\documentclass{beamer}
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\title[Chaos in chemotaxis]
      {Spatio-Temporal Chaos in Chemotaxis Models}

\author{Thomas Hillen}

\institute[University of Alberta ]{University of Alberta\\
\bigskip

{\em with K.J. Painter, (Edinburgh)}}

\date{ }
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\begin{frame}
\frametitle{A chemotaxis model with growth}

$u(x,t)$: cell density\\
$v(x,t)$: chemical signal\\

\begin{equation} \label{minimal}
\begin{array}{rcl}
u_t &  = & \nabla \left( D \nabla u - \chi u \nabla v \right) + r u (1-u)\,, \\
v_t & = & \Delta  v + u - v\,. 
\end{array}
\end{equation}

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\begin{frame}
\begin{itemize}
\item Consider on $[0,L]$ with homogeneous boundary conditions.
\pause
\item Initial conditions
\[ (u(x,0), v(x,0)) = (1, 1+\varepsilon(x)), \qquad |\varepsilon|<10^{-2}\]
\pause
\item Fix $\chi=10, D=1$ and vary $L$:
\end{itemize}

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\begin{frame}\frametitle{Fourier Modes of Step Function}

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\includegraphics[width=6cm]{step-Fourplot2d2.eps}
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