\documentclass[a4paper,12pt,pdftex]{exam}

\newcommand{\ptitle}{Random walk}
\input{../header.tex}
\firstpagefooter{Supervisor: Jan Grewe}{phone: 29 74588}%
{email: jan.grewe@uni-tuebingen.de}

\begin{document}

\input{../instructions.tex}


%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Random walk with memory.}

The movement pattern of some animals can be described as a random walk when
searching for food. In some cases this random walk is not completely
random. In fact, sometimes there is some memory involved. Whenever
there is a positive gradient in food gain between successive steps the
animal will continue in the very same direction as in the step before. When the
next step leads to a decrease in food gain the animal switches back to
a random walk and changes directions randomly.


\begin{questions}
  \question{} The accompanying dataset (random\_world.mat) contains a
  single variable. This is the world (10000\,m$^2$ area with
  10\,cm spatial resolution) in which there are randomly distributed
  food sources (Gaussian blotches of food).
  
  \begin{parts}
    \part{} Create a plot of the world using \code{imshow}.\\[0.5ex]
    \part{} Create a model animal (agent) that performs a pure random walk. The
    agent can walk in eight different directions (the cardinal and
    diagonal directions) with a stepsize of 10\,cm
    (approximately). Let the agent start at a random location in the
    world and count how much food it eats in 10000 steps (eaten food
    disappears from the world, of course). If the agent bumps into the
    borders of the world choose a different direction.\\[0.5ex]
    \part{} Plot a typical example walk. (You can also make an animation
    with MATLAB, see plotting chapter in the script).\\[0.5ex]
    \part{} Same as above, but create a model animal that has some memory,
    i.e. the direction is kept constant as long as there is a positive
    gradient in the food gain. Otherwise, a random walk is performed.\\[0.5ex]
    \part{} Plot a typical example walk also for this agent.\\[0.5ex]
    \part{} Compare the performance of the two agents. Create
    appropriate plots and apply statistical methods. You will need to
    run the simulations several times to get a good estimate of the
    neumbers.
    \part{} Can you think about better search strategies?
  \end{parts}
\end{questions}

\end{document}