\documentclass[11 pt]{article}

%pagestyle{empty} uncomment to eliminate page numbering

\setlength{\oddsidemargin}{-.45in}
%\setlength{\evensidemargin}{-.5in}
\setlength{\textwidth}{7.1in}
\setlength{\topmargin}{-0.95in}
\setlength{\textheight}{9.7in}

%Change second above dimensions to below in order to widen page for margin labels:
%\setlength{\oddsidemargin}{36pt}
%	\setlength{\evensidemargin}{74pt}
%\setlength{\textwidth}{350pt}
%\setlength{\topmargin}{27 pt}
%\setlength{\textheight}{522 pt}
%\setlength{\marginparwidth}{80 pt}
%\setlength{\marginparsep}{7pt}

 \pagestyle{empty} %(no page numbering} ; below is for adding footer


%Exam header and page total footer
\usepackage{fancyhdr}
%\pagestyle{fancy}
%\lfoot{}
%\cfoot{}
%\rfoot{\rfoot{{\bf{Page Total}} \dotfill  \pagescorebox}}
%\lhead{}
%\chead{MTH 203 B}
%\rhead{}

\usepackage{graphics}

% The epsfig package clashes with maple2e below. Use one or the other.
%\usepackage{epsfig}
%\usepackage{wrapfig}
%\input epsf.sty

%\usepackage{maple2e}
% \def\emptyline{\vspace{12pt}}
%\DefineParaStyle{Maple Output}
%\DefineParaStyle{Warning}
%\DefineCharStyle{2D Math}
%\DefineCharStyle{2D Output}

%\usepackage{bbding}
% Package is incompatible with marvosym and wasysym
%\usepackage{pstcol,pst-node,pstricks,pst-text,pst-grad,pst-3d}
\usepackage{pstcol,pstricks,pst-text,pst-grad,pst-node}
\usepackage{graphicx}
\usepackage{latexsym}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage[intlimits]{amsmath}
%\usepackage{wasysym}
%\usepackage{marvosym}
%\usepackage{pifont}

\usepackage{amsthm}
\usepackage{multicol}
\usepackage{psfrag}
%\usepackage{pxfonts} % combine with \varointctrclockwise\limits_{C_2} for counterclockwise line integral
%comment out wasysym, amstheorem, and theoremstyle[plain]

%***********************************
%**********THEOREM-LIKE ENVIRONMENTS
%***********************************
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem{theorem}{Theorem}
\newtheorem{proposition}{Proposition}

\newtheorem{example}{Example}
\newtheorem{activity}{Sample Application }
\newtheorem{project}{Project}

%\newtheorem*{weak_duality}{Weak Duality}
%\newtheorem*{strong_duality}{Strong Duality}
%\newtheorem*{duality_theorem}{The Duality Theorem}

\newtheorem{waypoint}{Waypoint}[definition]

\newenvironment{progress_ck}[1][.9\linewidth]
{ \vspace{.1in} \noindent \begin{center}\begin{minipage}[c]{#1}
$\Diamondblack$ \hrulefill $\Diamondblack$
\begin{waypoint}}
{\end{waypoint}  $\Diamondblack$ \hrulefill $\Diamondblack$ \end{minipage} \end{center}}


\newcounter{progress_ck_counter}
%**********************************

%COMMANDS
\newcommand{\be}{\begin{enumerate}}
\newcommand{\ee}{\end{enumerate}}

\newcommand{\bi}{\begin{itemize}}
\newcommand{\ei}{\end{itemize}}

\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\Nat}{\mathbb N}
\newcommand{\Q}{\mathbb Q}
\newcommand{\Z}{\mathbb Z}

\newcommand{\vecc}{\mathbf c}
\newcommand{\vecx}{\mathbf x}
\newcommand{\vecs}{\mathbf s}
\newcommand{\vece}{\mathbf e}
\newcommand{\veca}{\mathbf a}
\newcommand{\vecd}{\mathbf d}


