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\begin{center}
\parbox{10cm}{\small The title of this series of technical reports constitute
my homage to the legendary magazine \emph{Cahiers du cin\'ema}, and to the
role that it has played for many years in the creation of a widespread
critical sense regarding film.
}

\vspace{0.5cm}

{\small (C) Simone Santini, \copyrgt}
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\begin{center}
  \parbox{30em}{ These notes constituted the class material for part
    of the syllabus of the graduate course \dqt{Caracterizaci\'on de
      Redes y Topolog{\'\i}as Biol\'ogicas} read during the academic
    years 2017-18 and 2018-19. I gratefully acknowledge the
    contribution of all the students of these classes who, with their
    interaction, have contributed to improve these notes.}

  \vspace{10em}

  Author contact: \textbf{simone.santini@uam.es}
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\section{\sc ARS, reward, and dopamine in the evolution of life}

The starting point of these notes is a talk to which I heard few
years back in Milan, Italy. The speaker, Giuseppe Boccignone, showed
us the pair of images in Figure~\ref{wow}, which represent two paths
in two dimensions. It is not hard to see that the macroscopic
characteristics of these images are quite the same. Their most evident
macroscopic feature is that they are heavily clustered: there are a
bunch of points in a very restricted area and then suddenly the path
jumps to a different area where a new cluster of points is created.

\begin{figure}[bth]
  \begin{center}
    \VInsert{graphs/saccade.jpg}{0.3\columnwidth}
  \end{center}
  \caption{\capstyle Two images that, apparently, describe the same
    phenomenon. While the paths displayed in the two images are
    qualitatively similar, their origin is quite different. The path
    on the right is a recording of the saccadic eye movements of a
    person looking at the image to which it is superimposed. The path
    of the left is that of a spider monkey (a monkey common in
    the Yucatan peninsula of Mexico) looking for food. The two paths
    are example of a common behavior found throughout the animal
    kingdom: \emph{Area-Restricted Search}.}
  \label{wow}
\end{figure}

The images are so similar that one would have little trouble believing
that they have been created by two instances of the same physical
phenomenon. Yet, much to my amazement, Boccignone told us that this
was not the case.  The figure on the right shows the saccadic eye
movements of a person looking at the picture that you can faintly see
in the background; the picture on the left is the path followed by a
spider monkey of the Yucatan peninsula while looking for food. It has
nothing to do with the picture and was superimposed to it only to make
the point more forcefully.

To find such a similarity in completely unrelated activities of two
different species is as striking as it would be to find out that the
ritualistic chant of a remote tribe in Papua has the same harmonic
structure as the Goldberg Variations. Just as in the case of the tribe
we would like to look for an explanation (maybe a previous contact
with some Bach-loving explorer that has been incorporated into the
rituals of the tribe), so in this case it is not too far-fetched to
start looking for some common underlying mechanism.

We are encouraged in our endeavor by the fact that the two behaviors
do have indeed something in common: they are both examples of
\emph{search}. Search for visual information in one case, search for
food in the other. So, we are on a hunt for a common mechanism that
guides \emph{search} in a wide variety of species under the most
diverse circumstances. The mechanism must be very general, since it
should apply not only to different species but also to very different
levels of abstraction (from search for food in physical space to
search for information in conceptual space).

The behavior that we observe in these two examples is commonly known
as \emph{ARS (Area Restricted Search)}, a strategy that consists in
\dqt{a concentration of searching effort around areas where rewards in
  the form of specific resources have been found in the past. When
  resources are encountered less frequently, behavior changes such
  that the search becomes less sensitive, but covers more space}
\shortcite{hills:06}.

As we shall see, the same basic mechanism permits ARS in a variety of
cases and circumstances, from the foraging behavior of the nematode
\emph{C.elegans} to goal-directed cognition in people. You can have a
personal experience of ARS by looking at Figure~\ref{ARS-exp} and
following the instructions in the caption (read the caption before
looking at the pictures).

\begin{figure}[bth]
  \HInsert{graphs/ARS_exp.jpg}{\columnwidth}
  \caption{\capstyle Paying attention to where your eyes look, begin
    in the left figure and look for the upside-down triangle (there is
    only one). Once you have found it, move to the figure to the right
    and look for the upside-down triangle there. Go ahead and do that
    now, before you read the rest of the caption.  Where did you look
    first when you were looking for the triangle in the second figure?
    Did you look first near the black hearts? If you did, then you
    were performing ARS.  You focused the attention in the area where
    you expected (based on your previous experience) the reward (the
    upside-down triangle) to be found and then, once you found out
    that your \dqt{confidence} area did not have the resource you were
    looking for, you started a rapid scan of the rest of the image,
    until you found the sought-after triangle.}
  \label{ARS-exp}
\end{figure}

ARS is incredibly widespread. Some form of it has been found in all
major eumetazoan clades. To have an idea of what this entails, in
Figure~\ref{taxonomy} I have drawn a very partial taxonomy of the
animal kingdom.

\begin{figure}
  \begin{center}
    {\small
    $\displaystyle
    \xymatrix@R=1em@C=4em{
      & & \mbox{metazoa} \ar@{-}[dr] \ar@{-}[dl] \\
      & \mbox{porifera} & & \mbox{eumetazoa} \ar@{-}[dr] \ar@{-}[dl] \\
      &        & \mbox{bilatera} \ar@{-}[dll] \ar@{-}[dl] \ar@{-}[d] \ar@{-}[drr] & & \mbox{radiata} \\
\mbox{lophotrochozoa} \ar@{-}[dd]& \mbox{platyzoa} & \mbox{ecydozoa} \ar@{-}[dr] \ar@{-}[dl] \ar@{-}[d] & & \mbox{deuterostomia} \ar@{-}[d] \\
                      &  {} \ar@{..}[r] & \mbox{nemotoda}\ar@{..}[r] & {} & \mbox{chordata} \ar@{-}[dr] \ar@{-}[dl] \ar@{-}[d] \\
(mollusks)      &       &   & {} \ar@{..}[r] & \mbox{craniata} \ar@{-}[dr] \ar@{-}[dl] \ar@{..}[r] & {} \\
      &       &   &  \mbox{vertebrata} & & \mbox{myxini}
    }
    $
    }
  \end{center}
  \caption{\capstyle A (very small) fragment of the taxonomy of \emph{metazoa}
    (i.e.\ animals). The group of \emph{porifera} is composed of
    animals without tissue, and consists pretty much of sponges and
    little else. The clade of the eumetazoa contains all other
    animals, and in this whole clade ARS has been
    observed.}
  \label{taxonomy}
\end{figure}

The clade on the left of the root, the \emph{porifera} is composed of
animals that do not have a real tissue: sponges and little else. The
other clade, \emph{eumetazoa} contains all other animals, from worms
to mollusks to you and me. ARS can be observed, in some form or
another, in the whole eumetazoa clade. This broad presence indicates
that the mechanism behind ARS must have evolved quite early, since the
major divisions of the eumetazoa clade are very ancient, and it is
reasonable to assume that all forms of ARS derive from a mechanism
that was put in place before this division. ARS is, in other words,
one of the basic mechanisms of life.

