##### Exercise2

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

\begin{equation*} \begin{aligned} R_{k+1} \amp {}={} 0.8 R_k + pS_k \\ S_{k+1} \amp {}={} -0.6R_k + 2 S_k\text{.} \\ \end{aligned} \end{equation*}where \(p\) is a parameter.

If \(p\gt 0\text{,}\) describe the nature of the interactions between the two species.

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

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

Explain why the results of the last two parts of this exercise hold by referring to the nature of the interactions between the two species.