{\bf
Wolf ({\em Canis lupus}) range in northeastern Minnesota forms a striking 
spatial mosaic of non-overlapping 
territories\cite{Mech:WNI73,VanBallenberghe:WM-43-1}.
We propose a mechanism for the formation of these territorial
patterns, analysing a spatially explicit
mathematical model which incorporates wolf movement and scent marking.
No assumptions are made about the existence or extent of territories; 
territorial patterns arise naturally as stable steady-state solutions to 
the equations.  Model results reflect field observations: 
buffer zones where wolves are scarce arise between adjacent packs and 
near these buffer zones there are increased levels of 
scent marking.  
We show mathematically how reduced predation in 
the buffer zones can provide a refuge for prey species.  
This result reflects
field studies; spatial patterns in the
distriubtion of primary prey species, 
white-tailed deer ({\em Odocoileus virginianus}), are closely tied 
to those of the wolves---deer are found primarily in buffer the
zones\cite{Hoskinson:JWM-40-429,Rogers:JWM-44-253}.
Evidence for a similar situation with deer populations and
human societies is given by a case
study of Sioux and Chippewa Indians from 1780--1850\cite{Hickerson:MCA65,%
Mech:S-198-320}
}\\
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\section{Striking spatial patterns}

Radio-marking studies of wolves ({\em Canis lupus}) 
in northeastern Minnesota in the last 20 years have greatly 
facilitated the observation of wolf territories --- not an easy task.  
By following the movements of radio-marked individual members from a pack,
or cooperative extended family group, these studies have made it possible
to deduce distinct spatial patterns in wolf distribution.  Wolves typically
remain within well-defined territories\cite{Mech:WNI73,%
VanBallenberghe:WM-43-1} that overlap only along
their edges (Fig.\ 1).
These territories may effectively partition jurisdiction over 
spatially distributed resources such as prey.

Spatial patterns in the distribution of a primary prey species,
white-tailed deer ({\em Odocoileus virginianus}), are closely tied
to those of the wolves; deer are found primarily in the narrow 
corridors between territories of adjacent wolf 
packs\cite{Hoskinson:JWM-40-429,Rogers:JWM-44-253} (Fig.\ 2).
We now develop and analyse a spatially explicit model for the
wolf and deer populations with a specific view to understanding 
mechanisms for production of these distinct spatial patterns. 

\section{The ecology of wolf territoriality}

In northeastern Minnesota wolf packs basically remain in territories 
ranging from 125 to 310 km${^2}$ \cite{Peters:AS-63-628}.  
Territorial boundaries are usually avoided, 
primarily to limit confrontation with adjacent packs\cite{Mech:S-198-320}.  
The resulting buffer zones or `no-mans-lands' are about 2 km wide and 
may comprise as much as 25-40\% of the available area\cite{Mech:S-198-320,%
Mech:PPP75}.

Based upon many years of field observations,
Mech\cite{Mech:WOW91} claims that wolf territories are formed
and maintained by inter-pack aggression in conjunction with two 
warning systems: scent marking and howling, and that the 
result is a mosaic of territories covering the
wolves' range.  While howling may provide temporary information
on a pack's location, elaborate spatial patterns of scent marks serve to 
advertise precise information about territorial claims.  It is these
scent marks that we now consider.

As with other carnivores, the primary sensory modality for wolves is 
olfaction (or scent)\cite{Macdonald:SOM85}.  
As wolves travel, they typically leave 
olfactory signs.  Marks include (1) raised-leg urination (RLU), (2) squat 
urination, (3) defecation, and (4) scratching.  Behavioral studies 
indicate that it is the RLU which plays the most important role in leaving 
information for subsequent canine travellers\cite{Peters:AS-63-628,%
Merti-Millhollen:ZB-5-7}.  
Thus, although wolves typically use a spectrum of olfactory stimuli for 
territory maintenance, we concentrate here upon the RLU as the dominant one.  
The information available for subsequent travellers includes the identity 
of the pack from which the RLU 
originated, the approximate time since the RLU was made and
the reproductive status of the pack\cite{Rothman:AB-27-750}.  
Out of a typical pack of 5--15 
wolves\cite{Mech:WEB70}, only a few mature dominant members leave 
RLUs\cite{Peters:AS-63-628}.
However, these wolves leave RLUs quite regularly upon trails and
and a travelling wolf may expect to encounter one RLU every 
two to three minutes\cite{Peters:AS-63-628}.  
Observations indicate some aversion to 
the scent from RLUs made by neighboring packs\cite{Peters:AS-63-628}.
Marking frequency approximately doubles near territorial 
borders, giving rise to `bowl-shaped' distributions of RLUs
across territories, with the raised edges of the olfactory bowl located at the
territorial boundaries\cite{Peters:AS-63-628}.
%It is interesting that scent marking 
%increases over the winter months when the wolves are no 
%longer compelled to remain close to the den or rendezvous site, and are 
%therefore more able to infringe upon neighboring territories.