ARS might even be more basic than the eumetazoa: there are molecular
mechanisms in protozoa that could be precursors of ARS. The most
primitive example is the \dqt{run and tumble} movement of
\emph{E.coli} and \emph{Salmonella typhimurium}. The movement of these
bacteria is controlled by a flagellar motor. \emph{Runs} consist of
forward motion (longish stretch of resource search), while
\emph{tumbles} are made of random turns that keep the bacteria more or
less in the same place (exploiting local resources while they
last). Receptor proteins in the membrane bind these behaviors to
external stimuli \shortcite{}.  The mechanism on which ARS is based
is, in its essential structure, fairly consistent across the whole
spectrum of eumetazoans. Its fully formed presence in organisms with
limited learning capabilities, such as \emph{C.elegans} suggests that
learning is not involved or, to the extent that it is (such as in
mammals), it is based on a fully formed pre-learning machinery. This
is what makes ARS so interesting: it is a basic mechanism that very
different forms of life have adopted as a basic strategy to solve such
diverse problems as looking for food in a Petri dish or trying to
prove a mathematical theorem. Its omnipresence derives from the
optimality of ARS as a search strategy in cases in which resources are
\dqt{clumpy} and the information about the locations of the
\dqt{clumps} is limited (we'll see that in the next section). In the
case of foraging animals, the resource is food, and the reward is
finding something to eat. In the case of somebody looking at an image,
the resource is information about its content, and the reward is
understanding what the image is about

\footnote{Things are more complex in the case of looking at images:
  the actual paths that people's gaze follow depend on what they are
  looking for. Given an image of an interior, the actual paths are
  different if the subjects are asked e.g.\ \dqt{From what epoch is the
    interior?} or if they are asked e.g.\ \dqt{What are the people in
    the image doing?} This is not important for our present
  considerations: all these paths, different as they may be, exhibit
  the features of ARS.}

. In all these cases, the animal or person moves locally around the
same clump as long as it is rewarding to do so (viz.\ as long as
locally one finds more food or new information), then starts moving
rapidly to explore quickly new territory in search of a new clump.

\begin{figure}[bthp]
  \begin{center}

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      \put(0,0){
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        \put(8,5){\line(2,1){2}}
        \put(10,6){\line(2,-1){2}}
        \put(12,5){\line(0,-1){2.5}}
        \put(12,1.8){\makebox(0,0){NH$_2$}}

        \put(1.8,1){\makebox(0,0)[r]{HO}}
        \put(1.8,6){\makebox(0,0)[r]{HO}}

        \put(6,-2){\makebox(0,0){dopamine}}
      }
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        \multiput(1,3)(4,0){3}{\line(2,1){2}}
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        \put(8,-2){\makebox(0,0){GABA}}
      }
    \end{picture}
  \end{center}
  \caption{\capstyle Three neurotransmitters that will figure in our
    discussion. Simplifying things very (very!) much, glutamate has
    essentially an excitatory action: its release in a synapse will
    move the postsynaptic neuron closer to firing. GABA is, on the
    contrary, inhibitory, its release will make the postsynaptic
    neuron less likely to fire. Dopamine is a modulator of
    glutammate.}
  \label{dopamine}
\end{figure}

The \dqt{reward} that one is after can be something very concrete
(food) or something very abstract (the \dqt{Eureka} moment of proving
a theorem) but in all cases the nervous system makes the reward
substantial by encoding it as the release of a very specific chemical:
\emph{dopamine} (figure~\ref{dopamine}).  The organism in which the
molecular basis of ARS is best understood is the nematode
\emph{C.elegans} \shortcite{hills:04}. The neural circuitry consists
of eight sensory neurons presynaptic to eight interneurons that
co\"ordinate forward and backward movements. The sensory neurons alter
the turning frequency by releasing dopamine on the interneurons,
modulating the reception of glutammate (Figure~\ref{elegans}).

\begin{figure}[bthp]
  \begin{center}
    \VInsert{graphs/elegans_da.jpg}{15em}
  \end{center}
  \caption{\capstyle Dopaminergic action in \emph{C.elegans}. Glu and Da
    represent glutamatergic and dopaminergic presynaptic neurons,
    respectively. The release of dopamine from the dopaminergic
    neurons alters the postsynaptic neuron's response to glutammate. It
    is not known whether this is due to a presynaptic response of the
    glutamatergic neuron and/or to a postsynaptic response of the
    locomotory neurons (from \shortcite{hills:04}).}
  \label{elegans}
\end{figure}

External administration of dopamine increases the turning frequency,
while administration of a dopamine antagonist reduces it
\shortcite{hills:04}. A reasonable model of \emph{C.elegans} behavior
suggests that, while on food, the sensory neurons release dopamine,
which, via the action of the glutammate, leads to increased switching
behavior in the interneurons, resulting in more turns and,
consequently, in a trajectory that stays local. When off food, the
dopaminergic activity is reduced, and the interneurons reduce their
switching frequency, leading to less turns and more ground covered

Although \emph{C.elegans} is the only organism for which the
neuromolecular mechanism of ARS is well understood, there is strong
evidence of dopaminergic modultation of glutamatergic synapses
throughout the major clades of the eumetazoans
\shortcite{acerbo:02,cleland:97}. In insects, for example,
dopaminergic neurons in the abdominal ganglion are sparsely
distributed, but show large branching patterns, indicative of
neuromodulation \shortcite{nassel:96}.

The relation between dopamine and ARS has been documented throughout
the invertebrates, especially in the fruit fly \emph{Drosophila
  melanogaster} \shortcite{bainton:00}, in crustaceans
\shortcite{harriswarrick:95}, in \emph{Aplysia} \shortcite{due:04},
etc.  In all these cases, ARS is limited to food search which is,
clearly, the search problem for which ARS first evolved.

In vertebrates, the modification of behavior by dopamine increases in
complexity and begins to involve behavior not directly related to
food. For example, in frogs and toads dopamine modulation is involved
in the visuomotor focus on preys \shortcite{buxbaumconradi:99};
similar dopaminergic involvement in visuomotor coordination can be
found in rats and humans \shortcite{barrett:04,dursun:99,evenden:93}.
This finding is significant in that it indicates a strong relation
between ARS and \emph{inhibition of return}: the fact that viewers show
significant latency in revisitig objects or regions of a scene that
have already been investigated, united to the lingering of saccadic
movements in regions of intertest \shortcite{tipper:94}.

The important change in vertebrates is the extension of ARS-like
behavior to cover not only actions with an immediate reward, such as
the search for food, but also situations in which the reward is
projected or even in which the reward itself is a neural state. The
detachment from the immediate food rewards is what makes it possible
to adapt ARS to abstract functions such as \emph{goal-directed
  cognition}. It seems, in other words, that when new problems arose
that had the same abstract structure as search for food, animals,
rather than developing a new mechanism, co\"opted the dopaminergic
modulation that guided food search to work on the new problem.

The most important neural structure associated with goal-directed
cognition is the \emph{basal ganglia} and, more specifically, the
\emph{striatum} \shortcite{delong:90,reiner:94} (Figure~\ref{striatum}).

\begin{figure}
  \setlength{\unitlength}{1em}
  \begin{picture}(0,0)(0,0)
    \put(1,-4){\makebox(0,0)[l]{1. lateral medial}}
    \put(1,-6){\makebox(0,0)[l]{2. globus pallidus}}
    \put(1,-8){\makebox(0,0)[l]{3. striatum}}
  \end{picture}
  \begin{center}
    \VInsert{graphs/basal-ganglia.jpg}{15em}
  \end{center}
  \caption{\capstyle The basal Ganglia and, specifically the striatum
    have a large number of dopaminergic inputs from other parts of the
    brain, and are involved in ARS. While in simple animals like
    \emph{C.Elegans} ARS is in a simple pathway from sensory input to
    motor neuron, in mammals the input comes from other areas of the
    brain, and the output acts on the cortex, making it possible to
    use ARS for more abstract problems than the immediate search for
    food.}
  \label{striatum}
\end{figure}

Information enters the basal ganglia through the striatum, and a great
number of the inputs to the system are dopaminergic
\shortcite{reiner:98}. The structure of the basal ganglia and much of
their connectivity are maintained across vertebrates
\shortcite{salas:03}. The major change from anamniotes (fish and
amphibians) to amniotes is the proliferation of dopaminergic neurons
that input to the striatum \shortcite{reiner:98}, while the structure
of the striatum stays pretty much the same.

