Journal of Mathematical Biology

, Volume 72, Issue 1–2, pp 25–46 | Cite as

Territorial pattern formation in the absence of an attractive potential



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.


Advection–diffusion Animal movement Home range   Individual based models Mathematical ecology Partial differential equations Pattern formation Territoriality 

Mathematics Subject Classification

35B36 92B05 



This study was partly funded by NSERC Discovery and Accelerator grants (MAL, JRP). MAL also gratefully acknowledges a Canada Research Chair and a Killam Research Fellowship. We are grateful to Andrew Bateman and other members of the Lewis Lab for helpful discussions as well as two anonymous reviewers for helping improve the manuscript.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of SheffieldSheffieldUK
  2. 2.Department of Mathematical and Statistical Sciences, Centre for Mathematical BiologyUniversity of AlbertaEdmontonCanada

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