###### Exercise 8

Suppose that two species \(P\) and \(Q\) interact with one another and that we measure their populations every month. We record their populations in a state vector \(\xvec = \twovec{p}{q}\text{,}\) where \(p\) and \(q\) are the populations of \(P\) and \(Q\text{,}\) respectively. We observe that there is a matrix

such that the matrix transformation \(T(\xvec)=A\xvec\) is the transition function describing how the state vector evolves from month to month. We also observe that, at the beginning of July, the populations are described by the state vector \(\xvec=\twovec{1}{2}\text{.}\)

What will the populations be at the beginning of August?

What were the populations at the beginning of June?

What will the populations be at the beginning of December?

What will the populations be at the beginning of July in the following year?