Journal of Mathematical Biology

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

Territorial pattern formation in the absence of an attractive potential

Article

Abstract

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.

Keywords

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

Mathematics Subject Classification

35B36 92B05 

Notes

Acknowledgments

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.

References

  1. Adams ES (2001) Approaches to the study of territory size and shape. Annu Rev Ecol Syst 32:277–303CrossRefGoogle Scholar
  2. Arnold J, Soulsbury CD, Harris S (2011) Spatial and behavioral changes by red foxes ( Vulpes vulpes) in response to artificial territory intrusion. Can J Zool 89:808–815CrossRefGoogle Scholar
  3. Bateman AW, Lewis MA, Gall G, Manser MB, Clutton-Brock TH (2015) Territoriality and home-range dynamics in meerkats, Suricata suricatta: a mechanistic modelling approach. J Anim Ecol 84:260–271CrossRefGoogle Scholar
  4. Briscoe BK, Lewis MA, Parrish SE (2002) Home range formation in wolves due to scent marking. Bull Math Biol 64:261–284CrossRefGoogle Scholar
  5. Burt WH (1943) Territoriality and home range concepts as applied to mammals. J Mammal 24:346–352CrossRefGoogle Scholar
  6. Durrett R, Levin S (1994) The importance of being discrete (and spatial). Theor Pop Biol 46:363–394MATHCrossRefGoogle Scholar
  7. Einstein A (1916) The foundation of the general theory of relativity. Ann. Phys 354:769–822CrossRefGoogle Scholar
  8. Fulton W (1969) Mathematics lecture note series, W.A., Algebraic curvesBenjamin, New YorkGoogle Scholar
  9. Giuggioli L, Kenkre VM (2014) Consequences of animal interactions on their dynamics: emergence of home ranges and territoriality. Mov Ecol 2:20. doi: 10.1186/s40462-014-0020-7 CrossRefGoogle Scholar
  10. Giuggioli L, Potts JR, Harris S (2011a) Animal interactions and the emergence of territoriality. PLoS Comput Biol 7:1002008CrossRefGoogle Scholar
  11. Giuggioli L, Potts JR, Harris S (2011b) Brownian walkers within subdiffusing territorial boundaries. Phys Rev E 83:061138CrossRefGoogle Scholar
  12. Harris S (1980) Home ranges and patterns of distribution of foxes ( Vulpes vulpes) in an urban area, as revealed by radio tracking. In: Amlaner CJ, Macdonald DW (eds) Handbook of biotelemetry and radio tracking. Pergamon Press, Oxford, pp 685–690CrossRefGoogle Scholar
  13. Lewis MA, Murray JD (1993) Modelling territoriality and wolf–deer interactions. Nature 366:738–740CrossRefGoogle Scholar
  14. Lewis MA, White KAJ, Moorcroft PR (1997) Analysis of a model for wolf territories. J Math Biol 35:749–774MATHMathSciNetCrossRefGoogle Scholar
  15. Mallinson GD, de Vahl DG (1973) The method of the false transient for the solution of coupled elliptic equations. J Comput Phys 12:435–461MATHCrossRefGoogle Scholar
  16. Moorcroft PR (2012) Mechanistic approaches to understanding and predicting mammalian space use: recent advances, future directions. J Mammal 93:903–916CrossRefGoogle Scholar
  17. Moorcroft PR, Lewis MA (2006) Mechanistic home range analysis. Princeton University Press, PrincetonGoogle Scholar
  18. Moorcroft PR, Lewis MA, Crabtree RL (2006) Mechanistic home range models capture spatial patterns and dynamics of coyote territories in Yellowstone. Proc R Soc B 273:1651–1659CrossRefGoogle Scholar
  19. Murray JD (2002) Mathematical biology II: spatial models and biomedical applications, 3rd edn. Springer, New YorkGoogle Scholar
  20. Potts JR, Lewis MA (2014) How do animal territories form and change? Lessons from 20 years of mechanistic modelling. Proc R Soc B 281:20140231CrossRefGoogle Scholar
  21. Potts JR, Harris S, Giuggioli L (2012) Territorial dynamics and stable home range formation for central place foragers. PLoS One 7:0034033CrossRefGoogle Scholar
  22. Potts JR, Mokross K, Lewis MA (2014) A unifying framework for quantifying the nature of animal interactions. J R Soc Interface 11:20140333CrossRefGoogle Scholar

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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