\newcommand{\vecxs}{{\mathbf {x}}_s}
\newcommand{\vecxsh}{{\mathbf {x}}_{s,h}}
\newcommand{\vecxsp}{{\mathbf {x}}_{s,p}}
\newcommand{\vecb}{\mathbf b}
\newcommand{\vecu}{\mathbf u}
\newcommand{\vecv}{\mathbf v}
\newcommand{\vecw}{\mathbf w}
\newcommand{\vecn}{\mathbf n}
\newcommand{\vecy}{\mathbf y}
\newcommand{\vecz}{\mathbf z}
\newcommand{\onevec}{\ensuremath{\mathbf{1}}}
\newcommand{\zerovec}{\ensuremath{\mathbf{0}}}
\newcommand{\deltavec}{\boldsymbol{\delta}}

\newcommand{\bs}{\blacksquare}

\newcommand{\F}{\mathbf F}
\newcommand{\T}{\mathbf T}
\newcommand{\N}{\mathbf N}
\newcommand{\B}{\mathbf B}
\newcommand{\vecR}{\mathbf R}
\newcommand{\V}{\mathbf V}

\newcommand{\tor}{\boldsymbol{\tau}}


\newcommand{\dr}{d\mathbf r}
\newcommand{\dS}{d\mathbf S}

\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}

\newcommand{\x}{\phantom{ +x_1}}

\newcommand{\bv}{\mbox{BV}}
\newcommand{\BV}{\mbox{BV }}
\newcommand{\NBV}{\mbox{NBV }}

\newcommand{\LPC}{\emph{FuelPro Petroleum Company }}
\newcommand{\LP}{\emph{FuelPro }}

\newcommand{\CMC}{\emph{ConPro Manufacturing Company }}
\newcommand{\CM}{\emph{ConPro }}

\newcommand{\newpar}{\vspace{.15in}\noindent}
\newcommand{\noi}{\noindent}

\newcommand{\ds}{\displaystyle}
\newcommand{\cbox}{\framebox{\parbox[c][.5 in][c]{.5 in}{\makebox[.5in][l]{}} } \hspace{.025in}}
\newcommand{\pagescorebox}{\framebox{\parbox[c][.35 in][c]{.8 in}{\makebox[2in][l]{}} } \hspace{.025in}}

%MARGIN LABEL
\newcommand{\marginlabel}[1]{\mbox{}\marginpar{\raggedleft\hspace{0pt}#1}}

%MAPLE FONT
\newcommand{\mf}[1]{ {\ttfamily{\upshape{#1}}}}

%**********************************
%MAPLE CUSTOMIZED LIST COMMAND

\newenvironment{maplelist}{\renewcommand{\labelitemi}{$>$}\begin{itemize}}{\end{itemize}}

%\begin{list}{$\diamondsuit$}{\setlength{\leftmargin}{.5in}
%\setlength{\rightmargin}{.5in}}
%\item something here
%\item something else here
%\end{list}

%***********************************
%LP
%\begin{align}
% \mbox{maximize } z &=10x_1+8x_2  \\
%	\mbox{ subject to} \notag\\
%	x_1 \phantom{ +20x_2} &\leq 10 \notag\\
%	x_1+2x_2 &\leq 20  \notag\\
%	10x_1+2x_2 &\leq 120 \notag \\
%	x_1,x_2 &\geq 0.\notag
%	\end{align}	


%TABLEAU
%\begin{table}[h]
%\centering
%\begin{tabular}{|c||c|c|c|c|c||c|}
%\hline
%$z$ & $x_1$ & $x_2$ & $s_1$ & $s_2$ & $s_3$ & RHS \\
%\hline \hline
%1 & 0 & -8 & 10 & 0 & 0 & 100 \\ \hline
%0 &  1 & 0 & 1 & 0 & 0 & 10 \\ \hline
%0 &  0 & 2 & -1 & 1 & 0 & 10 \\ \hline
%0 &  0 & 7 & -10 & 0 & 1 & 20 \\ \hline
%\end{tabular}
%\caption{Tableau after first iteration}\label{Ta:petroleum_second_tableau}
%\end{table}


