% !TEX TS-program = pdflatex
% !TEX encoding = UTF-8 Unicode

% This file is a template using the "beamer" package to create slides for a talk or presentation
% - Talk at a conference/colloquium.
% - Talk length is about 20min.
% - Style is ornate.

% MODIFIED by Jonathan Kew, 2008-07-06
% The header comments and encoding in this file were modified for inclusion with TeXworks.
% The content is otherwise unchanged from the original distributed with the beamer package.

\documentclass{beamer}


% Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
%
% In principle, this file can be redistributed and/or modified under
% the terms of the GNU Public License, version 2.
%
% However, this file is supposed to be a template to be modified
% for your own needs. For this reason, if you use this file as a
% template and not specifically distribute it as part of a another
% package/program, I grant the extra permission to freely copy and
% modify this file as you see fit and even to delete this copyright
% notice.


\mode<presentation>
{
  \usetheme{Szeged}
\usecolortheme{beaver}
  % or ...

  \setbeamercovered{transparent}
  % or whatever (possibly just delete it)
}


\usepackage[english]{babel}
% or whatever

\usepackage[utf8]{inputenc}
% or whatever

\usepackage{amsmath}

\usepackage{times}
\usepackage[T1]{fontenc}
% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.

\newcommand{\ds}{\displaystyle}

\title[Synchrony in a Two Patch Model] % (optional, use only with long paper titles)
{Synchronous Oscillatory Solutions in a Two Patch Predator-Prey Model}

%\subtitle
%{Include Only If Paper Has a Subtitle}

\author[Becker, Shi, Shaw] % (optional, use only with lots of authors)
{Matthew Becker\inst{1} \and Prof. Leah Shaw\inst{2} \and Prof. Junping Shi\inst{1}}
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
%   affiliation.

\institute[William \& Mary] % (optional, but mostly needed)
{
  \inst{1}%
  Department of Mathematics\\
  College of William and Mary
  \and
  \inst{2}%
  Department of Applied Science\\
  College of William and Mary}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.

\date[June 9th, 2011] % (optional, should be abbreviation of conference name)
{CSUMS Presentation \\ June 9th, 2011}
% - Either use conference name or its abbreviation.
% - Not really informative to the audience, more for people (including
%   yourself) who are reading the slides online

%\subject{Theoretical Computer Science}
% This is only inserted into the PDF information catalog. Can be left
% out.



% If you have a file called "university-logo-filename.xxx", where xxx
% is a graphic format that can be processed by latex or pdflatex,
% resp., then you can add a logo as follows:

\pgfdeclareimage[height=0.7cm]{tribe-logo.png}{tribe-logo.png}
\logo{\pgfuseimage{tribe-logo.png}}



% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
%\AtBeginSubsection[]
%{
 % \begin{frame}<beamer>{Outline}
 %   \tableofcontents[currentsection,currentsubsection]
%  \end{frame}
%}


% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command:

%\beamerdefaultoverlayspecification{<+->}


\begin{document}

\begin{frame}
  \titlepage
\end{frame}

\begin{frame}{Outline}
  \tableofcontents
  % You might wish to add the option [pausesections]
\end{frame}


% Structuring a talk is a difficult task and the following structure
% may not be suitable. Here are some rules that apply for this
% solution:

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

% - A conference audience is likely to know very little of what you
%   are going to talk about. So *simplify*!
% - In a 20min talk, getting the main ideas across is hard
%   enough. Leave out details, even if it means being less precise than
%   you think necessary.
% - If you omit details that are vital to the proof/implementation,
%   just say so once. Everybody will be happy with that.

\section{Introduction}

\subsection{Predator-Prey Background}

\begin{frame}{Predator-prey model}
  % - A title should summarize the slide in an understandable fashion
  %   for anyone how does not follow everything on the slide itself.

  \begin{itemize}
  \item
    Predator-Prey models are differential equations that model the interaction between two species. One species gains from this interaction while the other is hurt.
  \item
    These models are used to study the dynamic behavior of interacting species.
  \item
    The dynamics of each population depends upon the other.
  \end{itemize}
\end{frame}

\begin{frame}{Two Patch Model}
\begin{itemize}

\item
There are two different populations in two different geographic populations, but the populations can disperse from one patch to the other.
\item
Prey and predator may have different diffusion coefficients.
\end{itemize}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.4\textwidth,keepaspectratio]{animal-migration-caribou.png}
%\caption{The figure shows the oscillations in population that predator-prey models attempt to recreate.  \label{fig1}}
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Lynx and Hare}
\begin{figure}[t]
\includegraphics[width=0.25\textwidth,keepaspectratio]{snowshoehare.png}
%\caption{The figure shows the oscillations in population that predator-prey models attempt to recreate.  \label{fig1}}
%\end{figure}
%\begin{figure}[t]
\hspace{0.2in}
\includegraphics[width=0.25\textwidth,keepaspectratio]{lynx.png}
%\caption{The figure shows the oscillations in population that predator-prey models attempt to recreate.  \label{fig1}}
\end{figure}
\end{frame}

\begin{frame}{Lynx and Hare}
  \begin{itemize}
    \item
      The Lynx-Hare model is the most well known applied predator-prey model.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.4\textwidth,keepaspectratio]{Lynx-Hare.png}
%\caption{The figure shows the oscillations in population that predator-prey models attempt to recreate.  \label{fig1}}
\end{center}
\end{figure}
\item
The Hudson Bay Company kept records of the populations over one hundred years. This is possibly the most pure example of predator-prey dynamics in nature.
  \end{itemize}
\end{frame}


\subsection{Synchrony}

\begin{frame}{Introduction}
\begin{itemize}
\item
Synchronization is the process in which oscillations attain the same frequency based on their coupling.
\item
Due to the diffusion, along with other factors, between two patches, the populations in each patch may become synchronized over time.
\end{itemize}
\end{frame}

\begin{frame}{Examples}
\begin{figure}[t]
\includegraphics[width=0.5\textwidth,keepaspectratio]{Synchronyexamples.png}
\caption{Coupled pendulums are typical coupled oscillators which may synchronize.}
\end{figure}
\end{frame}

\begin{frame}{Examples}
\begin{figure}[t]
\includegraphics[width=0.5\textwidth,keepaspectratio]{firefliessmokymountains.png}
\caption{Fireflies synchronize their flashes in order to extend the distance the light can be seen.}
\end{figure}
\end{frame}

\end{document}