Exercise 10

Suppose we have two species, $P$ and $Q$, where species $P$ preys on $Q$. Their populations, in millions, in year $k$ are denoted by $P_k$ and $Q_k$ and satisfy $$\begin{aligned} P_{k+1} & {}={} 0.8P_k + 0.2Q_k \\ Q_{k+1} & {}={} -0.3P_k + 1.5Q_k \\ \end{aligned}$$. We will keep track of the populations in year $k$ using the vector $\xvec_k=\twovec{P_k}{Q_k}$ so that $$\xvec_{k+1} = A\xvec_k = \left[\begin{array}{rr} 0.8 & 0.2 \\ -0.3 & 1.5 \\ \end{array}\right] \xvec_k$$.

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 $\xvec_0 = \twovec{38}{44}$. Express $\xvec_0$ as a linear combination of $\vvec_1$ and $\vvec_2$.

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

4. What is the general form for $\xvec_k$?

5. After a very long time, what is the ratio of $P_k$ to $Q_k$?