%LABELS
%Section: \label{S:Name}
%Theorem/Explore: \label{Th:Name}
%Table: \label{Ta:Name}
%Reference \cite{Burch}

%CUSTOMIZED ITEMIZED LIST SYMBOL.
%\begin{list}{default_label}{\setlength{\leftmargin}{.5in} \setlength{\rightmargin}{.5in}}

%CHANGE TO DOUBLE AND 1.5 SPACING
%\renewcommand{\baselinestretch}{1.5}

%****************************************
%MATH SYMBOLS AND SHORTCUTS
%****************************************

%LABELED EQUATION. TO REFER TO THE EQUATION LATER IN TEXT TYPE "(\ref{E:quadratic})".
%begin{equation} \label{E:quadratic}
% f(x)=x^2 \text{hi}
%\end{equation}

%USE OF SPLIT
%begin{equation} \label{E:quadratic}
% \begin{split} &f(x)=long formula \\
% & broken over two lines \end{split}
%\end{equation}

%LABELED ALIGN
%\item Consider the system of equations (\ref{E:system1})	
%	\begin{align}
%		3x_1-x_2+2x_3 & =-4  \notag \\
%		2x_1+x_2+x_3 & =-1  \notag \\
%		x_1+3x_2& =2.  \label{E:system1} \\ \notag
%	\end{align}

%ALIGNED FORMULA
%\begin{align*}
%f(x)&= x^2+2x+1 \\ \intertext{because..}
%   &=(x+1)^2
%\end{align*}

%TWO ALIGNED FORMULA COLUMNS
%\begin{alignat}{2}
%P_0(x)&=1, & \qquad P_1(x)&=x \notag \\
%P_2(x)&=\frac{1}{2}(3x^2-1), & \qquad P_3(x)&=\frac{1}{2}(5x^3-3x) \notag
 % \end{alignat}

%GATHERING FORMULAS: TWO EQUATIONS, WITH NO ALIGNMENT
% \begin{gather*}
% x^2+2x+1
%  (x+1)^2
% \end{gather*}

%SPLITTING A LONG FORMULA WITH MULTLINE
%\begin{multline*}
% x^5+x^4+x^3 \\
% +x^2+x+1
%\end{multline*}

%AUGMENTED MATRIX
%\left[  \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix}
%\left| \begin{matrix} 1 \\ 2 \end{matrix} \right. \right]

%MATRIX WITH ELLIPSES
%\begin{bmatrix}
%f_{11}^{-1} & 0 & \ldots & 0 \\
%0 & f_{22}^{-1} & \ldots  & 0  \\
%\vdots & & \ddots & \vdots  \\
%0 & 0 & \ldots & f_{nn}^{-1}
%\end{bmatrix}\nabla f(\vecx_k)

%PIECEWISE DEFINED FUNCTION
%p(t)=\begin{cases}
%		\frac{1}{2},	&\text{if $t \in [-1,1]$};\\
%		0,	&\text{otherwise,}
%	\end{cases}

%DOUBLE INTEGRAL
%\iint\limits_D \frac {\partial Q}{\partial x} - \frac {\partial P}{\partial y} \, dx \, dy

%CONTOUR AND DOUBLE INTEGRALS WITH CONTOUR/REGION BELOW INTEGRAL SIGN
% \oint\limits_C
% \iint\limits_D

% SCALED MATH SYMBOLS
%\mbox{ \raisebox{-.2in}{ \scalebox{2}[6]{$\int$}}}_{\scalebox{1.5}[1.5]{$\Real$}}

%EVALUATION OF DERIVATIVE VIA LEIBNIZ NOTATION (INCLUDES ONE-SIDED PARENTHESIS)
% \left. \frac{\partial z}{\partial x}\right|_{(x,y)=(1,-2)}

