Homework 2

Due Date: Friday, September 18

Most of this assignment concerns the logistic family of functions introduced by biologist Robert May. For the following problems, let \(f(x) = cx(1-x)\), where \(x\) is always between \(0\) and \(1\), and \(c\) is a constant between \(2\) and \(4\).

1. Write Mathematica code that draws cobweb plots for the function \(f\). Your cobweb plots should include the graph of the function \(f\), the line \(y=x\), and the line segments of the cobweb, showing at least \(20\) iterations. The notebook Homework2Commands.nb illustrates some commands that may be helpful.

   Demonstrate your code by drawing cobweb plots of the orbit of \(x_0=0.01\) for \(c=2.2\), \(c=2.8\), \(c=3.2\), and \(c=4\).

2. Consider the case where \(c = 7/2\), so \(f(x) = \dfrac{7}{2}\,x(1-x)\).
   1. Find the fixed points for \(f\), and use Theorem 1.5 to show that they are both repelling.
   2. Use the Solve command to find the \(2\)-cycle for \(f\). Express the points of the cycle as fractions.
   3. Use the Stability Test for Periodic Points to show that the \(2\)-cycle you found in part (b) is repelling.
   4. Does the function \(f\) have any attracting cycles at all? Justify your answer.
3. 1. Suppose that, for a certain value of \(c\), the point \(1/2\) is periodic under \(f\). Explain why, in this case, the cycle that contains \(1/2\) must be attracting.
   2. Use the NSolve command to find a value of \(c\) for which \(1/2\) is periodic with period \(3\).
   3. Use NSolve to find three different values of \(c\) for which \(1/2\) is periodic with period \(5\).
   4. Draw cobweb plots showing the orbit of \(1/2\) (not \(0.01\)) for each of the values of \(c\) you found in parts (b) and (c).