\section{Analysis and Development of Models}
\subsection{Model 1 (Behavioral)}
\begin{enumerate}
\item The {\em continuum} model, formulated in Section 4.2, provides a useful
approach to analyzing and computing the expected densities of wolves and deer.
However, the typical field data for interactions will be in the form
of observations of individual behaviors. We will rigorously connect
individual behavioral rules, based on field observations,
to the model equations [(\ref{eq:m1nd1})--(\ref{eq:m1nd5})].
This will be achieved by using a {\em correlated random walk} approach, which
determines the probability of moving in a given direction as
a function of the behavior (see Okubo (1980) for a review of theory and
Turchin (1991) for specific examples using field data).
Thus we will provide an essential link from the behavior of individuals
to the nonlinear functions $n_1(\cdot)\ldots n_4(\cdot)$
and coefficients such as $c_u$, $D_u$ and $t_u$,
which describe population flux terms in the continuum model equations.
Our results will likely suggest modifications to the nonlinear functions
shown qualitatively in Figure 3.
\item We will estimate parameters such as $k_u$, $l_p$, $m_p$ and $f_p$.
A two-pronged approach to parameter estimation will be employed.
First we will use existing observations and published field data
to give estimates of the parameters in
(\ref{eq:w1a})--(\ref{eq:qa}).
For example, Peters and Mech (1975) recorded abrupt changes in RLU frequency
when packs encountered scents from a neighboring pack. Such data may allow us
to make an
estimate for the order of magnitudes of $m_q$ and $m_p$. Second, when we visit
the field ecologist, Dr.\ Steven Minta, we will refine our estimates for parameters
by drawing upon his extensive field experience and by utilizing
his knowledge of field data which otherwise would be difficult to obtain.
He has also agreed to act as a liaison and put us in touch
with other field researchers who have unpublished
observations and data regarding wolf behavior.
\item We will explore the possibility of incorporating a term which describes the
effect of a `cognitive map' (Peters, 1979) upon the behavior of the wolves.
For example, in addition to equation (\ref{eq:w1}),
an equation for the `memory' of the wolf pack could be used,
allowing the memory to increase over time in regions of high wolf density and to decay
with first order kinetics. A term could be added to (\ref{eq:w1}),
describing the preferential movement, or `taxis', of wolves up a memory gradient.
In other words, wolves would move back into the most familiar areas.
\item We will compare the steady state solutions for (\ref{eq:w1})
and (\ref{eq:w1b})--(\ref{eq:hb}) (for $r_u=0$),
with zero-flux and constant boundary conditions
for $u$ and $h$, respectively. This comparison will aid in answering the
last half of question (1), Section 3.1, namely how to predict the territorial
behavior of wolves in the absence of surrounding packs. We expect that the
presence of deer in the second system will cause an expansion of the
wolf territory as individuals move towards areas where the deer are not yet
depleted. We would like to quantify this as a function of the model
parameters so as to highlight key processes.
\item The nature of wolf-wolf territorial interactions, described in
equations (\ref{eq:w1a})--(\ref{eq:qa}), leads us to expect that, if
two packs are mixed together at random,
they will spontaneously segregate and territories will arise,
even though there initially may be confusion. We will mathematically
study the nature of this territory formation.
As inter-pack altercations are fairly rare (Mech, 1970), we expect that
$k_u$ and $k_v$ typically will have small values.
Initially we will set
$k_u=k_v=0$ in (\ref{eq:w1a})--(\ref{eq:qa}), assuming that $u$ and $v$
are constant everywhere and that $p$ and $q$ are given by their resulting
steady-state solutions. Using regular perturbation theory and
linear and weakly nonlinear analysis techniques (see, for example,
Lewis and Murray (1991) (Appendix)),
we will predict the behavior as perturbations grow away from this homogeneous
solution. We expect that territories will result as a new, stable,
nonhomogeneous steady state solution evolves.
Subsequently we will follow a similar analysis
when $k_u$ and $k_v$ are small positive parameters. In this case, the
variables will no longer be initially at a uniform steady state, but will be slowly
changing. In our analysis we will use multiple time scale methods and
similar mathematical techniques. Our analytical results will be supplemented
and confirmed by the numerical solution of the model equations. Finite difference
methods will be employed for the numerical calculations.
