Global dynamics of the May-Leonard system with a Darboux invariant.
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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 get the global dynamics of the classical May–Leonard differential system in R³ when β = −1 − α, a non-integrable case. In this case it admits a Darboux invariant. We provide the global phase portrait in each octant and in the Poincaré 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.