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

Abstract

We study the global dynamics of the classic May-Leonard model in \(\mathbb{R}^3\). Such model depends on two real parameters and its global dynamics is known when the system is completely integrable. Using the Poincare compactification on \(\mathbb R^3\) we obtain the global dynamics of the classical May-Leonard differential system in \(\mathbb{R}^3\) when \(\beta =-1-\alpha\). 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 Poincar\'e ball, that is, the compactification of \(\mathbb R^3\) in the sphere \(\mathbb{S}^2\) at infinity. We also describe the \(\omega\)-limit and \(\alpha\)-limit of each of the orbits. For some values of the parameter \(\alpha\) 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 \(\omega\)-limit. The same holds for the sixth and eighth octants. For more information see https://ejde.math.txstate.edu/Volumes/2020/55/abstr.html