The balance between glutammate and dopamine in the striatum is key to
the proper functioning of ARS-like activities, and an imbalance
between the two neurotransmitters is suspected in a number of
pathologies affecting goal-directed cognition, including Parkinson's,
schizophrenia, and addiction. Many of these conditions can be regarded
as radicalization of ARS in one direction or another (too local or too
global) due to imperfect dopamine control (see Figure~\ref{diseases}).

\begin{figure}
  \begin{center}
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    \begin{picture}(20,8)(0,0)
      \put(0,3){
        \multiput(0,0)(0,2){2}{\line(1,0){20}}
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        \put(15,1){\makebox(0,0){{\large Diffused}}}
        \put(0.1,0.1){\makebox(0,0)[lb]{{\small too much DA}}}
        \put(19.9,0.1){\makebox(0,0)[rb]{{\small too little DA}}}
        \put(6,2.2){\vector(1,0){8}}
        \multiput(8,2.7)(0.5,0){8}{\line(1,0){0.25}}
        \put(10,2.9){\makebox(0,0)[b]{schizophrenia}}
      }
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      }
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        \put(0,-0.1){\makebox(0,0)[t]{addiction}}
      }
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        \put(0,0){\vector(0,1){0.8}}
        \put(0,-0.1){\makebox(0,0)[t]{parkinson's}}
      }
      \put(3,6){
        \put(0,0){\vector(0,-1){0.8}}
        \put(0,0.1){\makebox(0,0)[b]{OCD, TS}}
      }
      \put(16,6){
        \put(0,0){\vector(0,-1){0.8}}
        \put(0,0.1){\makebox(0,0)[b]{ADHD}}
      }
    \end{picture}
  \end{center}
  \caption{\capstyle The continuous arrow at the top represents the normal
    temporal progression of dopaminergic activity in ARS, modulating
    behavior from focused (i.e.\ local) search to diffuse
    (i.e.\ global). The placement of the pathologies is qualitative
    and not based on a model but on the fact that they are treated
    either with dopamine or with dopamine antagonists. Although
    dopamine seems to be a factor in these diseases, there are clearly
    more factors at play. Schizophrenia is the most emblematic case in
    which the mechanism is not well understood, as reflected by its
    ambiguous positioning via the dashed line. OCD is
    \emph{Obsessive-Compulsive Disorder}, TS \emph{Tourette Syndrome},
    ADHD \emph{Attention Deficit Hyperactivity Disorder} (from
    \cite{hills:06}).}
  \label{diseases}
\end{figure}

In the striatum, dopaminergic neurons modulate the glutammatergic
input at the tips of spiny neurons. The action of the dopaminergic
inputs appears to perform a neuromodulation of the strength of the
glutammatergic inputs (Figure~\ref{mammalian})
The mechanism is similar to that described for \emph{C.elegans} in
Figure~\ref{elegans}, but in the mammalian striatum the shape of
the spiny neuron has specialized for this function and the inputs,
several magnitudes higher in number, come primarily from connections
to cortical neurons rather than directly from sensory neurons as in
\emph{C.Elegans} \shortcite{white:86}. However, at the level of the
microcircuit, little has changed from nematodes to amniote
vertebrates. The origin of the dopaminergic input does, however, mark
a fundamental evolutionary shift in the activity range of ARS-like
behavior. While in \emph{C.elegans} or \emph{Drosophila} the afferent
dopaminergic signal is reliably related to the presence or absence of
food, in higher vertebrates the signal may represent the
\emph{expectation of a reward}. The critical transition here is from a
concrete (directly sensed) reward to its neural representation that
is, from a physical reward to the abstract idea of a reward
\shortcite{schultz:95}. As Hills puts it: \dqt{the evolutionary theory
  $[$of ARS$]$ is therefore completely consistent with the reward
  theory of dopamine, but adds the evolutionary hypothesis that the
  initial reward represented by the release of dopamine were
  food. Only later was this system co-opted to represent the
  expectation of a reward, which allows for goal-directed cognition}
\shortcite{hills:06}.

To conclude this brief \emph{excursus} of ARS, we consider a region of
relatively recent evolution that, outside of the basal ganglia, is
heavily involved in goal-directed behavior: the \emph{prefrontal
  cortex} (PFC). The PFC has clearly evolved much later than ARS;
nevertheless, it is heavily involved in goal-directed behavior via
massive connections to the striatum \shortcite{odonnell:95}. Dopamine
has been shown to be a factor in the sustained activation of the PFC
\shortcite{seamans:98,wang:04}. Most models of PFC see the r\^ole of
dopamine as holding objects in attention long enough for appropriate
behavior to be activated \shortcite{braver:00}. Consistently with ARS,
already known solutions mediated by the PFC are most typically tried
when a problem has to be solved in a new situation
\shortcite{eichenbaum:01}.

The context in which goal-directed cognition takes place includes
external and internal stimuli; ARS depends in part on the alignment of
external stimuli with previous expectations. This is likely to be
controlled by the connections between the PFC and the \emph{Nucleus
  Accumbens} (NAcc) in the striatum, which modulates attention, eye
movements, and the maintenance of working memory
\cite{bertolucci:90,floresco:99,schultz:04}; dopamine has been
identified as one of the main influences in the modulation of NAcc
activity \cite{floresco:96}: novel stimuli lead to increase in
dopamine in the NAcc and in the PFC \cite{berns:01}.

\begin{figure}
  \begin{center}
    \VInsert{graphs/mammalian_da.jpg}{18em}

    \vspace{-3em}
  \end{center}
  \caption{\capstyle The dopaminergic-glutammatergic interaction in the synapses
    of spiny neurons in the mammalian striatum. Glu and DA represent
    glutammatergic and dopaminergic pre-synaptic neurons, respectively
    (from \shortcite{hills:06}, redrawn from \shortcite{dani:04}).}
  \label{mammalian}
\end{figure}

So, in the evolution of vertebrates, we see a progressive extension of
the r\^ole of ARS, from the dopaminergic control of visuomotor control
in frogs and toads to the similarity mediated maintenance of ideas in
working memory \cite{schultz:95}. ARS appears therefore to be one of
the fundamental strategies in the animal kingdom, co-opted and adapted
to a number of situations, from the \dqt{run and tumble} behavior of
\emph{E.coli} to the way we focus on and later abandon ideas when we
think about a problem.

\separate

This brief explanation of the evolutionary basis of ARS has been
centered on its molecular mechanism, especially on the r\^ole of
dopamine as neuromodulator. From now on, however, our focus will
change: we shall try to understand the \emph{exterior} characteristics
of the behavior. ARS leads to a well identified patterns of motion
either in the physical space (in the case of foraging), in the visual
space (scanning an image), or in any number of abstract spaces. We
shall study mathematically these patterns of motion and try to
characterize them. Our methods will be based mostly on the study of
random walks, of diffusion, and on the kinds of anomalous diffusion to
which ARS leads.

\section{\sc Optimality of ARS}
\label{genetic}
The evolutionary success of ARS entails, according to the theory of
natural selection, that ARS is an optimal strategy---if not globally,
at least locally---for a large set of problems. In abstract terms, we
have a space with certain resources placed in different parts of it;
we need a strategy to navigate this space collecting the greatest
amount of resources. This must be done without information on the
placement of the resources. (If we can sense from afar where the
resources are located, we simply walk there and get them: no search
strategy is necessary.) The nature of the resources can be the most
diverse: in the case of foraging (to which we shall mostly make
reference), the resource is food; in the case of saccadic movements,
it is the visual information that we get from the visual field, and so
on. ARS is optimal if the resources are \dqt{patchy,} that is, if they
are organized in resource-rich patches separated by areas of small or
zero resource concentration. In the model that we shall develop in
this section, we assume that the resources are consumed in the course
of the activity, and that they are not replenished while the activity
goes on. Resources are consumed simply by moving on top of them
(assuming that, after walking on them, they would be consumed with a
certain probability would not substantially change the model).
In the example of food, this means that we have food distributed in
patches (a grove, a pond, a herd, a school of fish...) and the forager
moves inside the patch and between patches eating what it finds. We
make a number of hypotheses. Firstly, we assume that the food doesn't
move around or if it does (as is the case of animal preys) its
movement is not significant and food can be modeled as
static. Secondly, we assume that the forager will eat all the food it
can find as soon as it finds it (its eyesight is perfect and its
appetite endless). Finally, food doesn't grow back: once it has been
eaten at a particular location, that location will remain barren for
the rest of the forager's activity.

In the case of saccades, we assume (as is often the case) that there
are patchy areas in the visual field that are rich in information
useful to interpret the scene (relevant or telling objects, faces,
etc.). We also assume that once we have analyzed the information in a
given area of the visual field, that information is remembered and it
is not necessary to analyze it again. This is equivalent to the
hypothesis that the food doesn't grow back once it has been eaten.

\separate

In this section, we want to check whether ARS emerges as an optimal
solution to the foraging problem. We shall do this by implementing a
genetic algorithm based on a competition among individuals whose
characteristics are encoded in a string of bits called a \emph{gene}
\cite{mitchell:98}.

Individuals move around a \emph{foraging areas} under the guidance of
their gene and collect food. Their \emph{score}, which determines
their fitness for survival, is the amount of food they have collected.
The world in which these individuals move is a regular grid of patches
of food of $p\times{p}$ ($p\in{\mathbb{N}}$) \emph{pellets}, separated
by barren areas without food (see Figure~\ref{feedenvironment}). Each
pellet is an atomic unit of food, that it, it is either not consumed or
consumed entirely. The consumption of each pellet increases  the
survival fitness by one unit.

\begin{figure}[btp]
  \begin{center}
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        \put(6.5,4){\makebox(0,0)[lb]{$(x_0,y_0)$}}
      }
    \end{picture}
  \end{center}
  \caption{\capstyle The environment for the application of genetic
    algorithms to the evolution of ARS. The dotted squares represent
    the patches composed of $p\times{p}$ pellets of food, where $p$ is a
    program parameter. The distance between the patches, $q$, is
    determined by the desired density of food, $\rho$ (also a program
    parameter) through the relation
    $q=\lceil{p/\sqrt{\rho}}\rceil$. In the implementation, patches
    are generated dynamically the first time that the forager walks on
    them so that the foraging field is virtually infinite. }
  \label{feedenvironment}
\end{figure}

Each patch is separated from the other by being placed in the
lower-left corner of a larger square of $q\times{q}$ units
($q\in{\mathbb{N}}$, $q>p$) called a \emph{plot}. The density of the
food is $\rho=p^2/q^2$. The plots are created dynamically at run time
as the walker steps on them for the first time, so that the foraging
field is virtually infinite.
In all the tests discussed below, $p$ is kept fixed ($p=16$), $\rho$
is a parameter that varies from $\rho=0.01$ to $\rho=0.95$, and $q$ is
determined as $q=\lceil{p/\sqrt{\rho}}\rceil$. Each individual does a
random walk (specified by certain parameters, as described below)
starting at $(x_0,y_0)=(p/2,p/2)$ that is, in the center of the patch
whose lower-left corner is the origin.

\subsection{The walk parameters}
Each time the walker walks on a position containing a pellet, it
\dqt{eats} it, incrementing its score (which determines its
evolutionary fitness) by one. The pellet is removed, so that further
visits to the location will not provide any food
(Figure~\ref{eating}). The movement of each individual is a random
walk whose statistical features depend on whether the individual is
currently eating (status: on-food) or whether it has been without food
for some time (status: off-food). The individual doesn't go
\dqt{off-food} immediately as soon as it steps on a location with no
food: the individual has memory, so that it gradually changes its
status from on-food to off-food during a certain number of time
steps. The amount of time without food that it takes to go to the
status off-food is controlled by a parameter in the gene of the
individual.

\begin{figure}[tbhp]
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      \savebox{\patch}{
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  \end{center}
  \caption{\capstyle A walk of an individual through a patch of food: each
    square crossed by the individual is \dqt{eaten,} and is removed
    from the patch. In this case, the individual passes over 8
    patches: after the walk, its score is eight, and eight pellets are
    removed from the patch.}
  \label{eating}
\end{figure}

The general behavior of the random walk is the same regardless of
whether the individual is on-food or off-food (the only thing that
changes in the two cases is the numerical value of the
parameters). Consider a generic situation in which the parameters are
$(\alpha_0,l_0)$. The individual is coming from a direction $\theta$,
being currently at location $(x,y)$. The individual chooses a
deviation angle $\alpha$, selected with a Gaussian distribution
centered at $\alpha_0$, and a length $l$ selected with an exponential
distribution with average $l_0$, and performs a jump in a direction at
an angle $\alpha$ from its current direction, and for a length $l$
(Figure~\ref{jump}). The parameters $(\alpha_0,l_0)$ characterize the
statistics of the jump, they are encoded in the individual's gene, and
they take different values depending on whether the individual is
on-food or off-food. When the individual is on-food, the jumps are
done according to the parameters $(\alpha_f,l_f)$, while when the
individual has been for some time off-food, the jumps are done
according to $(\alpha_n,l_n)$. The switch from the on-food parameters
to the off-food is done gradually when the individual is in an area
without food.  The gene defines a memory threshold $\tau$ to switch
from the on-food to the off-food behavior. If $t$ is the number of
step that the individual has been off-food ($t=0$ if the individual is
on-food) then the parameters for the next jump will be

\begin{equation}
  \begin{array}{ll}
    \begin{array}{rc}
      (\alpha_f,l_f)  &
    \end{array}    & \mbox{on food} \\
    ~ \\
    \left.
    \begin{array}{rc}
      (\alpha_f + \frac{t}{\tau}(\alpha_n-\alpha_f), l_f + \frac{t}{\tau} (l_n-l_f)) & t < \tau \\
      (\alpha_n,l_n)          &t \ge \tau
    \end{array}
    \right\} & \mbox{off food}
  \end{array}
\end{equation}

