Equilibrium Configurations for a Territorial Model

摘要

We consider a territorial model based on Voronoi tessellations, which form a partition 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. Equilibria for this model correspond to centroidal Voronoi tessellations. 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. 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 also numerically explore how stable and unstable equilibrium configurations are related to each other. Results for three agents are verified through experiments using robots which move according to a related territorial algorithm.