Our analysis and numerics will help us to answer the first part of question (1),
Section 3.1, namely how wolf pack territories form, and how they are maintained.
\item Using similar techniques to those just described, we will analytically
and numerically determine how the addition of deer dynamics changes
territorial patterns in the composite model [(\ref{eq:m1nd1})--(\ref{eq:m1nd5})];
will determine the resulting distribution of deer.
Insight into how and why the deer populations are found in buffer zones between
the wolf packs will be provided (see question (2), Section 3.1). Due to the number
of model equations, we will employ a symbolic manipulation
package ({\em Mathematica}) for complex analytical details.
By setting $\gamma_u$ and $\gamma_v$ to zero, increasing $\lambda_p$,
$\lambda_q$, $\mu_p$ and $\mu_q$, and decreasing $\delta_h$, we can
change [(\ref{eq:m1nd1})--(\ref{eq:m1nd5})] from summer to winter conditions.
Thus, by determining how the territorial and deer dynamics depend upon these parameters,
we will gain insight into typical winter time events such as buffer zone trespasses,
wolf-wolf altercations, wolf starvation, territory changes and low deer populations
(see questions (3) and (4), Section 3.1). We will consider both the qualitative
response of the model to these seasonal
parameters, and the sensitivity of the model to
the parameters.
\item We will forge a link between the model and field observations
by evaluating whether the biologically determined parameters, given in (1),
do indeed yield a realistic description of wolf-wolf and wolf-deer interactions and
populations. If they do not, this will be an indication for us to refine the
model in conjunction with Dr.\ Minta.
When satisfied with the model equations, we will determine the
sensitivity of the wolf territory boundaries and deer dynamics to all model parameters
(see the first half of question (5), Section 3.1).
\item Applying numerical and analytical approaches, we will determine two-dimensional
spatial patterns that arise from the model equations
[(\ref{eq:m1nd1})--(\ref{eq:m1nd5})]. The situation where there are
three or more adjacent wolf packs will be considered. This
will entail the development and testing of numerical code to solve the two-dimensional
problem. This is a nontrivial problem and will require substantial programming effort
(see Lewis and Murray (1991) (Appendix) for the 2D numerical solution of
a system of partial differential equations with comparable nonlinear dynamics.)
The results will help us in answering the last half of question (5), Section 3.1.
\end{enumerate}
\subsection{Model 2 (Ecological)}
\begin{enumerate}
\item Estimation of birth rates, migration rates and survival rates for equations
(\ref{eq:SR1})--(\ref{eq:SM8}). We will use the same approach as described in
Section 5.1, item (2), employing existing field data as well as
drawing upon the experience of Dr.\ Minta.
\item We will determine and evaluate the stability of the steady state solutions to
(\ref{eq:SR1})--(\ref{eq:SR3}) as the migration rate ($m(H_t,R_t)$) is
varied. In the event that unstable oscillatory solutions give rise to
a limit cycles, we will analytically and numerically determine how
the migration rate affects the amplitude and period of the resulting cycles
(see, for example, May (1975), Beddington {\em et al.} (1975),
Whitley (1982) and Kelley and Peterson (1991) for a suitable
approach to the analysis). This computation and analysis
will aid us in answering question (6) in Section 3.1.
\item Employing a similar approach to that described above, we will
use equations (\ref{eq:SM1})--(\ref{eq:SM8}) to determine whether the
biannual deer migrations act as a stabilizing factor in wolf-deer interactions
(see question (7), Section 3.1).
\end{enumerate}
\section{Significance of Research}
This research will be the first serious mathematical attempt to model the
spatio-temporal dynamics of territorial pattern formation. We feel that this
research will have fundamental significance to theoretical biology; territorial
patterns provide the basis for social organization in many mammals and birds.
The research should also provide significant insight into
how territorial patterns affect predator-prey interactions and prey distributions.
Particular insight will be provided into dynamics of
the widely studied wolf-deer interactions in northeastern Minnesota.
The mathematical models to be used for this research are,
by necessity, both complex and nonlinear. Analysis will entail
new, significant applications of analytical and numerical methods
and will provide an interesting challenge to applied mathematicians.