Electron. J. Differential Equations, Vol. 2020 (2020), No. 55, pp. 1-19.

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

Regilene Oliveira, Claudia Valls

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 Poincare 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 3, 2020.
Math Subject Classifications: 37C15, 37C10.
Key Words: Lotka-Volterra systems; May-Leonard systems; Darboux invariant; phase portraits; limit sets; Poincare compactification.

Show me the PDF file (429 KB), TEX file for this article.

Regilene Oliveira
Departamento de Matemática
ICMC-Universidade de São Paulo
Avenida Trabalhador São-carlense, 400 - 13566-590
São Carlos, SP, Brazil
email: regilene@icmc.usp.br
  Claudia Valls
Departamento de Matemática
Instituto Superior Técnico
Universidade de Lisboa
1049-001 Lisboa, Portugal
email: cvalls@math.tecnico.ulisboa.pt

Return to the EJDE web page