Numer. Math. Theor. Meth. Appl. Vol. 3, No. 2, pp. 143-161 doi: 10.4208/nmtma.2010.32s.2 May 2010 Analysis of a Class of Symmetric Equilibrium Configurations for a Territorial Model Michael Busch ∗ and Jeff Moehlis Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA. Received 15 September 2009; Accepted (in revised version) 6 January 2010 Available online 8 March 2010 Abstract. 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. AMS subject classifications: 37N25, 37G10 Key words: Territorial behavior, Voronoi tessellations, bifurcation, center manifold reduction. 1. Introduction A territory is a geographical area that an individual animal consistently defends against other individuals from its own species, typically in an attempt to maximize its reproductive opportunities and/or to secure food resources for itself and its young [11]. Territoriality is common across nearly all major groups of organisms on the planet. While higher animals like vertebrates exhibit the most obvious territorial boundaries, lower animals like inver- tebrates, plants, fungi and possibly even bacteria are known to aggressively defend space through behaviors and chemicals. The recent paper [10] studied equilibrium configurations for a model for territorial behavior based on Voronoi tessellations, which captures interactions between agents in a simple way [9]; also see [3, 6, 7]. For this model, at a given time and for each agent, one calculates the set of points in the domain of interest which are closer to that agent than to any other. Such a partition of the domain is called a Voronoi tessellation, and the set of ∗ Corresponding author. Email addresses: (M. Busch), (J. Moehlis) http://www.global-sci.org/nmtma 143 c 2010 Global-Science Press