Suppose we have two species, \(P\) and \(Q\text{,}\) where species \(P\) preys on \(Q\text{.}\) Their populations, in millions, in year \(k\) are denoted by \(P_k\) and \(Q_k\) and satisfy

\begin{equation*} \begin{aligned} P_{k+1} \amp {}={} 0.8P_k + 0.2Q_k \\ Q_{k+1} \amp {}={} -0.3P_k + 1.5Q_k \\ \end{aligned} \text{.} \end{equation*}

We will keep track of the populations in year \(k\) using the vector \(\xvec_k=\twovec{P_k}{Q_k}\) so that

\begin{equation*} \xvec_{k+1} = A\xvec_k = \left[\begin{array}{rr} 0.8 \amp 0.2 \\ -0.3 \amp 1.5 \\ \end{array}\right] \xvec_k \text{.} \end{equation*}
  1. Show that \(\vvec_1=\twovec{1}{3}\) and \(\vvec_2=\twovec{2}{1}\) are eigenvectors of \(A\) and find their associated eigenvalues.

  2. Suppose that the initial populations are described by the vector \(\xvec_0 = \twovec{38}{44}\text{.}\) Express \(\xvec_0\) as a linear combination of \(\vvec_1\) and \(\vvec_2\text{.}\)

  3. Find the populations after one year, two years, and three years by writing the vectors \(\xvec_1\text{,}\) \(\xvec_2\text{,}\) and \(\xvec_3\) as linear combinations of \(\vvec_1\) and \(\vvec_2\text{.}\)

  4. What is the general form for \(\xvec_k\text{?}\)

  5. After a very long time, what is the ratio of \(P_k\) to \(Q_k\text{?}\)