  Territoriality is a phenomenon exhibited throughout nature. On the individual
level, it is the processes by which organisms exclude others of the same
species from certain parts of space. On the population level, it is the
segregation of space into separate areas, each used by subsections of the
population. Proving mathematically that such individual-level processes can
cause observed population-level patterns to form is necessary for linking these
two levels of description in a non-speculative way. Previous mathematical
analysis has relied upon assuming animals are attracted to a central area. This
can either be a fixed geographical point, such as a den- or nest-site, or a
region where they have previously visited. However, recent simulation-based
studies suggest that this attractive potential is not necessary for territorial
pattern formation. Here, we construct a partial differential equation (PDE)
model of territorial interactions based on the individual-based model (IBM)
from those simulation studies. The resulting PDE does not rely on attraction to
spatial locations, but purely on conspecific avoidance, mediated via
scent-marking. We show analytically that steady-state patterns can form, as
long as (i) the scent does not decay faster than it takes the animal to
traverse the terrain, and (ii) the spatial scale over which animals detect
scent is incorporated into the PDE. As part of the analysis, we develop a
general method for taking the PDE limit of an IBM that avoids destroying any
intrinsic spatial scale in the underlying behavioral decisions.
