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\Blatt{7}{WS 2008/2009, 8.1.2009}
         {Theoretische Biophysik I}
         {Prof. J.~L.~van~Hemmen}

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\item[9.]
{\bf Logistische Gleichung}
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Das logistische Wachstumsmodell einer Bevölkerung $x$ lautet in
differenzieller Form
\begin{equation}
  \label{eq:logit}
  \frac{\mathrm{d}}{\mathrm{d}t} x(t) = r x \left( 1 - \frac{x}{K} \right) \,.
\end{equation}
\begin{enumerate}
\item Iterpretieren Sie die Konstanten $r$ und $K$. Was ist die
  Geburtenrate, was ist die Todesrate?
\item Berechnen Sie die Fixpunkte der Differenzialgleichung
  \eqref{eq:logit} und untersuchen Sie sie auf Stabilität.
\item Lösen Sie die Differenzialgleichung \eqref{eq:logit} unter der
  Anfangsbedingung $x(0)=x_0$.
\item Welches Wachstum ergibt sich für kleine positive Zeiten $t$ und
  $x_0 \ll K$?
\end{enumerate}


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\footnotesize
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Besprechung der Übungen am Donnerstag 15.1.2009 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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