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\Blatt{5}{WS 2008/2009, 20.11.2008}
         {Theoretische Biophysik I}
         {Prof. J.~L.~van~Hemmen}

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\item[5.]
{\bf Symbiotischer Prozess}
\par\smallskip
  Bestimmen Sie die allgemeine Lösung des symbiotischen Prozesses mit dem
  Differenzialgleichungssystem ($\alpha>0,\, \beta>0$)
  \begin{eqnarray*}
    \dot{x}(t) & = & \alpha \, y(t) \\
    \dot{y}(t) & = & \beta \, x(t) 
  \end{eqnarray*}

\item[6.]
{\bf Gefechtsmodell}
\par\smallskip
  Bestimmen Sie die allgemeine Lösung des destruktiven Prozesses mit dem
  Differenzialgleichungssystem ($\alpha>0,\, \beta>0$)
  \begin{eqnarray*}
    \dot{x}(t) & = & -\alpha \, y(t) \\
    \dot{y}(t) & = & -\beta \, x(t) 
  \end{eqnarray*}
  für $x(t)>0$ bzw. $y(t)>0$, andernfalls $x(t)=0$ bzw. $y(t)=0$.
  Lösen Sie das Anfangswertproblem $x(0)=x_0$, $y(0)=y_0$. Wer
  "`gewinnt"'?

\item[7.]
{\bf Drei-Klassen-Gesellschaft}
\par\smallskip
mit Aufstiegschancen. Klasse 1 steigt mit Rate $r_1$ zu Klasse 2 auf,
Klasse 2 mit Rate $r_2$ zu Klasse 3. 

\begin{itemize}
\item Stellen Sie das zugehörige Differentialgleichungssystem auf.
\item Wie entwickelt sich die Gesellschaft?
\item Betrachten Sie eine Teilgesellschaft der Klasse 1. Wie
  entwickelt sich diese?
\end{itemize}




% \item[8.]
% {\bf Gras-Schafe-Wölfe-Modell}
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% Die Schafe $S$ fressen das Gras $G$ , die Wölfe $W$ die Schafe.
% \begin{eqnarray}
%   \dot{S} = 
% \end{eqnarray}





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\footnotesize
\rule[.5mm]{\textwidth}{0.1mm}\\
Besprechung der Übungen am Donnerstag 27.11.2008 um 8.30 Uhr im
Raum PH 227. \\
Übungsleitung: Dr. Moritz Franosch, mail@Franosch.org,
http://www.t35.ph.tum.de.

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