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\put(  5.985,  0.408){\circle*{0.001}}
\put( -0.200,  3.000){\makebox(0,0)[r]{$l_0$}}
\multiput(  3.000,  0.000)(0,0.25){8}{\line(0,1){0.125}}
\put(  3.000, -0.200){\makebox(0,0)[t]{$1/l_0$}}
\end{picture}        }
      \end{picture}
  \end{center}
  \caption{\capstyle Generation of a single jump of the random walk. In (a), a
    walker is coming to point $(x,y)$ following a trajectory with
    direction $\theta$. An angle $\alpha$ is chosen from the Gaussian
    angle distribution in (b), centered around the angle $\alpha_0$:
    the direction of the new jump is $\alpha$ from the previous
    direction (viz.\ $\theta+\alpha$ absolute direction). The length
    of the jump is drawn from the exponential distribution with
    average $l_0$ in (c).}
  \label{jump}
\end{figure}

The parameters of the jump are reset to $(\alpha_f,l_f)$ as soon as
the individual finds food. That is, there is a lingering memory that
food was around there even if currently no food is found---a memory
that fades away in a time $\tau$---but the absence of food is
forgotten as soon as new food is found. Any moral or philosophical
conclusion, be it positive or negative, that can be drawn from this
hypothesis is beyond the scope of these notes.

\subsection{Gene definition}
Each individual is therefore characterized by five parameters:
$(\alpha_f,l_f,\alpha_n,l_n,\tau)$. We represent each one as a 8-bit
value (these values are scaled in order to compute the actual values
of the parameters) and collect them in a 40-bit \dqt{gene.} We try to
keep related parameters in nearby positions of the gene (this is
believed to speed up the convergence of the algorithm). Calling $A_F$,
$L_F$, $A_N$, $L_N$, and $T$ the 8-bit representations of the
parameters we have the genetic representation of an individual in
Figure~\ref{gene}.

\begin{figure}[tbh]
  \begin{center}
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    \begin{picture}(25,3)(0,2)
      \multiput(0,4)(0,1){2}{\line(1,0){25}}
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      \put(5,2.8){\makebox(0,0)[t]{7}}
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      \put(15,2.8){\makebox(0,0)[t]{23}}
      \put(20,2.8){\makebox(0,0)[t]{31}}
      \put(25,2.8){\makebox(0,0)[t]{39}}
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      \put(0,2){\vector(1,0){1}}
      \put(1.1,2){\makebox(0,0)[l]{bits}}
    \end{picture}
  \end{center}
  \caption{\capstyle The genetic representation of an individual. The five 8-bit
    parameters are scaled to provide the jump paramterers
    $(\alpha_f,l_f,\alpha_n,l_n,\tau)$.}
  \label{gene}
\end{figure}

The jump parameters are derived from these 8-bit integers as

\begin{equation}
  \label{normal}
  \begin{array}{ccc}
    \displaystyle \alpha_f=2\pi\frac{A_F}{256} & \displaystyle l_f = \frac{L_F}{4} & \displaystyle \tau = \frac{T}{10} \\
    & \\
    \displaystyle \alpha_n=2\pi\frac{A_N}{256} & \displaystyle l_n = \frac{L_N}{4}
  \end{array}
\end{equation}

The scaling factors (except those for $\alpha_f$ and $\alpha_n$, which
are derived from geometric considerations) have been determined by
trial and error.

\begin{figure}[tbhp]
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        \put(16,0.5){\makebox(0,0)[l]{parent A}}
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  \end{center}
  \caption{\capstyle Double cut for the generation of offsprings. Two parents
    generate two offprings, mixing the bits of their genes as
    represented.}
  \label{offsprings}
\end{figure}

\subsection{The algorithm}
The genetic algorithm is pretty standard. A \emph{generation} is a set
of individuals. Each individual is placed in the environment in the
same initial position, and does a random walk of predetermined length,
according to the parameters encoded in its gene, and collecting
pellets of food as specified above. The environment is restored
between individuals, so that each one has the same initial supply of
food (this entails that there is no competition among the
individuals). A point is scored for each pellet that is eaten. As a
result, after all individuals have executed a random walk, individual
number $k$, characterized by gene $\gamma_k$ has a score $s_k$, with
$k=1,\ldots,G$, where $G$ is the number of individuals in a
generation.

There are several methods to create the following generation of
individuals. Since the performance of the algorithms seems to have
little dependence in the specific method used, we use one of the
simplest, based on the creation of an intermediate \emph{gene
  pool}. The gene pool is a set ${\mathcal{P}}$ of $P$ individuals
possibly replicated (generally $|P|=G$: the pool has the same size
as the generations) such that the number of \dqt{copies} of an
individual in the pool is proportional to its score. An easy algorithm
for generating a pool is the \emph{tournament}: we do $P$ comparisons
of pairs of individuals taken at random from the generation: the
individual with the highest score goes into the pool:

\parbox{\columnwidth}{
  {\tt
    \cb
    \> $P$ $\leftarrow$ $\emptyset$ \\
    \> \cmd{for} k=1 \cmd{to} $P$ \cmd{do} \\
    \> \> i $\leftarrow$ rnd(1,G) \\
    \> \> j $\leftarrow$ rnd(1,G) \\
    \> \> \cmd{if} $s_i\ge{s_j}$ \cmd{then} \\
    \> \> \> ${\mathcal{P}}$ $\leftarrow$ ${\mathcal{P}} \cup \{\gamma_i\}$ \\
    \> \> \cmd{else} \\
    \> \> \> ${\mathcal{P}}$ $\leftarrow$ ${\mathcal{P}} \cup \{\gamma_j\}$ \\
    \> \> \cmd{fi} \\
    \> \cmd{od}
    \ce
  }
}

In order to build the next generation, pairs of genes are taken at
random from the pool (with uniform distribution) and crossed to create
two new individuals that will go into the next generation (this
requires that $G$ be even). We use the method of the \emph{double cut}
to cross the genes. Two values $a,b\in[0,39]$ are chosen randomly. The
two offspring are then generated as in Figure~\ref{offsprings}, in
which we assume $a<b$. We also define a small mutation probability:
for each new gene, with a (small) probability $p$, we pick a random
bit and flip it.
Note that this method doesn't guarantee that the best individual of a
generation will pass unchanged to the next, so we actually use the
crossing to create $G-2$ individuals to which we add the two best
performers of the previous generation.

\subsection{Results}
Figure~\ref{paths} shows typical paths from the best individual for
various values of the density of food,
while Table~\ref{params} shows the value of the parameters for the
same individual.

