- #1

- 19

- 0

**The Problem:**

Consider a species in which no individuals live beyond 3 years. Divide the population into four age groups labelled by n = 0,1,2 and 3. Assume that only the second and third groups can reproduce. For each group n let bn denote the birthrate and [tex]d_n[/tex] the death rate. Find a matrix A such that P(t + 1) = AP(t), with:

[tex]\ P = \left(\begin{array}{cc}P_0\\P_1\\P_2\\P_3\end{array}\right)[/tex]

Where [tex]]P_n[/tex](t) denotes the population size of age group

*n*at time

*t*.

It's a question on an age cohort model. I've began with using the following basic information/equations:

**My initial work:**

[tex]\begin{array}{l | c|c|c|c |}\&0&1&2&3\\\hline\b_i&0&b_1&b_2&0\\\hline d_i&d_0&d_1&d_2&1\\\hline\end{array}[/tex]

(Where the first column is year zero, the second year one, the third year two, the fourth year three. [tex]b_i[/tex] indicates the birth rate, [tex]d_i[/tex] indicates the death rate, where the "1" in column four indicates that individuals die at age three).

I then created a 4x4 matrix as below:

A = [tex]\left(\begin{array}{cccc}o&b_1&b_2&0\\1-d_0&0&0&0\\0&1-d_1&0&0\\0&0&1-d_2&0\end{array}\right)[/tex]

**My attempt at a solution:**

After creating that matrix, I move on to the next section:

**Section 2: Show that the eigenvalues of A satisfy the following equation:**

[tex]\lambda^4 - b_1(1 - d_0)\lambda^2 - b_2(1-d_0)(1-d_1)\lambda = 0[/tex]

[tex]\lambda^4 - b_1(1 - d_0)\lambda^2 - b_2(1-d_0)(1-d_1)\lambda = 0[/tex]

To try to find the determinant of my above matrix, I took [tex]\lambda[/tex]I from the above matrix (A - [tex]\lambda[/tex]I) and set it equal to 0 to find the eigenvalues. But when I try to solve for these I keep getting 0. Now I'm thinking that perhaps my intial matrix (A) is incorrect? I'm trying to find the determinants of the matrix by the following method:

http://easyweb.easynet.co.uk/~mrmeanie/matrix/matrixea.gif [Broken]

Reducing the 4x4 to 3x3 and then to 2x2 and then finding the determinant of that. But as I said, it's coming out to be 0. (I haven't done any matrix algebra in a few years, so I'm a bit out of practice).

This question is part of my revision for a final exam, so I'd appreciate any help you can give me!

Last edited by a moderator: