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\Blatt{6}{WS 2008/2009, 11.12.2008}
         {Theoretische Biophysik I}
         {Prof. J.~L.~van~Hemmen}

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\item[8.]
{\bf Delay-Differenzialgleichungen}
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Die Walfangkommission will das Verhalten der Walpopulation
vorhersagen. Dazu benutzt sie folgende Differenzialgleichung:
\begin{equation} \label{eq:DGL}
  \frac{d}{dt} x(t) = -x(t) + x(t-T) g(x(t-T))
\end{equation}
mit der Vermehrungsrate
\begin{equation}
  g(y) = 1+ q \left[ 1 - \left( \frac{y}{k} \right)^z \right],
\end{equation}
wobei $q$, $z$, $k$ Konstanten, $x(t)>0$ und $T$ die Zeitverzögerung
ist. Das Ziel ist, eine stabile Population zu gewährleisten. Sie
sollen daher untersuchen, ob Oszillationen auftreten können.
\begin{enumerate}
\item Machen Sie das Modell plausibel. Zeigen Sie dazu, dass eine
  Erhöhung der Sterberate aufgrund von Walfang (ersetze $1 \to 1+
  \varrho$) wegnormiert werden kann, wenn dazu die anderen Konstanten
  geändert werden.
\item Bestimmen Sie die beiden Fixpunkte in \eqref{eq:DGL}.
\item Linearisieren Sie um den positiven Fixpunkt, um dessen
  Stabilität zu untersuchen.
\item Machen Sie den üblichen Exponential-Ansatz für lineare
  Systeme. Er liefert $\lambda = -1-b\exp(-\lambda T)$.
\item Gibt es rein reelle Lösungen für $\lambda$?
\item In welchem Bereich sind die Lösungen stabil? Zeigen Sie, dass
  für $T>T_c$ spontane Oszillationen auftreten, wobei
  \begin{equation}
    T_c = \frac{\pi - \arccos(1/b)}{\sqrt{b^2-1}} \,.
  \end{equation}
  Zeigen Sie, dass die Oszillationsperiode in der Umgebung $T \approx T_c$
  näherungsweise $2 \pi / \sqrt{b^2-1}$ beträgt. Hierbei ergibt sich
  $T_c$ aus der Bedingung "`Realteil von $\lambda$ gleich 0."'.
\item Interpretieren Sie Ihre Ergebnisse. Welche Empfehlung könnte die
  Walfangkommission aussprechen?
\end{enumerate}



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\footnotesize
\rule[.5mm]{\textwidth}{0.1mm}\\
Besprechung der Übungen am Donnerstag 8.1.2009 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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