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\end{picture}
    \end{tabular}
  \end{center}
  \caption{\capstyle A typical path of the best individual for various values of
    the density. Note the different macroscopic behavior for high
    density and for low density. In the case of high density, the path
    is practically always contained in some patch, and the off-food
    status is practically unused. The low density situation shows the
    characteristics of ARS.}
  \label{paths}
\end{figure}

\begin{table}
  \begin{center}
    \begin{tabular}{|l|c|c|c|c|c|}
      \hline
      \hline
      $\delta$ & $\alpha_f$ & $l_f$ & $\tau$ & $\alpha_n$ & $l_n$ \\
      \hline
      \hline
      0.95 & 0.00 & 136.94 &     57 &   34.25 &   33.50 \\
      \hline
      0.5 &  199.06 &  15.53 &      0 &    2.00 &   60.50 \\
      \hline
      0.1 &  309.18 &   1.75 &     18 &  285.18 &   58.00 \\
      \hline
      0.01 & 227.29 &  1.50 &    37 & 345.88 &  45.50 \\
      \hline
    \end{tabular}
  \end{center}
  \caption{\capstyle The values of $A_F,L_F,T,A_N,L_F$ for the best individual
    for different values of the food density. The values are
    normalized as in (\ref{normal}) to obtain the walk parameters
    $(\alpha_f,l_f,\tau,\alpha_n,l_n)$.}
  \label{params}
\end{table}

\begin{figure}[tbp]
  \begin{center}
    \begin{tabular}{cc}
      \hspace{-3em}

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    \end{tabular}
  \end{center}
  \caption{\capstyle The behavior of $\langle{X^2(t)}\rangle$ as a function of
    $t$ for various values of the resource density $\rho$. Together
    with the curve, the figure shows its best approximation as
    $\langle{X^2(t)}\rangle\sim{x^\nu}$. The coefficient $\nu$ of the
    best approximation varies with $\rho$.
    On the significance of this variation, see the text.}
  \label{variances}
\end{figure}

We note that in the case of high food density the random walk does not
exhibit the characteristics of ARS (this observation will be made more
formal later on), simply because the individual is always, or almost
always, on food. As the density decreases, the behavior becomes more
characteristic of ARS.

In order to study the characteristics of these walks, we consider the
square average of the displacement from the initial position:

\begin{equation}
  \langle X^2 \rangle \dfeq \langle (x-x_0)^2 + (y-y_0)^2 \rangle.
\end{equation}

Let the individual be fixed (viz.\ let it be the best performing
individual). We execute, with this individual, $N$ random walks on the
environment with the prescribed density, each of length $T$. Let

\begin{equation}
  w^i = [p_0^i,\ldots,p_t^i,\ldots,p_T^i]
\end{equation}

be the $i$th random walk, where $p_t^i=(x_t^i-x_0,y_t^i-y_0)$. We are
interested in knowing how far the individual has gone, on average,
from its initial position, at time $t$. That is, we are interested in
studying the function

\begin{equation}
  \langle X^2(t) \rangle = \frac{1}{N} \sum_{i=1}^N (p_t^i)^2
\end{equation}

Figure~\ref{variances} shows the behavior of $\langle{X^2(t)}\rangle$
as a function of $t$ for various values of the density $\rho$,
together with the best approximation of the form
$\langle{X^2(t)}\rangle\sim{x^\nu}$, where the exponent $\nu$ depends
on $\rho$:

\begin{center}
  \begin{tabular}{|c|c|c|c|c|}
    \hline
    $\rho$ & 0.01 & 0.1 & 0.5 & 0.9\\
    \hline
    $\nu$  & 3.78 & 1.8 & 1.1 & 0.95 \\
    \hline
  \end{tabular}
\end{center}

Figure~\ref{varplot} shows the behavior of the
exponent $\nu$ as a function of $\rho$. As density approaches 1 or 0,
that is, as the environment becomes more homogeneous (either with a
lot of food or very little food), the exponent approaches 1, that is,
we approach a situation in which
$\langle{X^2(t)}\rangle\sim{x^1}$. This, as we shall see, is the
behavior characteristic of Brownian motion, as well as of several
other types of random walks. This should not come as a surprise: when
food can be found everywhere, the individuals have no particular
reason to modify their behavior, and will simply move to and fro in an
haphazard manner: they will do a random walk. The same is true if
there is very little food. The patches are so small and far apart that
the in-patch behavior will last for a very short time and will not
change significantly the characteristics of the walk, which will be a
random walk from patch to patch looking for resources.

When the food is patchy, on the other hand, the random walk of the
individual doesn't follow the standard Brownian model. In the next
section I shall consider the foraging walk more closely from a
mathematical point of view. We shall begin, in the next section by
considering the switching from the on-patch behavior to the off-patch,
without taking onto account the spatial characteristics of the
environment. Then, in the following section, we shall study random
walks in search of a model that fits the characteristics of a forager
on patchy resources. We shall see that such a model is given by the
so-called \emph{Levy walks}.

\begin{figure}[thbp]
  \begin{center}

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  \end{center}
  \caption{\capstyle Exponent of the best approximation
    $\langle{X^2(t)}\rangle\sim{x^\nu}$ as a function of $\rho$. The
    squares are the values that have been calculated from the
    simulation (in each case the best gene as resulting from the
    genetic algorithm has been used), the continuous line is a spline
    interpolation \cite{press:86}. The value $\nu\sim{1}$ is
    characteristic of standard Brownian motion. For very low and high
    $\rho$, the walk is essentially Brownian: when the density is very
    low, the forager wanders long distances and spends comparatively
    little time on each patch. When $\rho\sim{1}$, there is no ARS, as
    the food is everywhere. The region $0.1<\rho<0.4$ is that in which
    ARS is clearly taking place.}
  \label{varplot}
\end{figure}

\section{\sc Should I stay of Should I go?}

In the next section, I shall analyze the global characteristics of ARS
exploration considering it as a random walk such as those that emerge
from our genetic experiment. Before that, in this section I shall
consider a more basic problem. Suppose that you are in a patch.  For a
while, you stay there happily eating, as there is plenty of
food. After a while, the food begins to dwindle, the resources of the
patch begin to be exhausted. When is it a good time to leave? You are
confronted with two contrasting criteria. On the one hand, staying
implies that you can continue eating without having to make a possibly
long journey without food before you find another patch. In the long
run, you want to spend more time in a patch and less between
patches. On the other hand, the new patch that you will find has lots
and lots of food so it might be a good idea for you to move now to
greener pastures instead of half starving in this half barren
patch. When is it a good time to leave? This is the question I want to
answer in this section.  I shall consider a very simple model: I
analyze only the time that an individual spends on a patch ($t_p$) and
the time that it spends between patches ($t_b$), and how to optimize
them for maximum foraging efficiency. In this, I follow essentially
the techniques developed for \emph{optimal foraging theory}
\cite{stephens:86}.

Suppose that a forager searches for food for a certain (long) amount of time. It
spends a total time $T_b$ looking for the next patch, and a total time
$T_p$ staying on a patch and eating (all the symbols used in this
section are shown in Table~\ref{forasymbols})

\begin{table}[bht]
  \begin{center}
    \begin{tabular}{|l|p{25em}|}
      \hline
      \multicolumn{1}{|c|}{Symbol} &  \multicolumn{1}{|c|}{Meaning} \\
      \hline
      $T_b$  & Total time spent looking for a patch, \\
      $T_p$  & total time spent on a patch, \\
      $t_b$  & average time spent looking for a patch, \\
      $t_p$  & average time spent on a patch, \\
      $G$    & total resource gain, \\
      $g$    & average gain per patch, \\
      $g(t)$ & gain per patch as a function of the time spent on a patch, \\
      $R$    & rate of reward: average gain per time unit, \\
      $\pi$  & ${=g/t_p}$, profitability of a patch (resource per unit time when on the patch), \\
      $\lambda$ & average number of patches found per unit time, \\
      $P$    & number of types of patches, \\
      $p_i$  & probability of using a resource of type $i$. \\
      \hline
    \end{tabular}
  \end{center}
  \caption{\capstyle Symbols used in this section.}
  \label{forasymbols}
\end{table}

