##### 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 $$A=\mattwo{\frac12}{\frac12}{-p}2\text{.}$$

1. 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 $$A\text{.}$$

2. 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 $$A\text{.}$$

3. For what values of $$p$$ is the origin a saddle? What can you say about the populations when this happens?

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

in-context