text: 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 n units of time (years or generations), r is the growth constant, and K is a constant related to the size of food supply. This is called the logistic equation.
  
  If P[n-1] is small in relation to K, then the value of P[n] is approximately r*P[n-1]. For r > 1 this means the population grows exponentially as long as there is plenty of food. For 0 <= r < 1, it shrinks exponentially.
  
  If we replace P[n] by the ratio Q[n]=P[n]/K, then we can replace the equation above by the simpler
  
  Q[n_] := r*Q[n-1](1 - Q[n-1])
  
  If we define
  
  f[x_] := r*x*(1-x)                                                                          (1.2)
  
  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 r = 2, any initial value of x strictly between 0 and 1, will lead to a sequence converging to x = 1/2, the point at which f[x] = x. [In general the value of x for which f[x] = x is called the fixed point. Solving this equation gives x = (r-1)/r.]
  • For r = 1/2, the population always converges to 0, meaning that the species dies out.
  • For r = 3.5, the sequence does NOT converge to the fixed point x = 5/7, but instead eventually oscillates between two values on either side of it.
  
  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 y = f[x], i.e., the value of f'[x], at the fixed point. A simple calculation shows that f'[(r-1)/r] = 2-r.