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March  2009, 11(2): 347-352. doi: 10.3934/dcdsb.2009.11.347

On the number of limit cycles for three dimensional Lotka-Volterra systems

Mats Gyllenberg 1, and  Ping Yan 1,

1. 

Rolf Nevanlinna Institute, Department of Mathematics and Statistics, P.O. Box 68, FIN-00014 University of Helsinki, Finland, Finland

Received  October 2007 Revised  September 2008 Published  December 2008

For three-dimensional competitive Lotka-Volterra systems, Zeeman identified 33 stable equivalence classes. Among these, only classes 26-31 may have limit cycles. We construct two limit cycles without a heteroclinic cycle (classes 30 and 31 in Zeeman's classification). