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.
|Publication status||Published - 5 Dec 2007|
Bibliographical noteSponsorship: This work was supported by National Science Foundation grant NSF-0434328, an Alfred P. Sloan Research Fellowship in Mathematics (JM), and a Lloyds Tercentenary Foundation Fellowship (DAWB).
- Voronoi tesselations
- territorial behaviour