109 lines
3.9 KiB
TeX
109 lines
3.9 KiB
TeX
\begin{frame}%[label=this one]
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\frametitle{Système proies/prédateurs (système)}
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\framesubtitle{Description du système}
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\begin{columns}
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\column{.7\textwidth}\centering
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\includegraphics[width=.7\textwidth]{raster/system-predator-prey}\\
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\mkCitation{S. S. Mader, Biology 6th edition, 1998}
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\bigskip
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\begin{itemize}
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\item Les proies se reproduisent spontanément
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\item Les prédateurs meurent spontanément
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\item Les prédateurs chassent les proies
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\begin{itemize}
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\item Les proies meurent (chassées)
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\item Les prédateurs peuvent se reproduire (efficacité de la chasse)
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\end{itemize}
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\end{itemize}
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\end{columns}
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\end{frame}
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\begin{frame}%[label=this one]
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\frametitle{Système proies/prédateurs (modèles)}
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\framesubtitle{Modèles et formalismes}
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\centering
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\begin{tikzpicture}[overlay,%
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node distance=13mm and 20mm,
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model/.style={draw,circle,on grid=true,fill=white,minimum size=24pt},
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abstraction/.style={-Stealth,thick}
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]
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\node[model, left=of current page.center] (m1) {$\model_1$};
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\node[model,above=of current page.center] (m2) {$\model_2$};
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\uncover<2->{
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\node[model,below=of current page.center] (m3) {$\model_3$};
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}
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\uncover<3>{
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\node[model,right=of current page.center] (m4) {?};
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}
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\uncover<4->{
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\node[model,right=of current page.center] (m4) {$\model_4$};
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}
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\draw[abstraction] (m1) to node[anchor=south,midway,sloped]
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{\tiny trajectoire moyenne} (m2);
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\uncover<2->{
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\draw[abstraction] (m1) to node[anchor=north,midway,sloped,swap]
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{\tiny aggrégation de l'espace} (m3);
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}
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\uncover<3>{
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\draw[abstraction,dashed] (m2) to (m4);
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\draw[abstraction,dashed] (m1) to (m4);
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\draw[abstraction,dashed] (m3) to (m4);
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}
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\uncover<4->{
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\draw[abstraction] (m2) to node[anchor=south,midway,sloped,swap]
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{\tiny aggrégation de l'espace} (m4);
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\draw[abstraction] (m3) to node[anchor=north,midway,sloped]
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{\tiny trajectoire moyenne} (m4);
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}
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\end{tikzpicture}
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\begin{tikzpicture}[overlay]
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\uncover<1-2>{
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\node[anchor=south east] (m1eq) at (m1.north west) {\tiny$
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\left\{
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\arraycolsep=1.4pt%\def\arraystretch{2.2}
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\begin{array}{rcl}
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\displaystyle\frac{dU_V}{dt} &=& r U_V (1 - \displaystyle\frac{U_V}{K})
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- g U_P U_V\\[1.5ex]
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\displaystyle\frac{dU_P}{dt} &=& n U_V U_P - \mu U_P
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\end{array}
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\right.%}
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$};
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\node[anchor=north] at (m1eq.south) {\includegraphics[height=2cm]{vector/lv-graph}};
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\node[anchor=south east] (m2eq) at ($(m2.north west)+(0,7pt)$) {\tiny$
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\arraycolsep=1.4pt%\def\arraystretch{0}
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\begin{array}{rl}
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V + E &\overset{b}{\longrightarrow} 2\,V \\
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P + V &\overset{p_1}{\longrightarrow} 2\,P \\
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P + V &\overset{p_2}{\longrightarrow} P + E
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\end{array}
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\quad
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\begin{array}{rl}
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V &\overset{d_V}{\longrightarrow} E \\
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P &\overset{d_P}{\longrightarrow} E
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\end{array}
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$};
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\node[anchor=west] at (m2eq.east) {\includegraphics[width=2cm]{vector/lv-gil-graph}};
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}
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\uncover<2>{
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\node[anchor=north] (m3eq) at (m3.south) {\tiny$
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\left\{
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\arraycolsep=1.4pt%\def\arraystretch{2.0}
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\begin{array}{rcll}
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\displaystyle\frac{du_V}{dt} &=& r u_V (1 - \displaystyle\frac{u_V}{K}) - g u_P u_V \\
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&+& D_V (\nabla^2 u_V + u_V \nabla^2 u_P - u_P \nabla^2 u_V)
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\\[1.5ex]
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\displaystyle\frac{du_P}{dt} &=& n u_V u_P - \mu u_P \\
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&+& D_P (\nabla^2 u_P + u_P \nabla^2 u_V - u_V \nabla^2 u_P)
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\end{array}
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\right.%}
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$};
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}
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\uncover<4>{
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\node[anchor=west] at (m4.east) {\structure{[Lugo et McKane 2008]}};
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}
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\end{tikzpicture}
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\end{frame}
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