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\centerline{Workshop \#8, cellular-automata, outline for TF, not students}
\centerline{APM 115: mathematical modeling}

\bigskip
cellular automata (CA) of a predator-prey (rabbit-wolf) model
\bigskip

\begin{itemize}
\item Give the Matlab program simulating the Lotka-Volterra equations
  (ODEs) to the students, have them run it and observe the
  oscillations.
\item Give them next the CA model and Matlab code that corresponds to
  the above equations, and adds the spatial dimensions.  Before giving
  them the code, remove the parts that calculate the total number of
  wolfs and rabbits as function of time and remove the part that
  simulates random motion of rabbits and wolves.
\item The model considers only the four near neighbors of a given
  cell, not including diagonal cells.  Have the student come up with,
  or justify, the following CA rules as an attempt to reproduce the
  Lotka-Volterra dynamics.  In particular, which rule corresponds to
  which term in which equation and why?
  \begin{itemize}
  \item An empty cell with rabbit neighbor(s) turns into a rabbit with
    a probability $a$.
  \item A rabbit with wolf neighbor(s) has three possible outcomes:
    (1) empty (rabbit eaten\ldots, probability $b$); (2) rabbit eaten and
    wolf replaces it (wolf reproduced, probability $c$) and (3)
    nothing (probability $1-b-c$).
  \item A wolf cell has a probability $d$ to turn into an empty cell
    (predator death).
  \item The model assumes a ``double periodic'' domain.
  \end{itemize}
\item Have the students run the CA code for these rules.  The model
  uses for loops and if statements for simplicity (rather than the
  more efficient manipulation of logical arrays).
\item Have the students add the calculation of, and plot, the global
  sum of predator and prey in the CA model as function of iteration
  number (equivalent to time).  Can you adjust the various specified
  probabilities of the CA model to obtain a similar result to the ODE
  models?
\item Have the students implement random motion of rabbits and wolves:
  After each step, move each rabbit and wolf to a randomly selected
  nearby location, if that location is empty.  If not empty, leave
  rabbit or wolf unmoved.
\end{itemize}

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