110 lines
3.9 KiB
TeX
110 lines
3.9 KiB
TeX
\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}
|