\begin{frame}%[label=this one]
  \frametitle{Système proies/prédateurs (système)}
  \framesubtitle{Description du système}

  \begin{columns}
    \column{.7\textwidth}\centering
    \includegraphics[width=.7\textwidth]{raster/system-predator-prey}\\
    \mkCitation{S. S. Mader, Biology 6th edition, 1998}

  \bigskip
  \begin{itemize}
      \item Les proies se reproduisent spontanément
      \item Les prédateurs meurent spontanément
      \item Les prédateurs chassent les proies
        \begin{itemize}
          \item Les proies meurent (chassées)
          \item Les prédateurs peuvent se reproduire (efficacité de la chasse)
        \end{itemize}
    \end{itemize}
  \end{columns}

\end{frame}

\begin{frame}%[label=this one]
  \frametitle{Système proies/prédateurs (modèles)}
  \framesubtitle{Modèles et formalismes}
  \centering
  \begin{tikzpicture}[overlay,%
    node distance=13mm and 20mm,
    model/.style={draw,circle,on grid=true,fill=white,minimum size=24pt},
    abstraction/.style={-Stealth,thick}
    ]
    \node[model, left=of current page.center] (m1) {$\model_1$};
    \node[model,above=of current page.center] (m2) {$\model_2$};
    \uncover<2->{
      \node[model,below=of current page.center] (m3) {$\model_3$};
    }
    \uncover<3>{
      \node[model,right=of current page.center] (m4) {?};
    }
    \uncover<4->{
      \node[model,right=of current page.center] (m4) {$\model_4$};
    }
    \draw[abstraction] (m1) to node[anchor=south,midway,sloped]
    {\tiny trajectoire moyenne} (m2);
    \uncover<2->{
      \draw[abstraction] (m1) to node[anchor=north,midway,sloped,swap]
      {\tiny aggrégation de l'espace} (m3);
    }
    \uncover<3>{
      \draw[abstraction,dashed] (m2) to (m4);
      \draw[abstraction,dashed] (m1) to (m4);
      \draw[abstraction,dashed] (m3) to (m4);
    }
    \uncover<4->{
      \draw[abstraction] (m2) to node[anchor=south,midway,sloped,swap]
      {\tiny aggrégation de l'espace} (m4);
      \draw[abstraction] (m3) to node[anchor=north,midway,sloped]
      {\tiny trajectoire moyenne} (m4);
    }
  \end{tikzpicture}

  \begin{tikzpicture}[overlay]
    \uncover<1-2>{
      \node[anchor=south east] (m1eq) at (m1.north west) {\tiny$
        \left\{
          \arraycolsep=1.4pt%\def\arraystretch{2.2}
          \begin{array}{rcl}
            \displaystyle\frac{dU_V}{dt} &=& r U_V (1 - \displaystyle\frac{U_V}{K})
            - g U_P U_V\\[1.5ex]
            \displaystyle\frac{dU_P}{dt} &=& n U_V U_P - \mu U_P
          \end{array}
        \right.%}
      $};
      \node[anchor=north] at (m1eq.south) {\includegraphics[height=2cm]{vector/lv-graph}};
      \node[anchor=south east] (m2eq) at ($(m2.north west)+(0,7pt)$) {\tiny$
        \arraycolsep=1.4pt%\def\arraystretch{0}
        \begin{array}{rl}
          V + E &\overset{b}{\longrightarrow} 2\,V    \\
          P + V &\overset{p_1}{\longrightarrow} 2\,P  \\
          P + V &\overset{p_2}{\longrightarrow} P + E
        \end{array}
        \quad
        \begin{array}{rl}
          V     &\overset{d_V}{\longrightarrow} E     \\
          P     &\overset{d_P}{\longrightarrow} E
        \end{array}
      $};
      \node[anchor=west] at (m2eq.east) {\includegraphics[width=2cm]{vector/lv-gil-graph}};
    }
    \uncover<2>{
      \node[anchor=north] (m3eq) at (m3.south) {\tiny$
        \left\{
          \arraycolsep=1.4pt%\def\arraystretch{2.0}
          \begin{array}{rcll}
            \displaystyle\frac{du_V}{dt} &=& r u_V (1 - \displaystyle\frac{u_V}{K}) - g u_P u_V \\
                                         &+& D_V (\nabla^2 u_V + u_V \nabla^2 u_P - u_P \nabla^2 u_V)
            \\[1.5ex]
            \displaystyle\frac{du_P}{dt} &=& n u_V u_P - \mu u_P \\
                                         &+& D_P (\nabla^2 u_P + u_P \nabla^2 u_V - u_V \nabla^2 u_P)
          \end{array}
        \right.%}
      $};
    }
    \uncover<4>{
      \node[anchor=west]  at (m4.east) {\structure{[Lugo et McKane 2008]}};
    }
  \end{tikzpicture}
\end{frame}
