Activity4.4.2
Suppose we have two species \(R\) and \(S\) that interact with one another and that we record the change in their populations from year to year. When we begin our study, the populations, measured in thousands, are \(R_0\) and \(S_0\text{;}\) after \(k\) years, the populations are \(R_k\) and \(S_k\text{.}\)
If we know the populations in one year, they are determined in the following year by the expressions
\begin{equation*} \begin{aligned} R_{k+1} \amp {}={} 0.9 R_k + 0.8 S_k \\ S_{k+1} \amp {}={} 0.2 R_k + 0.9 S_k\text{.} \\ \end{aligned} \end{equation*}We will combine the populations in a vectors \(\xvec_k = \twovec{R_k}{S_k}\) and note that \(\xvec_{k+1} = A\xvec_k\) where \(A = \left[\begin{array}{rr} 0.9 \amp 0.8 \\ 0.2 \amp 0.9 \\ \end{array}\right] \text{.}\)

Verify that
\begin{equation*} \vvec_1=\twovec{2}{1},\qquad \vvec_2=\twovec{2}{1} \end{equation*}are eigenvectors of \(A\) and find their respective eigenvalues.
Suppose that initially \(\xvec_0 = \twovec{2}{3}\text{.}\) Write \(\xvec_0\) as a linear combination of the eigenvectors \(\vvec_1\) and \(\vvec_2\text{.}\)
Write the vectors \(\xvec_1\text{,}\) \(\xvec_2\text{,}\) and \(\xvec_3\) as a linear combination of eigenvectors \(\vvec_1\) and \(\vvec_2\text{.}\)
When \(k\) becomes very large, what happens to the ratio of the populations \(R_k/S_k\text{?}\)
If we begin instead with \(\xvec_0 = \twovec{4}{4}\text{,}\) what eventually happens to the ratio \(R_k/S_k\) as \(k\) becomes very large?
Explain what happens to the ratio \(R_k/S_k\) as \(k\) becomes very large no matter what the initial populations are.
After a very long time, by approximately what factor does the population of \(R\) grow every year? By approximately what factor does the population of \(S\) grow every year?