The logistic equation is a mathematical model for population growth with crowding, which, though simple in form, simulates phenomena of amazing complexity.

As is well known, a small population introduced into an empty
environment (think bacteria in a Petri dish) tends to grow at an exponential
rate. If **P _{n}** is the population after

**P _{n} = R^{n} * P_{0} **

where **P _{0}** is the initial population and

Expressed another way, **P _{0}** is specified and

**P _{n+1} = R * P_{n} **

This model works fairly well until the population becomes so
large that crowding becomes significant. How should we model crowding? Let's
add another factor to the recursion equation. Let's specify a maximum
population, call it **C**, for Capacity, so **P _{n}** cannot
exceed

**P _{n+1} = R * (1 - P_{n }/C) * P_{n}
**

The new effective growth rate is " **R* (1 - P _{n }/C)**
". Notice that with the assumption that:

**0 < 1 - P _{n }/C < 1**

and thus reduces the effective growth rate. When **P _{n}**
is small, that is,

How does one determine **C**? One doesn't. One eliminates
it. Let:

**p _{n} = P_{n} / C **

So **p _{n}** is the population as a fraction of
capacity,

**p _{n+1} = R * (1 - p_{n}) * p_{n}
**

Using Excel one can easily explore the behavior of populations governed by such an equation:

1) Label column A as "Rate"
( **A1 = "Rate"** ).

2) Let **A2 = 2** (for now).

3) Let **A3 = "=A2"**.

4) Click, drag and copy **A3**
down to **A100**.

( So column A is 2's all the way down. Yes, we could do fixed cell addressing, but this is easier.)

5) Label column B as
"Population" ( **B1 = "Population"** ).

6) Let **B2 = 0.01** (a small
initial population for now).

7) Let **B3 =
"=A2*B2*(1-B2)"**

**( **the logistic recursion
equation)

8) Click, drag and copy **B3**
down to **B100**.

Notice! The
population rapidly converges to a value of 0.5 - half capacity. If you change the initial population ( **B2
= 0.9** or **B2 = 0.3 **- remember the initial population must be < 1
), it still converges rapidly to a steady state of half capacity.

Now add a graph of column B only. (It's easier for us visually oriented primates to understand than a table of numbers).

9) Click on column **B**.

10) Click on "**Insert -
Chart - Scatter -** choose one which connects the points - **Finish**"

And you can see it approach 0.5. Change the initial population and watch the graph change instantly - this is computational experimental science!

Try a growth rate of **A2 = 1.5**. The population
approaches a smaller steady state population.

Try **A2 = 2.5**. It approaches a higher steady state population.
And each of these approaches the same steady state independent of the initial
population - try it for different **B2**'s and verify this.

Unlike the exponential model - which approaches infinity - this model simulates an approach to a steady state population - at least for those rates tried so far - which is far more realistic.

**Try A2 < 1** and it rapidly approaches zero -
extinction. Makes sense - negative growth leads to extinction.

**Try A2 = 3.0** and ... hmmmmmmmm ... it seems to bounce
back and forth and approach a limit. Try different B2's, and it still behaves
the same.

Try **A2 = 3.1** or **3.3** and it now bounces back
and forth between two levels - it's cyclic with period 2. Try different B2's,
and it still behaves the same.

Try **A2 = 3.5** and it now bounces back and forth
between four levels - it's cyclic with period 4. Try different B2's, and it
still behaves the same.

Try **A2 = 3.9.
It's CHAOS!!!** There's no limiting pattern at all. Try different
initial populations - try B2 = 0.43 and B2 = 0.44. Very different patterns! Try
A2 = 3.91 with the same B2. Very different patterns! This is mathematical
chaos. Not only is there no pattern to the long term behavior, but very small
changes in the initial conditions result in completely different behaviors. The
logistic equation is completely deterministic, and yet it results in apparently
"random" and unpredictable behavior. And since one can never know the
initial conditions with complete accuracy, and even a small change in the initial
conditions results in completely different behaviors, we have an example of a
deterministic phenomenum which is inherently unpredictable.

"A butterfly flapping its wings in China will change the weather in the USA two weeks later."

ŠJames S. Freeman, 2002