%OVERSET DEFINITION
%\overset{\text{def}}{=}

%DELIMITERS
% \left\{   \right\}

%SCALED, CENTERED EQUATION
%\begin{equation}\label{E:convexity_FOC_unconstrained_ConPro_Hessian}
%\mbox{
%{\scalebox{1.5}{ $\ds
%H_f(x_1,x_2)=\begin{bmatrix} \frac{-350x_2^{\frac{1}{3}}}{x_1^{\frac{3}{2}}} & \frac{700}{3\sqrt{x_1}x_2^{\frac{2}{3}}} \\
%\frac{700}{3\sqrt{x_1}x_2^{\frac{2}{3}}} & \frac{-2800\sqrt{x_1}}{9x_2^{\frac{5}{3}} } \end{bmatrix}$
%}}
%}
%\end{equation}

%GENERAL NLP
%\begin{equation}\label{E:diff_Gradient_general_NLP}
%\mbox{minimize (or maximize) } f(\vecx) \mbox{ where }\vecx \in S.
%\end{equation}

%GENERAL LP
%\begin{align}
% \mbox{minimize } w  \\
%	\mbox{ subject to} \notag\\
%	\vecy A&\leq w\vece^t \notag \\
%	\vecy\cdot\vece &=1 \notag  \\
%	\vecy &\geq \zerovec. \label{E:Duality_Steves_strategy}
%\end{align}


%********************************************************
%BOXES
%********************************************************

%BOX
%The second argument in the parbox (1in} controls the box width; first argument controls height
%\framebox{\parbox[c][.2in][c]{1in}{\makebox[.5in][l]{}} }

%DOTFILL TO CHECKBOX
%\dotfill \cbox

%SAMPLE TRUE-FALSE QUESTION
%	\item \cbox
%	If $A$ and $B$ are matrices and $AB=0$, then $A=0$ or $B=0$
%	\vspace{2.3in}

%********************************************************
%FIGURES
%********************************************************

%FIGURE WITH GRAPHICX PACKAGE
%\includegraphics{scale=.4}{mypic.png}

%FIGURE WITH GRAPHICS PACKAGE
%\begin{figure}[h]
%\hfil {  \includegraphics{c:/images/OR/blending_feasible_region.eps} }  \hfil
%\caption{Feasible region for Fishback Petroleum LP}\label{Fi:blending_feasible_region}
%\end{figure}
%scale with \scalebox{.5}{ }

%FIGURE WITH PSFRAG TAGS: Make sure Illustrator text format is 12 pt. Times New Roman, regular font
%\begin{figure}[h]
%\centering
%\psfrag{x1_axis_label}[Bc][Bc]{$x_1$}
%\psfrag{x2_axis_label}[Bc][Bc]{$x_2$}
%\psfrag{f_axis_label}[Bc][Bc]{$f$}
%\scalebox{.5}{\includegraphics{c:/images/OR/nonlinear/optimality_sufficient_SOC_quad_form_INDEF.eps} }
%\caption{Quadratic form $\ds f(x_1,x_2)=x_1^2+4x_1x_2+x_2^2$}
%\label{Fi:optimality_sufficient_SOC_quad_form_INDEF}
%\end{figure}

%FIGURE WITH EPSFIG PACKAGE: height SCALES PROPORTIONALY (SEE LATEX COMPANION, P. 317)
% \epsfig{file=c:/images/Lanczos/roof_function.eps,height=1cm.}

%TWO FIGURES SIDE BY SIDE, SCALED, AND LABELLED SEPARATELY
%\begin{figure}[h]
%\centering
%\raisebox{1.4in}{I)}
%\scalebox{.6}{\includegraphics{c:/mth300/Exams/exam2_gradientfield3.eps} }
%\raisebox{1.4in}{II)}
%\scalebox{.6}{\includegraphics{c:/mth300/Exams/exam2_gradientfield1.eps} }
%\end{figure}

