Analysis of a Class of Symmetric Equilibrium Configurations for a Territorial Model

Motivated by an animal territoriality model, we consider a centroidal Voronoi tessellation algorithm from a dynamical systems perspective. In doing so, we discuss the stability of an aligned equilibrium configuration for a rectangular domain that exhibits interesting symmetry properties. We also demonstrate the procedure for performing a center manifold reduction on the system to extract a set of coordinates which capture the long term dynamics when the system is close to a bifurcation. Bifurcations of the system restricted to the center manifold are then classified and compared to numerical results. Although we analyze a specific set-up, these methods can in principle be applied to any bifurcation point of any equilibrium for any domain.