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Four small limit cycles around a Hopf singular point in 3-dimensional competitive Lotka-Volterra systems. (English) Zbl 1334.34072
Subject of the paper are three-dimensional competitive Lotka-Volterra systems and the possible number of limit cycles around the unique equilibrium in the positive octant. As shown earlier, among the 33 stable equivalence classes of Zeeman only six may have limit cycles, namely at least two. The authors present a detailed review of the relevant literature and also point out mistakes in some papers on the subject. Then, as the main result of this paper, they prove the existence of four small-amplitude limit cycles for two examples of each of the two Zeeman classes 27 and 26. Moreover, they prove two conjectures established by Gyllenberg and Yan, referring to the case where a heteroclinic cycle exists on the boundary of the carrying simplex. The proofs use center manifold theory to reduce the system to a two-dimensional one. The calculation of center manifolds, normal forms, and focal values up to order four, requires extensive use of computer algebra systems such as various Maple programs.

##### MSC:
 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 34C23 Bifurcation theory for ordinary differential equations 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 92D25 Population dynamics (general)
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