%ARRAY OF FIGURES
%\begin{tabular}{ccc}
%\includegraphics{<1>}&\includegraphics{<2>}&\includegraphics{<3>}\\
%\includegraphics{<4>}&\includegraphics{<5>}&\includegraphics{<6>}
%\end{tabular}


%
%**********TEXT AND FIGURE SIDE BY SIDE*******************
%
%\begin{figure}[h]
%\begin{tabular}{lr}

%\raisebox{.8in}{\parbox[b]{3.3in}{1. Suppose $f$ is the function whose contour diagram is shown to the right. Determine
%whether the following partial derivatives are
%positive, negative, or zero. Briefly justify each of your responses. (12 points)

%\vspace{.4in}
%\noindent
%(a) $f_x(P)$

%}}

%&

%\centering
%\psfrag{xaxislabel}[Bc][Bc]{$x$}
%\psfrag{yaxislabel}[Bc][Bc]{$y$}
%\psfrag{z1}[Bc][Bc]{$z=6$}
%\psfrag{z2}[Bc][Bc]{$z=5$}

%\scalebox{.7}{\includegraphics{exam2_W2007_partial_signguess_lost_part.eps} }

%\end{tabular}
%\end{figure}

%******************************************
%MATRICES AND TABLES
%******************************************

%MATRIX COMMANDS
%\begin{bmatrix}a & b \\ c & d \end{bmatrix}

%TABLE COMMANDS
%\begin{tabular}{c|c} $x$ & $y$ \\ \hline a & b \\ c & d \\ e & f \ \end{tabular}

%MULTICOLUMN TABLE
%\begin{table}[h]
%\begin{center}
%\begin{tabular}{|c|c|c|c|c|c|c|}
%\hline
%& \multicolumn{6}{c|}{Genotype of Parents}\\ \hline
%Genotype of Offspring & AA-AA & AA-Aa & AA-aa & Aa-Aa & Aa-aa & aa-aa \\ \hline
%AA &  &  &  & &  &    \\ \hline
%Aa &  &  &  & &  &    \\ \hline
%aa &  &  &  & &  &    \\ \hline
%\end{tabular}
%\end{center}
%\end{table}

%MULTIVARIABLE CALCULUS 2-WAY TABLE
%\begin{table}[h]
%	\begin{center}
%	\begin{tabular}{|c||c|c|c|}
%	\hline
%	$x \diagdown y$ & $.8$ & $1$ & $1.2$ \\ \hline \hline
%	$1.8$ & $5.53$ & $5.85$ & $5.60$\\ \hline
%	$2$   & $6.11$ & $6.14$ & $6.16$ \\ \hline
%	$2.2$ & $6.69$ & $6.42$ & $6.73$ \\ \hline
%	\end{tabular}
%	\end{center}
%	\caption{Table for function $z=f(x,y)$.}
%	\end{table}

%VARIABLE SPACED COLUMNS IN A TABLE
%\begin{tabular}{p{2.5in} p{2.5in}}
%	$\det(BA)$ &   $\det(A^{-1})$ \vspace{2in} \\
%	$\det(3B)$ &   $\displaystyle \det \left( \begin{bmatrix} 1 & 2 & 4 \\ 2 & 0 & -2 \\ 1 & 2 & 2 \end{bmatrix} \right)$
%	\end{tabular}

%TABLE/GRAPH ON ONE LINE COMMANDS
%\begin{figure}[h]
%\centering
%\begin{tabular}{cc}
%	\raisebox{.75in}{\begin{tabular}[b]{c|c} $x$ & $y$ \\ \hline a & b \\ c & d \\ e & f  \end{tabular} } &	
%	\hspace{1in} \includegraphics[2in,2in]{c:/mth310/project/Hamilton.bmp}
%\end{tabular}
%\caption{{\scriptsize Table and graph for $f$.}}
%\end{figure}

