Integrability and global dynamics of the May-Leonard model

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 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