Global dynamics of the May-Leonard system with a Darboux invariant

Abstract

We study the global dynamics of the classic May-Leonard model in ℝ3. Such model depends on two real parameters and its global dynamics is known when the system is completely integrable. Using the Poincaré compactification on ℝ3 we obtain the global dynamics of the classical May-Leonard differential system in ℝ3 when β = -1 - α. In this case, the system is non-integrable and it admits a Darboux invariant. We provide the global phase portrait in each octant and in the Pointcaré ball, that is, the compactification of ℝ3 in the sphere S2 at infinity. We also describe the ω-limit and α-limit of each of the orbits. For some values of the parameter α we find a separatrix cycle F formed by orbits connecting the finite singular points on the boundary of the first octant and every orbit on this octant has F as the ω-limit. The same holds for the sixth and eighth octants.