirrels gets closer to
*E*, and even become negative if the population grows larger than
*E*. If the number of squirrels is exactly *E*, the
population does not grow at all. We call *E* a fixed point (or
equilibrium point) of our dynamical system. The new model looks like:

*p*_{n+1} - *p*_{n} =
( - 1)
*p*_{n} (1 -
).
The left hand side denotes the increase of the population from year
*n* to year *n+1*. The term
- 1 is the growth
rate, as given in the exponential growth model. If
*p*_{n} is a lot smaller than *E*, the
term is
very small, and 1 -
is almost
1. Therefore, the right hand side is almost equal to
( - 1)
*p*_{n}, which means that there is still
exponential growth. However, if *p*_{n} gets
close to *E*, the term 1 -
tends to 0 and the population growth comes to a stop. The growth is
exactly 0 when *p*_{n} = *E*,
and becomes negative as *p*_{} becomes larger
than *E*. Our new model is known as the *logistic growth*
model.

**Up:** *Logistic Growth Model*

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Created: Apr 1 1998 ---
Last modified: Wed Apr 8 18:08:57 1998