##### 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 $$A = \left[\begin{array}{rr} 0.9 \amp 0.8 \\ 0.2 \amp 0.9 \\ \end{array}\right] \text{.}$$

1. 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.

2. Suppose that initially $$\xvec_0 = \twovec{2}{3}\text{.}$$ Write $$\xvec_0$$ as a linear combination of the eigenvectors $$\vvec_1$$ and $$\vvec_2\text{.}$$

3. Write the vectors $$\xvec_1\text{,}$$ $$\xvec_2\text{,}$$ and $$\xvec_3$$ as a linear combination of eigenvectors $$\vvec_1$$ and $$\vvec_2\text{.}$$

4. When $$k$$ becomes very large, what happens to the ratio of the populations $$R_k/S_k\text{?}$$

5. 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?

6. Explain what happens to the ratio $$R_k/S_k$$ as $$k$$ becomes very large no matter what the initial populations are.

7. 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?

in-context