\addcontentsline{toc}{section}{BUDGET JUSTIFICATION}
\section*{BUDGET JUSTIFICATION}
\subsection*{Personnel}
The proposed research project addresses
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 co-principal investigator
(Co-PI) and a substantial
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 [(\ref{eq:m1nd1})--(\ref{eq:m1nd5}), (\ref{eq:SR1})--(\ref{eq:SR3})
and (\ref{eq:SM1})--(\ref{eq:SM8})] 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 has been involved in very many similar projects
and collaborates with a large number of biologists.
The Co-PI, being jointly affiliated with
Applied Math and Zoology departments at the University of Washington, 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 (see below for further details).
\item We will be mathematically analyzing large
systems of equations, such as [(\ref{eq:m1nd1})--(\ref{eq:m1nd5}),
(\ref{eq:SR1})--(\ref{eq:SR3}) and (\ref{eq:SM1})--(\ref{eq:SM8})], 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.
Both the PI and the Co-PI are very familiar with
the application of asymptotic methods to nonlinear
biological problems and the Co-PI
is experienced in using symbolic manipulators to
aid in analysis.
\item Whereas numerical solutions of difference equation
models such as (\ref{eq:SM1})--(\ref{eq:SM8}) are typically straightforward,
two-dimensional numerical solutions
of a large nonlinear PDE system such as
(\ref{eq:m1nd1})--(\ref{eq:m1nd5}) (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 the Co-PI 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
biology. Currently the University of Washington Applied Mathematics Department
has nine graduate students who are specializing in mathematical biology
(including four women). One of the graduates has shown a keen
interest in working on this project; she is currently enrolled in advanced
applied mathematics and mathematical biology courses which will provide a
solid foundation for the research.
\subsection*{Consulting}
The consultant, Dr.\ S. Minta, is involved in modelling the spatial
and temporal aspects of animal interactions (see,
for example, Minta, in press), has a wide range of
experience in field studies of predator-prey interactions,
and recently compiled a bibliography
on wolf interactions and compatibility with ungulates, livestock
and humans (with over 2000 entries and over 1000 annotations).
He is thus well qualified to assist us in two capacities:
\begin{enumerate}
\item
He can provide a critical appraisal of our behavioral and ecological models,
drawing upon long-term involvement with spatio-temporal animal interactions
studies and upon his extensive field experience.
\item
He can direct us to sources of field data on wolf-deer interactions to which we
would otherwise not have access.
\end{enumerate}
We anticipate meeting with Dr.\ Minta three times per year for 2 days per
meeting (a total of six days per year). The Co-PI will travel from Seattle
to visit Dr.\ Minta at Whitefish, Montana (see Travel, below).
\subsection*{Computing}
The University of Washington Department of Applied Mathematics has a
network of 14 DecStation computers (including 4 DS2100's,
4 DS3100's and 6 DS5000's), with 10 Gigabytes of disk space, which are linked
to color and monochrome printers.
Software available on this network
of computers includes the NAG FORTRAN library, Matlab, Splus, Mathematica,
various public domain subroutine libraries, TeX and LaTeX,
the Frame desk-top publishing system and a variety of compilers, including FORTRAN.
This system is for the exclusive use of the Applied Mathematics Department.
Maintenance and development of the system, by the department's full-time
systems programmer, depends critically upon the ongoing support provided
by research funding for hardware and software.
In view of the major computational component of
this proposal, we are requesting a modest level of support:
\begin{itemize}
\item
Year 1:
\begin{itemize}
\item Purchase of a Digital Equipment Corporation 19 inch greyscale DecStation
5000 Model 200 8 MB Workstation (\$6000).
This will be used for developing code to solve the behavioral (PDE) model in one and
two spatial dimensions, for running simulations of the PDE model, for solving the
ecological difference equation models, for plotting solutions graphically, for
symbolic manipulations needed in the mathematical analysis and for word processing
and typesetting papers.
\item Purchase of a Tektronix 19 inch greyscale XP23 X-terminal
which will be linked by ethernet to the DS5000 (\$1750).
This will be used primarily by the graduate student to
access the DecStation and assist in the tasks outlined above.
\item Purchase of a {\em Mathematica} (a symbolic manipulator) licence for the
DS5000 (\$1200).
\item A yearly license to use the NAG library numerical software routines on the
DS5000 (\$600). The NAG library provides subroutines to solve nonlinear PDE systems
in one spatial dimension. Although the majority of code for two-dimensional
solutions will have to be custom-made, certain NAG subroutines
can be incorporated and will assist greatly in producing accurate and efficient code.
\end{itemize}
\item
Years 2 and 3:
A yearly license to use the NAG library numerical software routines on the DS5000
(\$600).
\end{itemize}
The hardware, software, and networking already in place
form a backbone onto which we can easily integrate the requested
DS5000, X-terminal and software.
The PI can gain access to the networked system and the DS5000 via an existing
high resolution, color, Tektronix XP29 X-terminal which is already in his office.
The PI currently has a Zenith TurboSport 386 Portable microcomputer (with graphics
capabilities) which can be used to access the above computing
facilities directly in order to demonstrate the numerical solution of model equations
when the Co-PI visits the consultant in Whitefish, Montana.
\subsection*{Travel}
In order to be advised effectively by the consultant (see Consultant, above)
the Co-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
(with the aid of the portable Zenith for illustrating numerical solutions,
see Computing, above) 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 \$600 per visit. This
includes return air fare to Missoula, Montana, return ground travel Whitefish,
Montana, and two night's accomodation.
\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).
\subsection*{Yearly Adjustments}
Yearly adjustments in salary are given as 3.9\% in the 1993 calendar year and
6.0\% in the 1994 and 1995 calendar years. These are given by the University
of Washington guidelines.
Yearly adjustments for Supplies and Publication Costs are given as 5\%.
Fringe benefits are calculated as 23\% of the
proportion of the PI's and Co-PI's salary covered by the grant proposal and
as 1\% of the graduate student's salary.