We propose a {\em mechanism} for generating the territorial patterns
using a mechanistic, spatially explicit mathematical model
with simple rules for scent-marking and movement.
Making no underlying assumptions about the size
and extent of the wolf territories themselves, we show how the
territorial patterns actually arise naturally
as stable steady-state solutions to the equations.
Our key modelling assumption is that wolf movement and behavior is
mediated by the absence/presence of foreign RLUs. In the {\em absence}
of foreign RLUs, (i) movement
is primarily dispersive as individuals search for food and other resources,
and (ii) RLU-marking occurs at low levels. The {\em presence} of
foreign RLUs elicits two responses: (i) movement towards
an organizing center for the pack
such as a den, above-ground rendezvous site or familiar
location\cite{Peters:BEW79}; and (ii) increase in RLU-marking
frequency\cite{Peters:AS-63-628}.
The RLU marks, themselves, are assumed to lose intensity with age.
The expected location of an RLU-marking wolf is described by a
probability density function denoting the chance
of finding it at point $\ul{x}$ and time $t$. For any given pack,
these are summed over the number of RLU-marking wolves to yield the
{\em expected} density of RLU-marking wolves,
henceforth referred to as the expected
density of wolves in a pack.
For a model of two adjacent interacting wolf packs, pertinent state
variables thus are the expected densities of wolves in packs 1 and 2,
($u(\ul{x},t)$ and $v(\ul{x},t)$);
and of RLUs from packs 1 and 2, ($p(\ul{x},t)$ and $q(\ul{x},t)$).
Components of the wolf movement
are expressed mathematically by:
\begin{equation}
\tpa{u}=-\nabla\cdot\left\{\ul{J}_{d_u}+\ul{J}_{c_u}
\right\},
\label{eq:w1a}
\end{equation}
and
\begin{equation}
\tpa{v}=-\nabla\cdot\left\{\ul{J}_{d_v}+\ul{J}_{c_v}
\right\},
\label{eq:w2a}
\end{equation}
where
$\ul{J}_{d_u}$ and $\ul{J}_{d_v}$ describe dispersive movement
via diffusion and
$\ul{J}_{c_u}$ and $\ul{J}_{c_v}$ describe movement towards organizing
centers situated at $\ul{x}_u$ and $\ul{x}_v$ via convection towards
$\ul{x}_u$ and $\ul{x}_v$, respectively.
Specific forms for the fluxes in (\ref{eq:w1a})--(\ref{eq:w2a})
were derived under the assumptions that
(i) dispersive movement is modelled via simple diffusion
($\ul{J}_{d_u}=-d\nabla u$, $\ul{J}_{d_v}=-d\nabla v$);
and (ii) convective movement is at a
rate proportional to the expected density of foreign RLU
($\ul{J}_{c_u}=-cuq\frac{(\ul{x}-\ul{x}_u)}{|\ul{x}-\ul{x}_u|}$,
$\ul{J}_{c_v}=-cvp\frac{(\ul{x}-\ul{x}_v)}{|\ul{x}-\ul{x}_v|}$, $c>0$).
Equations for RLU-marking and decay are:
\begin{equation}
\tpa{p}=u\left(l+mq\right)-fp,\label{eq:pa}
\end{equation}
and
\begin{equation}
\tpa{q}=v\left(l+mp\right)-fq,\label{eq:qa}
\end{equation}
where $l>0$ denotes low level continual RLU-marking,
$m>0$ describes increased
RLU-marking levels in the presence of foreign RLUs, and $f>0$
represents first order RLUs decay kinetics.
Analysis indicates the existence of a stable, spatially heterogeneous
steady-state solution to (\ref{eq:w1a})--(\ref{eq:qa}).
Describing territorial patterns and corresponding levels of
the spatially-distributed RLUs,
this solution (Fig.\ 3) can be found analytically by integrating
steady-state versions of (\ref{eq:w1a})--(\ref{eq:qa}) subject to
zero-flux boundary conditions for $u$ and $v$.
The solution predicts the partitioning of available space into territories
and the formation of territorial boundaries near which there are low levels
of expected wolf density and high levels of expected RLU density
(Fig.\ 3). This clearly reflects the
field observations regarding territorial patterns, interpack buffer zones
and bowl-shaped RLU densities.
Thus the simple rules for scent-marking
and movement given here are sufficient to explain territorial patterns.
Here, scent-marks are crucial to the formation of territories;
if scent-marking is not included
($l,m=0$ in (\ref{eq:pa})--(\ref{eq:qa}))
then $p,q\rightarrow 0$ and equations (\ref{eq:w1a}) and
(\ref{eq:w2a}) are reduced to diffusion equations so that
the territories do not form.
It can also be shown that
a necessary condition for the bowl-shaped RLU patterns
with interior maximums for $p$ and $q$ (Fig.\ 3)
is that foreign RLUs elicit a sufficiently large
increase in the RLU-marking rate.
Mathematically, this condition is
met when $m$ exceeds a critical value.
We now consider an extension of the above model which can be used to
predict the striking negative correlation between wolf territories and the
white-tailed deer distribution (Fig.\ 2).
For the sake of illustration we assume: (i) wolf
territories are stationary and stable (Fig.\ 3);
(ii) Holling Type I functional response of deer to
wolf predation;
(iii) Beverton-Holt density-dependent population dynamics.
Denoting $h(\ul{x},t)$ as the expected density of deer,
we have
\begin{equation}
\tpa{h}=-\psi (u(\ul{x})+v(\ul{x}))h,
\label{eq:HTI}
\end{equation}
where $u(\ul{x})$ and $v(\ul{x})$ denote the expected densities of pack 1 and
pack 2 wolves, $\psi$ is the predation rate and $0\leq t \leq T$,
the year being $T$ time units long.
Steady-state solutions, yielding constant
values when deer densities are compared from
one spring to the next, satisfy
the Beverton-Holt dynamics:
\begin{equation}
h(0,\ul{x})=\frac{\lambda h(T,\ul{x})}{1+(\lambda-1)h(T,\ul{x})/K},
\label{eq:BH}
\end{equation}
where $\lambda>1$ and $K$ measure the population growth rate and
carrying capacity of the deer in the absence of predation.
Solving (\ref{eq:HTI}) subject to the condition (\ref{eq:BH})
gives steady-state solutions as
\begin{equation}
h(0,\ul{x})=\max\{0,K \left(\lambda-\exp\{[u(\ul{x})+v(\ul{x})]\psi T\}\right)/(\lambda-1)\}.
\label{eq:deer_dist}
\end{equation}
Thus deer are found primarily where the combined wolf densities are lowest
--- in the buffer zones between packs (Fig.\ 4).
As noted by Mech\cite{Mech:S-198-320},
evidence for a similar situation involving deer and
human societies is given by Hickerson's study
of Sioux and Chippewa Indians from 1780--1850,
``The Virginia deer and intertribal buffer zones in the upper
Mississippi Valley''\cite{Hickerson:MCA65}.
Warfare between the two tribes gave rise to a buffer zone
which was normally unoccupied. In turn,
reduced hunting pressure in this buffer zone
effectively provided a refuge for the deer. However,
when a long truce was maintained, the buffer zone was
destroyed and famine ensued.