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,

  1. 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 \(\xvec_{k+1} = A\xvec_k\text{.}\)

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

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

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

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

  6. What does this mean about the population of rodents?

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