If the total gain of the activity is $G$, then the rate of reward
(reward per unit time), is

\begin{equation}
  \label{totalgain}
  R = \frac{G}{T_b+T_p}
\end{equation}

This equation is inconvenient as it depends on the total times $T_p$
and $T_b$ and on the total gain $G$ (if the time spent goes to
infinity, both the numerator and the denominator go to infinity). One
can derive a more convenient equation, independent on the actual
foraging time, by considering the average on-patch time $t_p$, the
average time between patches $t_b$ and the average gain per patch
$g$. The rate at which patches are discovered, that is, the number of
patches discovered per unit time, is $\lambda=1/t_b$. The total number
of patches discovered during foraging is therefore $\lambda{T_b}$. The
total gain and the total time spent on patches depend on this number,
that is:

\begin{equation}
  \begin{aligned}
    G &= \lambda T_b g \\
    T_p &= \lambda T_b t_p
  \end{aligned}
\end{equation}

Introducing these values in (\ref{totalgain}) we have

\begin{equation}
  R = \frac{\lambda T_b g}{T_b + \lambda T_b t_p} = \frac{\lambda g}{1 + \lambda t_p}
\end{equation}

or, in terms of average times

\begin{equation}
  R = \frac{g/t_b}{1 + t_p/t_b} = \frac{g}{t_b+t_p}
\end{equation}

This equation is known as the \emph{Holling disk} equation \cite{holling:59}

\footnote{The name \dqt{disk equation} has nothing to do with the
  properties of the equation. Holling developed his model by studying
  the behavior of a blindfolded researcher assistant who was given the
  task of picking up randomly scattered sandpaper disks.}

. Define the profitability of a patch as $\pi=g/t_p$, that is, the
gain per unit of time spent on the patch. With this definition we have:

\begin{equation}
  R = \frac{\pi}{{\displaystyle 1+\frac{t_b}{t_p}}}
\end{equation}

When the patches become more and more dense, then $t_b\rightarrow{0}$, and

\begin{equation}
  \label{pilimit}
  \lim_{t_b\rightarrow{0}} \frac{\pi}{{\displaystyle 1+\frac{t_b}{t_p}}} = \pi
\end{equation}

This simple model can be extended in several ways. One very useful one
is to consider that a patch has \emph{diminishing returns}: as the
forager spends time on a path, its resources become depleted, so it
becomes harder to get rewards, and the profitability of the patch is
reduced.

That is, the reward $g$ is a function of $t$, $g(t)$, that tells us
how much reward one accumulates while foraging on a patch for a time
equal to $t$. The profitability is also a function of time:
$\pi(t)=g(t)/t$. Physical considerations place certain constraints on
these functions. The gain is positive and cumulative (you never lose
what you have gained), which implies $g(t)\ge{0}$ and
$g'(t)\ge{0}$. The first inequality also entails $\pi(t)\ge{0}$. On
the other hand, it is reasonable to assume that, as time goes by and
the resources become depleted, it will take longer and longer to amass
the same amount of reward; this entails $\pi'(t)\le{0}$. These two
relations imply $\lim_{t\rightarrow\infty}\pi(t)=C\ge{0}$. The
condition $\pi'(t)\le{0}$ imposes conditions on $g'(t)$. From
$\pi(t)=g(t)/t$, we have

\begin{equation}
  \pi'(t) = \frac{1}{t^2}\Bigl[g'(t)t-g(t)\Bigr] \le 0
\end{equation}

that is

\begin{equation}
  \label{gprime}
  g'(t)\le \frac{g(t)}{t} = \pi(t)
\end{equation}

I assume certain regularity conditions. In particular, that $\pi(t)$
decreases without \dqt{bumps}, that is, that $\pi'$ is monotonically
increasing, which entails that $\pi''\ge{0}$. Similarly, I assume
$g''(t)\le{0}$. Note that in the most common case the patch will
become depleted, that is,

\begin{equation}
  \lim_{t\rightarrow\infty} g(t) = g_\infty > 0
\end{equation}

but the analysis applies to the more general case in which $g(t)$ goes
to infinity slower than a linear function.

Given the average between-patches time $t_b$, we are interested in
finding the optimal time that the forager should spend on a patch
($t_p$) to maximize the reward $R$. The idea is that if you spend too
little time on a patch, then you don't take full advantage of its
resources, and spend comparatively too much time without patches, in
an area where you have no reward: your rate of gain will be reduced.

On the other hand, since the resources get depleted as we stay on a
patch, if you spend too much time there you shall waste your time on a
depleted patch that won't yield too much, while it would be more
convenient to invest some time ($t_b$) to find a new patch with better
yield. Given the equality

\begin{equation}
  R(t) = \frac{g(t)}{t_b+t}
\end{equation}

compute the derivative

\begin{equation}
  \frac{\partial R}{\partial t} = \frac{g'(t)(t_b+t)-g(t)}{(t_b+t)^2}
\end{equation}

It is $\partial{R}/\partial{t}=0$ if $g'(t)(t_b+t)-g(t)=0$, that is, if

\begin{equation}
  g'(t)=\frac{g(t)}{t_b+t}=R
\end{equation}

This result is known as the \emph{Charnov's Marginal Value Theorem}
\cite{charnov:76}.

This equation has a simple geometric interpretation. The average gain
$R$ results in a straight line in a $t$-$g$ diagram
(Figure~\ref{patchdiag}).

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\multiput(5.493061,-1)(0,0.766667){12}{\line(0,1){0.383333}}
\put(5.493061,-0.8){\makebox(0,0)[t]{$t_p$}}
\put(-4.506939,-0.8){\makebox(0,0)[t]{$-t_b$}}
\put(10.986123,8.444444){\makebox(0,0)[tl]{$g(t)$}}
\put(10.986123,10.328708){\makebox(0,0)[tl]{$R$}}
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\put(-3.805551,0.800925){\makebox(0,0)[br]{$R$}}
\end{picture}

  \end{center}
  \caption{\capstyle A graphical illustration of Charnov's Marginal Value
    Theorem. The patch time $t_w$ that maximizes the rate of gain $R$
    occurs when the line with slope $R$ is tangent to the function
    $g(t)$, that is, when the instantaneous rate of gain on the patch
    equals the average rate of gain $R$.}
  \label{patchdiag}
\end{figure}

The patch gain is a curve that stays at zero for a time $t_b$ and then
grows as $g(t)$; the optimal $t_p$ occurs when the slope of the curve
$g(t)$ is equal to the average rate of gain. Note that there are two
ways in which the environment can change so that $R$ increases. First,
the profitability of the patch may increase, that is, the curve $g(t)$
can be pulled upward (Figure~\ref{increase_patch}.a). Second, the
patches may become dense, reducing $t_b$
(Figure~\ref{increase_patch}.b). As $t_b\rightarrow{0}$ the rate $R$
approaches $\pi$ (the profitability of the patch), as per (\ref{pilimit}).

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\multiput(5.493061,-1)(0,0.766667){12}{\line(0,1){0.383333}}
\put(5.493061,-0.8){\makebox(0,0)[t]{$t_w$}}
\put(-4.506939,-0.8){\makebox(0,0)[t]{$-t_s$}}
\put(10.986123,8.444444){\makebox(0,0)[tl]{$g(t)$}}
\put(10.986123,10.328708){\makebox(0,0)[tl]{$R$}}
\put(-1.534264,0){\line(1,1){12.520387}}
\multiput(3.465736,-1)(0,0.600000){12}{\line(0,1){0.300000}}
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\end{picture}
      (a) & & (b)
    \end{tabular}
  \end{center}
  \caption{\capstyle As a consequence of Charnov's Marginal Value
    Theorem, there are two ways to increase the average rate of gain
    $R$: one can either increase the profitability of a patch,
    i.e.\ raising the gain curge $g(t)$ as in (a), or make the patches
    more dense, thereby decreasing the average beyween-patches time
    $t_b$, as in (b).}
  \label{increase_patch}
\end{figure}

\example

Suppose we are looking for a low-priced hotel in Paris. We have
several web sites available, and our strategy is to log in to one,
start checking prices looking for the cheapest price for a while, then
move to another one looking for a new price or, simply, stop looking
and accept the lowest price we have found. The question is: how long
should we stay on the site and keep looking before we move on?

