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\Blatt{4}{WS 2008/2009, 13.11.2008}
         {Theoretische Biophysik I}
         {Prof. J.~L.~van~Hemmen}

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\item[4.]
{\bf Populationsmodelle mit Altersverteilung}
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Sei $n(t,a)$ die Populations\emph{dichte} bezüglich des Alters in
Abhängigkeit der Zeit $t$, d.h. die Zahl der Individuen pro
Altersintervall zum Zeitpunkt $t$.
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\item[(a)] 
  Falls $\mu(a)$ die altersabhängige Sterberate ist, dann gilt
  \begin{equation}
    \frac{\partial n(t,a)}{\partial t} + \frac{\partial
      n(t,a)}{\partial a} = - \mu(a) n(t,a) \,.
  \end{equation}
  Warum?
\item[(b)] 
  Leiten Sie für $t<a$ bei gegebener Anfangsverteilung $n(0,a)$ die
  zeitliche Entwicklung der Populationsdichte
  \begin{equation}
    n(t,a) = n(0,a-t) \exp\left( - \int_{a-t}^a \mu(s) \, \mathrm{d}s \right)
  \end{equation}
  her.
\item[(c)]
  Wir nehmen an, dass altersabhängig Kinder geboren werden mit einer
  Rate $b(a)$, also
  \begin{equation}
    n(t,0) = \int_0^\infty n(t,a) b(a) \, \mathrm{d}a \,.
  \end{equation}
  Wie entwickelt sich dann die Population, diesmal für beliebige $t>0$
  und $a>0$?
\item[(d)] Wir betrachten nun den Spezialfall eines
  Erneuerungsprozesses (\emph{renewal process}): Bei Ableben eines
  Individuums wird dieses Individuum im selben Moment durch ein neues
  mit Alter 0 ersetzt (\emph{reset or renewal to age 0}). Zeigen Sie,
  dass dann wie erwartet die Größe der Gesamtpopulation $n(t) =
  \int_0^\infty n(t,a) \, \mathrm{d}a$ konstant bleibt. Zeigen Sie
  außerdem, dass für große Zeiten für die Dichte der Neugeborenen
  $g(t) := n(t,0)$ gilt:
  \begin{equation}
    g(t) = \int_0^\infty \mu(a) \exp\left( - \int_0^a \mu(s)
      \mathrm{d}s \right) g(t-a) \, \mathrm{d}a \,.
  \end{equation}

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\footnotesize
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Besprechung der Übungen am Donnerstag 20.11.2008 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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