http://www.gsd.uab.cat ON THE GLOBAL FLOW OF A 3–DIMENSIONAL LOTKA–VOLTERRA SYSTEM JUSTINO ALAVEZ–RAM ´ IREZ 1 , GAMALIEL BL ´ E 1 , V ´ ICTOR CASTELLANOS 1 AND JAUME LLIBRE 2 Abstract. In the study of the black holes with Higgs field appears in a natural way the Lotka–Volterra differential system ˙ x = x(y - 1), ˙ y = y(1 + y - 2x 2 - z 2 ), ˙ z = zy, in R 3 . Here we provide the qualitative analysis of the flow of this system describing the α–limit set and the ω–limit set of all orbits of this system in the whole Poincar´ e ball, i.e. we identify R 3 with the interior of the unit ball of R 3 centered at the origin and we extend analytically this flow to its boundary, i.e. to the infinity. 1. Introduction and statement of the main results Breitenlohner et al. in their study of the black holes with Higgs field reduced the relevant terms to the following Lotka–Volterra polynomial dif- ferential system in R 3 : (1) ˙ x = x(y − 1), ˙ y = y(1 + y − 2x 2 − z 2 ), ˙ z = zy, see for more details page 441 of [4]. They, analyzing the local motion around the z–axis (i.e. x=y=0), which is formed by singular points, obtained infor- mation about the growing of the mass of a black hole with Higgs field. The Lotka–Volterra systems are the differential systems of the form ˙ x k = x k f k (x 1 ,...,x n ), for k =1,...,n. The name of such systems is due to the fact that Lotka and Volterra were the first in considering them in dimension 2 for studying problems coming from the ecology, see [12, 17]. Later on Kolmogorov in [11] generalizes these systems, and then some authors called them Kolmogorov systems. There are many natural phenomena that can be modeled by the Lotka–Volterra systems such as the time evolution of conflicting species in biology [14], chemical reactions [9], plasma physics [13], hydrodynamics [5], economics [16], etc. In this article we want to describe the α–limit set and the ω–limit set of all orbits of system (1) in R 3 . For a precise definition of the α–limit set and of the ω–limit set of an orbit, see for instance section 1.4 of [7]. Key words and phrases. α–limit, ω–limit, phase portrait, Lotka–Volterra system, black hole, Higgs field. 1 This is a preprint of: “On the global flow of a 3–dimensional Lotka–Volterra system”, Justino Alavez-Ram´ ırez, Gamaliel Bl´ e, V´ ıctor Castellanos, Jaume Llibre, Nonlinear Anal., vol. 75, 4114– 4125, 2012. DOI: [10.1016/j.na.2012.03.002]