AM107 AM107  Homework 1
Assigned: Thursday, September 7, 2000
Due:  Thursday, September 14, 2000

Problem 1:  Edelstein-Keshet, p.29, Problem 1.

Problem 2:  Edelstein-Keshet, p.34, Problem 19.
 (Be aware that this problem appears to contain a typo:
  the first equation in part (b) is missing a sigma.
  This will affect the purported answers in the remaining parts.)

Problem 3: An excursion involving Fibonacci's Rabbits.
After visiting Fibonacci's rabbits once a month for some time, a scientist produced the following report:

"All the rabbits in Fibonacci's garden have an a percent chance of survival from one of my visits to the next.  And rabbits in their second and third months have a b percent chance of producing a single offspring.  Rabbits in their first month do not yet produce offspring.  I have never seen a rabbit in Fibonacci's garden past it's third month --- apparently these rabbits are rounded up and become mascots in an elementary school classes across the nation."
a) Draw a life graph for the rabbits described above.

b) Write down the corresponding recurrence relation (matrix).

c) Suppose all rabbits survive (a=1) and half reproduce (b=1/2). What is the eigenvalue of largest magnitude for this system? What are the implications of this value for the population of rabbits.

d) Suppose instead that b=0.6. Without computing the exact value of this leading eigenvalue (or you can if you want to, yikes!), describe the long term behavior of the the rabbit population? [hint: recall the intermediate value theorem]