%***********************************************
%BOX ENVIRONMENTS
%******************************************

%TEXT IN A BOX VIA \framebox[width][position]{text} WHERE position = l,r,c
%\framebox[2in][l]{Name}

% Fill-in-the-blank box: \framebox{\parbox[c][.2in][c]{3 in}{\makebox[.5in][l]{}} }

%TEXT IN A BOX VIA \fbox
%\setlength{\fboxrule}{2pt}\setlength{\fboxsep}{2mm}
%\fbox{Text in a box}

%RAISEBOX / ROTATEBOX
%\raisebox{lift}{contents}  \rotatebox{-90}{contents}

%SCALEBOX
%\scalebox{fraction}{contents}
%\scalebox{horz fraction }[vert fraction]{contents}

%TRUE-FALSE CHECKBOX
%\newcommand{\checkbox}{\framebox{\parbox[c][.125 in][c]{.125 in}{\makebox[.125in][l]{}} } }

%PARBOX: POSITIONED RELATIVE TO BASELINE c, t, b
%\parbox[pos]{width}{text}

%MINIPAGE: CREATES A MINIVERSION OF THE PAGE WITHIN THE PAGE; CAN BE USED INSIDE FIGURE AND TABLE ENVS
%\begin{minipage}[pos]{width} text
%\end{minipage}

%**************************************************
%MISC.
%**************************************************

%\footnote{text}

%\marginpar[left-text]{right-text} USE SCRIPTSIZE AND CHANGE TEXTWIDTH AT START OF FILE

%WEB URL
%\texttt{http://www.phy.syr.edu/courses/java-suite/crosspro.html}

%MAPLE FILE NAME
%\textbf{extrema.mws}

% POSITIONING
% TO CREATE ENVIRONOMENT: \begin{  center, flushleft, flushright,}
% WITHIN ANOTHER ENVIRONMENT \centering, \raggedright, \raggedleft

%***************************************************
%CONFIDENTIALITY AGREEMENT
%***************************************************

%By signing my name below, I promise that I have not collaborated in any
%way with any other person on this exam and that all the work is entirely
%my own.  In addition, I have not used any written resources other than
%my text or course notes.\\

%\noindent Signature: \underline{\hspace{7.8cm}}\\

%***************************************************

%COOL SYMBOLS   Use with \scalebox{scale factor}{symbol} All but Smiley and last are in wasysym, marvosym, and % %bbding package

\usepackage{wasysym}
\usepackage{marvosym}
\usepackage{pifont}


%TELEPHONE
%{\large{\Telefon}}

%FAX
%{\large{\FAX}

%CELLULAR
%{\large{\Mobilefone}}

%POINTING HAND:
\newcommand{\handpointright}{ \scalebox{2}{$\mbox{\ding{43}}$}}

%PENCIL: Use above with \ding{46}

%CHECKMARK
% \ding{52} %(looks nicer) OR
%\newcommand{\ckmark}{ \mbox{\scalebox{2}{$\ensuremath{\mathbf{\ckmark}}$}}}

%SMILEY FACE: \Smiley or
\newcommand{\smileyface}{ \scalebox{2}{$\mbox{\Smiley}$}}

%CIRCLED NUMBERS (1)-(10): \ding{n} n=172-181 (in pifont}


%***************************************************
%************END OF PREAMBLE************************
%***************************************************


%***************************************************
%************EXAM HEADER****************************
%***************************************************
%\begin{document}
%\noindent{{\LARGE {{\textbf {Final Exam}} }}}
%\hspace{2in} {\Large {{\textbf {Name: }} }}\\
%\noindent{{  {\bfseries{ Fall 2007, Dr. Fishback}} }} \\
%\vspace{.3in}
%\noindent
%Total Score (100 Points Possible): \dotfill
%	\framebox{\parbox[c][.4 in][c]{.7 in}{\makebox[.4in][l]{}} }
%\vspace{-.1in}
%\hrule

