### Abstract

We consider a territorial model based on Voronoi tessellations. Such tessellations form a partitioning of a planar region by enclosing each agent in a polygon such that every point within the polygon is closest to that agent instead of any other. For rectangular domains and for small population sizes, we show that there can be distinct coexisting stable equilibrium configurations, including the possibility of stable equilibria that are not related by symmetry. By considering randomly distributed initial positions, we give a statistical characterization of the basins of attraction for these equilibria in the case of a square domain. Furthermore, we show that the final territory that an agent occupies can have a wide range of sizes, which suggests that an individual can obtain a competitive advantage or disadvantage due entirely to its initial position. Finally, by treating the ratio of the length of the shorter side to the length of the longer side of the rectangle as a bifurcation parameter, we numerically explore how stable and unstable equilibrium configurations are related to each other.

Translated title of the contribution | Equilibrium configurations for a territorial model |
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Original language | English |

Pages (from-to) | 1234 - 1260 |

Number of pages | 27 |

Journal | SIAM Journal on Applied Dynamical Systems |

Volume | 8 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jul 2009 |

### Keywords

- territorial behavior
- Voronoi tessellations
- symmetry

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## Cite this

Votel, R., Barton, D. A. W., Gotou, T., Hatanaka, T., Fujita, M., & Moehlis, J. (2009). Equilibrium configurations for a territorial model.

*SIAM Journal on Applied Dynamical Systems*,*8*(3), 1234 - 1260. https://doi.org/10.1137/070710123