In these notes I will use mathematical notation similar to the syntax of Mathematica.

1. The logistic equation, January 19.

Consider the population model defined by

** P[n_] := r*P[n-1]*(K - P[n-1])/K** (1.1)

where ** P[n] **is the population after

If ** P[n-1**] is small in relation to

If we replace ** P[n]** by the ratio

`Q[n_] := r*Q[n-1](1 - Q[n-1])`

If we define

then we can rewrite the definition of ** Q[n]** as

`Q[n_] := f[Q[n-1]]`

Thus studying the long term behavior of a population governed by (1.1)
is equivalent to studying the sequence of numbers obtained by what happens
by repeatedly applying the function f of (1.2) to an initial value of ** x**.

A cobweb
diagram is a graphical device for studying this. In class we experimented
with a few values of the growth constant r and initial values of ** x.**
This is what we found.

- For
, any initial value of`r = 2`strictly between 0 and 1, will lead to a sequence converging to`x`, the point at which`x = 1/2`. [In general the value of x for which`f[x] = x`is called the`f[x] = x`*fixed point*. Solving this equation gives.]`x = (r-1)/r` - For
, the population always converges to 0, meaning that the species dies out.`r = 1/2` - For
, the sequence does NOT converge to the fixed point`r = 3.5`, but instead eventually oscillates between two values on either side of it.`x = 5/7`

This raises the following question: For which values of the growth constant
r does the sequence converge to the fixed point ** x=(r-1)/r**
and why? This seems to be related to slope of the parabola defined by

*This page was last revised on February 2, 1998.*