0. Consider a discrete-time age-structured population model with 2 states, Young and Old. In one time step, each individual in state Y creates 1 other individual in state Y per time step and ages to state O. Each individual in state O creates one individual in state Y and then dies.
(a) What is the transition matrix for this model?
(b) Show that this matrix is regular.
(c) Because the matrix is non-negative and regular, there is an asymptotic ratio of Young to Old individuals in this population. It is approached regardless of the initial proportions. What is this asymptotic ratio?
1. (i) Which of the following non-negative matrices are "regular"? Explain. Do your calculations by hand and show all your work.
(a) 1 1 (b) 0 1 (c) 1 1 0
0 1 1 0 0 0 1
1 0 0
(ii) How is the regularity or otherwise reflected in what happens in cases (a) and (b) if you multiply a vector by the matrix over and over again?
2. Write down and explain the A matrix for the model
your group developed in class on Day 5, Tuesday Feb 1.
NOTE: Please provide this answer on a separate sheet
because I (not the grader) will grade it.
3. Haberman, Exercise 35.3, p142. THIS IS NOW OPTIONAL, FOR EXTRA CREDIT.
4. Haberman, Exercise 35.5, p142. Use a modified
version of the Maple worksheet
we used in class to do this, and perhaps it is best
if you use the new LinearAlgebra package: see this. (Or you may start your own Maple worksheet from scratch
if you wish.) Turn in hard copy (View->Zoom Factor to 75% please)
of your Maple work as an appendix to your hand-written answer.
Note: if I'm not mistaken this population goes towards extinction and doesn't double.
If you find the same, either discuss the time it takes to halve
instead of double, or increase the birth rate enough so that
it does grow and double.
