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]