Homework 2 for MAT 494 Due on Th. Sept. 27, 2007
1): A population starts with 100 members of age zero. Assume that each member of age zero
produces one offspring and that 2/3 survive to age one. All members of age one produce 3
offspring and then die. Find the corresponding Leslie matrix, its dominant eigenvalue and a
corresponding eigenvector. Describe the stable age distribution and the asymptotic behavior
of the population.
2): For the given Leslie matrix find the dominant eigenvalue (if there is one and you can use
MATLAB), the corresponding eigenvector and the stable age distribution. Use MATLAB to
simulate the Leslie model (end time=10) for three different initial conditions.
A=
1 2 1
0.5 0 0
0 0.5 0
.
3): Consider the Usher model with
A=
0 0 73
0.04 0 0
0 0.39 0.65
.
a). Compute eigenvectors and eigenvalues of the model with MATLAB.
b). What is the intrinsic growth rate? the stable stage distribution?
c). Express the initial vector X0= (100,10,1) as a sum of the eigenvectors.
d). Use your answer in part (d) to give a formula for the population vector Xn.
4): Exercise 3, page 62.
5): Exercise 4, page 62.
6): Exercise 11, page 63.
7): Exercise 16, page 65.
8): Exercise 9, page 104.
9): Exercise 11, page 105.
10): Exercise 15, page 105.