Activity 4.4.2

Suppose we have two species $R$ and $S$ that interact with each 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 $S_0$; after $k$ years, the populations are $R_k$ and $S_k$.

If we know the populations in one year, suppose that the populations in the following year are determined by the expressions $$\begin{aligned} R_{k+1} & {}={} 0.9 R_k + 0.8 S_k \\ S_{k+1} & {}={} 0.2 R_k + 0.9 S_k\text{.} \\ \end{aligned}$$ This is an example of a mutually beneficial relationship between two species. If species $S$ is not present, then $R_{k+1} = 0.9R_k$, which means that the population of species $R$ decreases every year. However, species $R$ benefits from the presence of species $S$, which helps $R$ to grow by 80% of the population of species $S$. In the same way, $S$ benefits from the presence of $R$.

We will record the populations in a vector $\xvec_k = \twovec{R_k}{S_k}$ and note that $\xvec_{k+1} = A\xvec_k$ where $A = \left[\begin{array}{rr} 0.9 & 0.8 \\ 0.2 & 0.9 \\ \end{array}\right]$.

1. Verify that $$\vvec_1=\twovec{2}{1},\qquad \vvec_2=\twovec{-2}{1}$$ are eigenvectors of $A$ and find their respective eigenvalues.

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

3. Write the vectors $\xvec_1$, $\xvec_2$, and $\xvec_3$ as linear combinations of the eigenvectors $\vvec_1$ and $\vvec_2$.

4. What happens to $\xvec_k$ after a very long time?

5. When $k$ becomes very large, what happens to the ratio of the populations $R_k/S_k$?

6. 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?

7. If we begin instead with $\xvec_0 = \twovec{4}{4}$, what eventually happens to the ratio $R_k/S_k$ as $k$ becomes very large?