When we are on a page, the important events are the prices that we
look at so, for the sake of convenience, we take the time that it
takes to move from one hotel to the next one on the same page as the
time unit so that at time $t=n$ we have looked at $n+1$ prices (we
look at the first price at $t=0$). The actual length depends of what
we are looking for: if we look just for the best price, the interval
is very small; if we look for price subject to certain constraint
(hotel with a bar, with a sauna, etc.), it will take longer. In any
case, we look at a new hotel (one the same page) per unit of time. In
these units, let $t_b$ be the time that it takes to get set on a page
(including typing the address, logging in, etc.)

We begin by determinig the expected value of the minimum price that we
have observed if we have observed $n$ prices. In this simple example
we shall favor simplicity over plausibility and we shall assume that
the prices of teh hotels are uniformly distributed in the interval
$[\mu-a,\mu+a]$.

To begin with, we shall answer an even simpler question: given $n$
observations $X=\{x_1,\ldots,x_n\}$ of $n$ random variables
independent and uniformly distributed in $[-1,1]$, which is the
expected value of $\min(X)$? The cumulative distribution and the
density for each of the $x_i$ are

\begin{equation}
  \Phi(x) =
  \begin{cases}
    0 & x \le 1 \\
    \frac{x+1}{2} & -1<x<1 \\
    1 & x\ge 1
  \end{cases}
\end{equation}

and

\begin{equation}
  \phi(x) =
  \begin{cases}
    \frac{1}{2} & -1\le{x}\le{1} \\
    0           & \mbox{otherwise}
  \end{cases}
\end{equation}

respectively. The probability density for the minimum is given by
(\ref{mindense}):

\begin{equation}
  \phi_{\min}(x) = n[1 - \Phi(x)]^{n-1}\phi(x) =
  \begin{cases}
    \frac{n}{2^n}[1-x]^{n-1} & (-1\le{x}\le{1}) \\
    0                      & \mbox{otherwise}
  \end{cases}
\end{equation}

and its expected value is

\begin{equation}
  \begin{aligned}
    m(n) &= \int_{-1}^{1} u \phi_{\min}(u) du = \frac{n}{2^n} \int_{-1}^{1} u(1-u)^{n-1} du \\
         &= \frac{n}{2^n} \left[ \int_{-1}^{1} (1-u)^{n-1}du - \int_{-1}^{1}(1-u)^ndu\right] \\
         &= \frac{n}{2^n} \left[ - \int_{2}^{0} u^{n-1}du + \int_{2}^{0}u^n du\right] \\
         &= \frac{n}{2^n} \left[ \frac{2^n}{n} - \frac{2^{n+1}}{n+1} \right] \\
         &= \frac{1-n}{1+n}
  \end{aligned}
\end{equation}

Scaling and shifting one obtains, for variables distributed in $[\mu-a,\mu+a]$:

\begin{equation}
  m(n) = \mu - a \frac{1-n}{1+n}
\end{equation}

Note that if we only observe one price, the expected value of the
minimum is $\mu$. We we observe more prices, the expected value
decreases, with

\begin{equation}
  \lim_{t\rightarrow\infty} m(t) = \mu-a
\end{equation}

Let us consider that we start exploring at the time when we observe
the first price, and that everything that comes before that is
preparation that is included in the time $t_b$. At time $t$, we have
looked at $t+1$ prices, and the expected value for the minimum price
is $m(t+1)$. The gain that we have obtained is the difference between
the first price that we saw (expected value equal to $\mu$) and the
current one:

\begin{equation}
  \label{gdef}
  g(t) = m(1) = m(t+1) = a \frac{t}{t+2}
\end{equation}

Up to now, I have considered $t$ as a discrete variable (the number
of prices observed); from now on we regard it as a continuous
variable, so I can take the derivative:

\begin{equation}
  g'(t) = \frac{2a}{(t+2)^2}
\end{equation}

I now apply Charnov's Marginal value theorem: the optimal value to
spend on the site is given by

\begin{equation}
  g'(t) = \frac{2a}{(t+2)^2} = \frac{at}{t+2} \frac{1}{t+t_b} = \frac{g(t)}{t+t_b}
\end{equation}

which has solution $\tau_1=\sqrt{2t_b}$

\footnote{Dimensionally, the equation is sound: the factor $t+2$ in
  (\ref{gdef}) entails that the term 2 has the dimensions of a time.}

. The corresponding rate of reward is

\begin{equation}
  \label{t1bum}
  R_1 = g'(\tau_1) = \frac{2a}{(\sqrt{2t_b}+2)^2} = \frac{a}{(\sqrt{t_b}+\sqrt{2})^2}.
\end{equation}

Note that the time that we spend on a site grown sub-linearly with
the time we spend looking for the site: if we spend twice as long
looking for the web page, the time we should spend on the web page
grows like $\sqrt{2}$, that is, we should stay about 40\% longer.

~~\eoe

\example

The considerations of the previous example are valid for the first
\dqt{patch}, that is, for the first site that we visit. Suppose now
that we want to visit a second site (the time necessary to do the
switch is assumed to be $t_b$). Now we already have a minimum, the one
that we found in the first patch, namely

\begin{equation}
  m_1 \dfeq \mu - a\frac{\tau_1}{\tau_1+2} = \mu - a \frac{\sqrt{t_b}}{\sqrt{t_b}+\sqrt{2}} \dfeq \mu - a\alpha
\end{equation}

with

\begin{equation}
  \alpha \dfeq \frac{\sqrt{t_b}}{\sqrt{t_b}+\sqrt{2}} < 1
\end{equation}

While we explore the second site, as long as the minimum price that we
find there is greater than $m_1$, our gain is zero. Assume that the
second patch has the same average price as the first, but a larger
spread, that is, in the second patch the prices are uniformly
distributed in $[\mu-b,\mu+b]$, with $b>a$. Also, define
$\gamma=b/a>1$ (I shall need it later).

The current minimum after we have explored the second site for a time
$t$ is

\begin{equation}
  \tilde{m}(t) = \mu - b\frac{t}{t+2}
\end{equation}

The expected minimum in the second site is the same as the optimal
price in the first at a time $t_0$ such that $\tilde{m}(t_0)=m_1$,
that is

\begin{equation}
  t_0 = \frac{2a\alpha}{b-a\alpha} = \frac{2\alpha}{\gamma-\alpha}
\end{equation}

The gain in the second site is given by the savings we obtain over the
previous minimum, that is

\begin{equation}
  g_2(t) =
  \begin{cases}
    0 & t<t_0 \\
    b\frac{t}{t+2} - a\alpha & t \ge t_0
  \end{cases}
\end{equation}

and

\begin{equation}
  g_2^\prime(t) =
  \begin{cases}
    0 & t<t_0 \\
    b\frac{2b}{(t+2)^2} - a\alpha & t \ge t_0
  \end{cases}
\end{equation}

The situation is depicted schematically in figure~\ref{patchdiag2}.

\begin{figure}
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