Physics 374
Fall 2005
Pro. E. F. Redish

     Several decades ago, scientists looking to model a species's population growth in an ecosystem devised the following Non-Linear Logistic Difference Equation:
					(Subscript[x, n + 1]) = r (Subscript[x, n])(1-(Subscript[x, n]))
					
where x is a fraction between zero and one representing the size of the population.  An x of zero means extinction while an x of one implies a maximum population.  If (Subscript[x, n]) is the population fraction in, say, this month, then the population fraction next month is (Subscript[x, n + 1]).  r is a growth rate parameter.  Note that all the terms in the Non-Linear Logistic Difference Equation have already been scaled such as to make them dimensionless.  (We'll work on this later in the semester.) 
     Use Mathematica tables to examine the population predictions made by the Non-Linear Logistic Difference Equation.  Always start with an initial value of (Subscript[x, 1]) = .4.  What happens to the population fraction over time for a growth rate of r = .5?  For r = 1.5?  For r = 2.7?  What about for r = 3.1?  What about r = 3.5?  Can you find the next "period doubling"?  What has happened by the time r reaches 3.6?  Print out enough material to clearly support your case.
     If you happen to be interested in this type of thing, I'd recommend reading the chapter on Mitchell Feigenbaum in Abraham Pais's excellent book, The Genius of Science.  

(Problem by Tom Bing)