Integrability and global dynamics of the May-Leonard model
Blé, Gamaliel (UJAT(México). División Académica de Ciencias Básicas)
Castellanos, Víctor (UJAT(México). División Académica de Ciencias Básicas)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Quilantán, Ingrid (UJAT(México). División Académica de Ciencias Básicas)

Date:	2013
Abstract:	We study when the celebrated May-Leonard model in R3, describing the competition between three species and depending on two positive parameters a and b, is completely integrable; i. e. when a+b = 2 or a = b. For these values of the parameters we shall describe its global dynamics in the compactification of the positive octant, i. e. adding its infinity. If a + b = 2 and a 6= 1 (otherwise the dynamics is very easy) the global dynamics was partially known, and roughly speaking there are invariant topological half-cones by the flow of the system. These half-cones have vertex at the origin of coordinates and surround the bisectrix x = y = z, and foliate the positive octant. The orbits of each half-cone are attracted to a unique periodic orbit of the half-cone, which lives on the plane x + y + z = 1. If b = a 6= 1 then we consider two cases. First, if 0 < a < 1 then the unique positive equilibrium point attracts all the orbits of the interior of the positive octant. If a > 1 then there are three equilibria in the boundary of the positive octant, which attract almost all the orbits of the interior of the octant, we describe completely their bassins of attractions.
Grants:	Ministerio de Ciencia e Innovación MTM2008-03437 Agència de Gestió d'Ajuts Universitaris i de Recerca 2005/SGR-550
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	May-Leonard model ; Lotka-Volterra systems ; First integrals ; Global dynamics ; Poincaré compactification
Published in:	Nonlinear Analysis: Real World Applications, Vol. 14 (2013) , p. 280-293, ISSN 1468-1218

DOI: 10.1016/j.nonrwa.2012.06.004

Postprint 23 p, 558.3 KB

The record appears in these collections:
Research literature > UAB research groups literature > Research Centres and Groups (research output) > Experimental sciences > GSD (Dynamical systems)
Articles > Research articles
Articles > Published articles