###### Exercise10

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

\begin{equation*} \begin{aligned} P_{k+1} \amp {}={} 0.8P_k + 0.2Q_k \\ Q_{k+1} \amp {}={} -0.3P_k + 1.5Q_k \\ \end{aligned}\text{.} \end{equation*}

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

\begin{equation*} \xvec_{k+1} = A\xvec_k = \left[\begin{array}{rr} 0.8 \amp 0.2 \\ -0.3 \amp 1.5 \\ \end{array}\right] \xvec_k\text{.} \end{equation*}
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 $$\xvec_0 = \twovec{38}{44}\text{.}$$ Express $$\xvec_0$$ as a linear combination of $$\vvec_1$$ and $$\vvec_2\text{.}$$

3. 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 $$\vvec_2\text{.}$$

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

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

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