%\vspace{.1in}
%\noindent {\bfseries {\underbar{Directions:}}}
%{\small{
%\bi
%\item Show your work! If a question asks for an explanation, keep your response brief and to the point.
%\item Unless otherwise stated, you may use a graphing calculator to assist you on any question, provided you state explicitly
%how you are doing so. However, using symbolic manipulating capabilities of calculators
%such as the TI-89 or TI-92 is not permitted.
%\item If a question asks for an {\underbar{exact}} value, then a calculator approximation is insufficient to receive full
%credit. (For example, $\sqrt{2}$ is exact, whereas $1.414\ldots$ is not.)
%\item The use of a note sheet or external aid of any other type on this quiz is not permitted.
%\item If you have a cell phone {\large{\Mobilefone}}, please turn it completely off at this time.
%\ei
%}}

%***************************************************
%************ACTIVITY HEADER****************************
%***************************************************

\begin{document}
\begin{center}
{\Large {{\textbf {A Game Theory Model of a Predator-Prey Habitat}} }} \\
\end{center}

%\section{A Game Theory Model of a Predator-Prey Habitat}\label{S:GameTheory}

\newpar	
In a desert environment, snakes prey upon rodents in two different habitats, a vegetated setting, hereafter referred to as the \emph{bush}, and the \emph{open}, or exposed, area.\footnote{Based upon research of A. Bouskila, as discussed in A Habitat Selection Game of Interactions Between Rodents and Their Predators,
	\emph{Ann. Zool. Fennici.}, \textbf{38}, 55-70. } Each animal can develop a mixed strategy, choosing one habitat over the other a certain fraction of the time, with payoffs resulting from inter-species interactions. For example, when snakes consume rodents, the species incur positive and negative payoffs, respectively. Likewise, in the absence of such consumption, snake's payoffs are negative rodent's positive.

\newpar 
We will denote $A$ as the payoff matrix for the snake and $B$ as the payoff matrix for the rodent, with the first index of each matrix corresponding to the bush and the second corresponding to the open. For example, $[A]_{12}$ represents the snake's payoff should it reside in the bush and the rodent reside in the open. We expect this quantity to be negative as it signifies a lack of inter-species encounters. Similarly, $[B]_{11}$ denotes the rodent's payoff when both it and the snake select to live in the bush.

\newpar
In addition, we define $\vecx$ and $\vecy$ to be the respective species mixed strategies. The entries of $A$ and $B$ depend upon the probabilities of inter-species encounters as well as certain energy gains and losses resulting form such encounters as well as fixed energy costs resulting from choosing one habitat over the other. While there are many different ways to interpret energy costs, we consider it be a net measure of increased or decreased reproduction levels.

\newpar
The Table \ref{Ta:predator_prey_field_data}summarizes the various parameters that determine the entries of $A$ and $B$. These parameters are based upon a statistical analysis of collected field data.

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

\be
\item Use the data above to construct the payoff matrices, $A$ and $B$, for the snake and rodent, respectively. The matrix $A$ should have positive diagonal entries and negative entries elsewhere. Conversely, the matrix $B$ should have negative diagonal entries and positive entries elsewhere. Explain why the signs of the entries make sense, given the fact the snake preys upon
    the rodent.
\item If the snake lives in a habitat different from the rodent, which choice yields the snake a larger energy gain?
\item If the rodent lives in a habitat different from the rodent, which choice results in a smaller energy loss?
\item Use the payoff matrices to construct a bimatrix game in which each species attempts to maximize its net energy gain. Determine the mixed strategy of each species that corresponds to the Nash equilibrium. What are the species' respective payoffs?
\item Suppose the snake deviates from its equilibrium strategy and chooses the bush 10\% less. What is its new payoff, assuming the rodent's strategy remains the same?
\item Suppose the rodent deviates from its equilibrium strategy and chooses the open 10\% less. What is its new payoff, assuming the snake's strategy remains the same?

\ee 









\end{document}
