The bulk of my research will focus on foraging theory and its
applications in engineering. Behavioral ecologists use foraging
theory to model and analyze animals' strategies for investing
resources to find and process food. A popular guide to foraging
theory is the 1986 book, *Foraging Theory*, by D. Stephens
and J. Krebs, or the more recent 1999 book, *Models of
Adaptive Behavior: An approach based on state*, by A. I. Houston and
J. M. McNamara. In *Foraging Theory*, Stephens and Krebs use
a well-known model of the expected rate of caloric "point" gain
that a predator will receive given the assumption of Poisson
distributed prey arrival times. Using this model, known as
*Holling's Disc Equation*, Stephens and Krebs are able to
give algorithms predicting the optimal behavior of a predator.
In principle, the optimal conditions that Stephens and Krebs predict
could be used for control of an automatic agent facing tasks which
have a certain point value. For example, Stephens and Krebs give a
"prey algorithm" that predicts which prey should be processed when
encountered and which prey should be ignored. This algorithm should
then also be able to predict which targets an unmanned aerial
vehicle (UAV) should take the time to process and which targets the
UAV should ignore completely. However, there are some problems with
the derivation of the these algorithms that do not make them
directly applicable to engineering. One major problem is that
stochastic and deterministic methods have been improperly mixed in
the derivations of these algorithms. On top of this, even assuming
validity of the algorithms, Stephens and Krebs place strong
restrictions on environmental parameters that prevent the
application of many of the algorithms to the general class of
engineering problems.
Thus, it is the goal of my research with Professor Kevin Passino to
develop a proper stochastic model for rate of point gain and to show
proofs of the optimality conditions on this new model that make no
assumptions about the environmental parameters. In particular, the
prey algorithm, which chooses which prey types to process and which
prey types to ignore, and the patch algorithm, which chooses when to
leave an area of diminishing returns, will be constructed for the
optimal case for arbitrary environmental parameters. Moreover, we
are going to evaluate the potential to derive vehicle control
polices by optimizing performance metrics appropriate for UAV
applications.
This will have completed and extended the work of Stephens and Krebs
to a much more general case. However, there are a number of
questions not even addressed by Stephens and Krebs that will have a
significant impact on engineering design. One question of interest
to me is the one posed by R. Gendron and J. Staddon in their 1983
*American Naturalist* paper, "Searching for cryptic prey:
The effect of search rate." In this paper, Gendron and Staddon use
the same *Holling's Disc Equation* to show that there exists
an optimal non-maximal prey search rate in the presence of prey that
camouflage themselves and thus require more time for detection. The
choice of search rate is important for engineering due to sensor
sensitivity to speed. For example, low bandwidth sensors will be
ineffective at high speeds, so the agent's optimal speed will most
likely not be the maximum speed of the agent but some lesser speed
where the sensors can operate properly. However, the Gendron and
Staddon paper is only meant to explain an apparent speed modulation
in observed predators in nature; it does not propose a general
framework for predicting what an optimal speed might be in a given
environment. This is an important question that I will answer in the
course of my research.
Next, neither Stephens and Krebs nor Gendron and Staddon have a
satisfactory theoretical treatment of how an animal copes with
imperfect information. In many engineering cases, the agent will not
know the expected arrival rate of its tasks a priori and will have
to estimate this "on the fly." If the agent is additionally
modulating its speed with changes in perceived arrival rate, those
changes in speed will naturally cause more changes in the perceived
arrival rate. Thus there is a circular relationship that gives the
trajectory of the agent's speed dynamics that need to be
well-understood.
It is worth noting that while the UAV case may be an obvious
application of this research, it is not the only application into
which this research will provide insights. Servers servicing
networks that bring in tasks can be thought of in terms of foragers.
Even personal computers processing local tasks can be thought of as
foragers. In fact, in the interesting case of a personal computer
that does useful processing in its idle time, the computer may act
like a forager that actually gains points when searching. This is an
interesting case that could not be handled with the assumptions
currently held by Stephens and Krebs since they assume that a
forager always suffers a cost for searching. In the case where
searching brings some positive gain, then the optimal choice of prey
may be to ignore all tasks if no combination of tasks achieves the
same point gain as the idler task.
Foraging theory also applies to the social case, where multiple
foragers sharing one space may interact. In "social foraging
theory," best summarized in the book, *Social Foraging
Theory*, by L. Giraldeau and T. Caraco, evolutionary game theory is
used to analyze the optimal "designs" created by evolution.
Understanding the dynamics of social agents will undoubtedly give
tremendous insight on optimal control schemes for decentralized
controllers for engineering applications. I am particularly
interested in the theoretical requirements for emergent cooperation
between agents and how to design an optimal mix of heterogenous
agents to accomplish a task. Our current plan is to apply the theory
to the design of optimal robust cooperation for a group of UAV's
performing cooperative search for targets.
These questions surrounding foraging theory will account for the
bulk of my research during my fellowship tenure. Each of the major
issues here will be the subject of a journal paper. The goal is to
publish the full details of the topics mentioned here in a book
expected to be titled, *Foraging Theory for Engineering*,
authored by engineering Professor Kevin Passino, biology Professor
Tom Waite, engineering graduate student Burt Andrews, and myself. We
currently have more than 80 pages written for this book.
My interest in these topics was recreational at first. The sources
of my interest were a number of books that I read outside of school
which discussed complex biological systems. As I began to understand
some of the origins behind the emergent structures and behaviors of
these systems, it became clear to me that understanding their
dynamics would be extremely useful to future work in engineering.
Controls engineering is being pushed into the multi-agent arena that
is preferably decentralized. Before building these systems, it will
be extremely useful to understand already built working examples of
similar systems that we already find in biology. Additionally, I
find it immensely appealing that applying engineering methodology to
the analysis of biological systems may one day provide some sort of
new insights to practicing biologists. If biology can assist
engineering, engineering should be able to assist biology, and being
a part of that would be extremely rewarding to me.
Now, in order to make sure I have the appropriate background to
study biology in this way, I am taking a number of biology and
ecology courses while pursuing my graduate degree. In order to
understand some of the more complex dynamics in these systems, I am
taking a number of fairly advanced mathematics courses. Together
with my engineering background, I feel that I will be adequately
equipped to be productive in a field that is very interesting to me.

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