###### Exercise 10

Consider a small rodent that lives for three years. Once again, we can separate a population of females into juveniles, yearlings, and adults. Suppose that, each year,

Half of the juveniles live to be yearlings.

One quarter of the yearlings live to be adults.

Adult females produce eight female offspring.

None of the adults survive to the next year.

Writing the populations of juveniles, yearlings, and adults in year \(k\) using the vector \(\xvec_k=\threevec{J_k}{Y_k}{A_k}\text{,}\) find the matrix \(A\) such that \(\xvec_{k+1} = A\xvec_k\text{.}\)

Show that \(A^3=I\text{.}\)

What are the eigenvalues of \(A^3\text{?}\) What does this say about the eigenvalues of \(A\text{?}\)

Verify your observation by finding the eigenvalues of \(A\text{.}\)

What can you say about the trajectories of this dynamical system?

What does this mean about the population of rodents?

Find a population vector \(\xvec_0\) that is unchanged from year to year.