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On the first Lyapunov coefficient formula of 3D Lotka-Volterra equations with applications to multiplicity of limit cycles. (On the first Liapunov coefficient formula of 3D Lotka-Volterra equations with applications to multiplicity of limit cycles.) (English) Zbl 1460.92169

Summary: This paper provides the first Lyapunov coefficient formula of 3D Lotka-Volterra equations. This formula gives applications to stability of positive equilibrium and to detecting sub/super criticality of Hopf bifurcation. For 3D competitive Lotka-Volterra equations, combining this formula with the Poincaré-Bendixson theorem, we obtain criteria on multiplicity of limit cycles among Zeeman’s classes 27-31, and present a series of examples to admit at least two limit cycles, which are rigorously proved by the first Lyapunov coefficient formula, rather than by symbolic computation using Maple. A new Hopf bifurcation that all \(2 \times 2\) principal minors of the community matrix are positive is found, and numerical simulation reveals its global limit cycle bifurcations are plenty.

MSC:

92D25 Population dynamics (general)
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems

Software:

Matlab; BifTools; Maple
PDFBibTeX XMLCite
Full Text: DOI

References:

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