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\section*{BUDGET JUSTIFICATION}
\subsection*{Personnel}
The proposed research project involves three separate researchers:
Mark Lewis (Assistant Professor at the University of Utah, Salt Lake
City), James Murray (Professor at the University of Washington, Seattle)
and Steve Minta (consultant). Upon the advice of NSF officer Dr.\
M. Steuerwalt (NSF Applied Mathematics) we are submitting two separate
grant proposals. This proposal concerns research under the supervision
of Mark Lewis (PI). The other proposal (number DMS--9209737),
submitted by Professor Murray, concerns work under his supervision and
work by the consultant, Steve Minta. Accordingly, justification is
provided here for the work under the supervision
of the PI for this grant (Mark Lewis); budget justification for the remainder
of the project is provided in proposal number DMS--9209737.
In this proposed research project we address
a complex biological problem involving
intra- and inter-specific behavioral and
ecological interactions. Our preliminary
models (Section 4), by necessity,
involve complex systems of nonlinear
equations. We anticipate that successful
modelling of the biological problem
and subsequent mathematical and numerical
analysis will entail a large time
commitment from the principal investigator
(PI).
Reasons are threefold:
\begin{enumerate}
\item A key element in the project will be the further development
of our preliminary models.
This process, which we now briefly review,
will be one of iterative refinement (see also Section 5):
model parameters for [(9)--(13), (14)--(16) and (17)--(24)]
will be estimated from the field
observations of relevant processes (such as fecundity, mortality, movement etc.);
these parameters will be substituted into the model system; analytical results
regarding the spatio-temporal distribution and abundance of wolves and deer
will be compared with field data;
discrepancies between the analytical predictions and field data
will suggest areas for improvement
and the model will then be revised so as to reflect the biology more accurately.
Further analysis and comparison may suggest new areas for improvement.
Whereas comparison between analytical results and field data
will concern the distribution and abundance of species,
estimation of model parameters will arise
from field observations of specific processes.
Thus a `curve-fitting' approach (where one merely tries to
mimic the observed data) will be avoided. This kind of
effective refinement of our models into a form where important biological detail
is not sacrificed for mathematical simplicity may be a lengthy process.
The PI is currently
involved in several math-biology interdisciplinary projects (including some field work).
The interdisciplinary nature of this process will involve periodic visits
to the biological consultant, Dr.\ Minta at Whitefish, Montana.
(Dr.\ Minta's salary is to be provided
by proposal number DMS--9209737.).
\item We will be mathematically analyzing large
systems of equations, such as [(9)--(13), (14)--(16) and (17)--(24)],
using asymptotic methods to determine their behavior.
These methods are typically
algebraically intricate and time consuming,
even when applied to fairly simple systems.
However, it is only through this kind of
{\em mathematical} analysis of the
model equations that we can gain real insight
into the fundamental processes that
control the dynamic pattern formation.
The PI is very familiar with
the application of asymptotic methods to nonlinear
biological problems and
is experienced in using symbolic manipulators to
aid in analysis.
\item Whereas numerical solutions of difference equation
models such as (17)--(24) are typically straightforward,
two-dimensional numerical solutions
of a large nonlinear PDE system such as
(9)--(13) (see Section 5.1) are
very challenging and time-consuming.
The PI has been involved with many
research problems that involve intensive numerical work
as well as analytic work and is familiar and experienced
with numerical methods for solving nonlinear PDEs.
\end{enumerate}
The proposed research project will require a thorough understanding of applied
analysis, numerical analysis and the construction of mathematical models.
The work will be of an interdisciplinary nature, involving background
knowledge of quantitative ecology and animal behavior.
The project is thus very suitable for the training of graduate students
in the area of applied mathematics who wish to specialize in mathematical
and computational biology. Currently the University of Utah Mathematics Department
has a strong mathematical biology program and one of the graduates has shown a keen
interest in working on this project; he is currently enrolled in advanced
applied mathematics and mathematical biology courses which will provide a
solid foundation for the research. It is anticipated that the summer work by
the graduate student will entail mathematical modelling and numerical computations.
\subsection*{Travel}
In order to be advised effectively by the consultant
the PI will visit for two days at approximately four month intervals
at the consultant's office in Whitefish Montana.
We feel that the resulting intensive collaboration
will play a crucial role in
ensuring that the biological questions are clearly addressed by our research and
that the biological component of the project is maintained.
The interdisciplinary nature of this collaboration is best suited to
an inter-personal `brainstorming' approach as opposed to protracted
communication by mail or telephone; the first few sessions will be crucial
for agreeing upon the conceptual modelling perspective.
We have estimated travel expenses to be approximately \$764 per visit. This
includes return air fare to Missoula, Montana, return ground travel Whitefish,
Montana, and two night's accommodation.
\subsection*{Materials and Supplies}
The funds requested for Materials and Supplies are primarily for: photocopying,
long-distance telephone calls, mailing and expendables (such as stationary and
paper for laser printers).
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