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.

Keywords

• lotka-volterra systems
• may-leonard systems
• darboux invariant
• phase portraits
• limit sets
• poincare compactification