###### Exercise 10

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

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

Show that \(\vvec_1=\twovec{1}{3}\) and \(\vvec_2=\twovec{2}{1}\) are eigenvectors of \(A\) and find their associated eigenvalues.

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 \(\vvec_2\text{.}\)

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 \(\vvec_2\text{.}\)

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

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