Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 55, pp. 1-19.

Title: Global dynamics of the May-Leonard system with a Darboux invariant
       
Authors: Regilene Oliveira (ICMC-Univ. de Sao Paulo, Brazil)
         Claudia Valls (Univ. de Lisboa, Lisboa, Portugal)

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.
 
Submitted March 1, 2019. Published June 03, 2020.
Math Subject Classifications: 37C15, 37C10.
Key Words: Lotka-Volterra systems; May-Leonard systems;
           Darboux invariant;  phase portraits; limit sets;
           Poincare compactification