| Summary: | 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.
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