\section{A Game Theory Model of a Predator-Prey Habitat}\label{S:PredatorPreyGameTheory}
\newpar
In the desert, snakes prey on small rodents in both open and vegetated habitats.
\footnote{Based upon Bouskila, \cite{Bouskila}, (2001).} Should the rodent and snake elect to reside in the same habitat,
we expects the two species to have negative and positive payoffs, respectively, due to potential inter-species encounters. Similarly, when the rodent and snake reside in different habitats, the payoffs are positive and negative, respectively.

\newpar
Let habitats 1 and 2 correspond to the vegetated and open areas. (Hereafter, we simply refer to the vegetated area as the ``bush.'') Corresponding to the snake, we define the 2 by 2 payoff matrix $A$, where $[A]_{ij}$ represents the snake's payoff when it resides in habitat $i$ and the rodent in habitat $j$. For the rodent, we denote its payoff matrix $B$, where $[B]_{ij}$ denotes the rodent's payoff when it resides in habitat $i$ and the snake in habitat $j$. We expect that the diagonal entries of $A$ to be nonnegative and those of $B$ nonpositive.

\newpar
Payoff for each species is defined in terms of dimensionless ``energy gains'' or ``energy losses,''
 whose relative sizes reflect the extent to which they affect a specie's future reproductive output. Entries for the two payoff matrices depend upon various parameters, such as probabilities of inter-species encounters in the two habitats. Table \ref{Ta:predator_prey_data} lists the various parameter and probability values as estimated by data collected in the field.

\begin{table}[h]
\caption{Field data used to determine payoff matrices}\label{Ta:predator_prey_data}
\begin{tabular}{|l|c|c|}
\hline
Quantity & Label & Value \\ \hline
Energy loss for rodent if captured by snake & $d$ & 1500 \\ \hline
Energy gain for snake if it captures rodent & $e$ & 1000 \\ \hline
Energy required of rodent to live in either habitat & $\alpha$ & 7 \\ \hline
Energy required of snake to live in either habitat & $\beta$ & 7 \\ \hline
Probability snake captures rodent in the bush & $P_{sb}$ & .032 \\ \hline
Probability snake captures rodent in the open &  $P_{so}$ & .009 \\ \hline
Energy gain for rodent that survives predation by snake in the bush & $\rho_b$ & 12 \\ \hline
Energy gain for rodent that survives predation by snake in the open & $\rho_o$ & 17 \\ \hline
\end{tabular}
\end{table}

\be
\item Use the values of $e$, $\beta$, $P_{sb}$, and $P_{so}$ to construct
the snake's payoff matrix, $A$.
\item Use $d$, $\alpha$, $P_{sb}$, $P_{so}$, $\rho_b$, and $\rho_o$ to construct
the rodent's payoff matrix, $B$.
\item Determine the mixed strategy Nash equilibrium \index{matrix game!mixed strategy Nash equilibrium}
for this bimatrix game model of the predator-prey habitat.
\item Suppose snake's average payoff must increase by 10\% from its equilibrium value and that the
rodent's average loss must decrease by 10\%. What fraction of time
should each species spend in each habitat to bring this change about?
\ee


