Lab 4:  Following chaos
The logistic equation in discrete time

Ask the students to write the recursion equation in discrete time in Mathematica 
format, with an initial population size of 1.0 individuals.  Have them plot
using ListPlot the population size for 200 generations with r=0.1 and k=100
and with r=2.1 and k=100.

n[t_,r_,k_]:=n[t,r,k] = n[t-1,r,k] + r n[t-1,r,k] (1- n[t-1,r,k]/k)

n[0,r_,k_]=1.0

ListPlot[Table[n[t,0.1,100],{t,0,200}]]

ListPlot[Table[n[t,2.1,100],{t,0,200}]]

Have the whole lab draw the bifurcation diagram on the board by plotting N vs r, 
showing the points cycled through once the stable limit cycle is reached.
You'll want to draw the x-axis stretch near 2.5-3. 
Give different groups of students the following values of r:

	Group 1:  0.5,  2.1,  2.5,  2.565, 2.7, 3.01
	Group 2:  1.5,  2.2,  2.52, 2.57, 2.8, 3.05
	Group 3:  1.7,  2.3,  2.55, 2.572, 2.9, 3.1
	Group 4:  1.9,  2.4,  2.56,  2.6, 3.0, 3.2
	
The last values should only be plotted up to t=30
(the population size becomes very large and negative).

ListPlot[Table[n[t,2.56,100],{t,0,200}]]

Tables may help the students define the points cycled through
(e.g. the following is very close to an eight point cycle):

N[Table[n[t,2.56,100],{t,150,200}],5]