##### Exercise3

Suppose that the populations of two species interact according to the relationships

\begin{equation*} \begin{aligned} R_{k+1} \amp {}={} \frac12 R_k + \frac12S_k \\ S_{k+1} \amp {}={} -pR_k + 2S_k \\ \end{aligned} \end{equation*}where \(p\) is a parameter. As we saw in the text, this dynamical system represents a typical predator-prey relationship, and the parameter \(p\) represents the rate at which species \(R\) preys on \(S\text{.}\) We will denote the matrix \(A=\mattwo{\frac12}{\frac12}{-p}2\text{.}\)

If \(p = 0\text{,}\) determine the eigenvectors and eigenvalues of the system and classify it as one of the six types. Sketch the phase portraits for the diagonal matrix \(D\) to which \(A\) is similar as well as the phase portrait for \(A\text{.}\)

If \(p=1\text{,}\) determine the eigenvectors and eigenvalues of the system. Sketch the phase portraits for the diagonal matrix \(D\) to which \(A\) is similar as well as the phase portrait for \(A\text{.}\)

For what values of \(p\) is the origin a saddle? What can you say about the populations when this happens?

Describe the evolution of the dynamical system as \(p\) begins at \(0\) and increases to \(p=1\text{.}\)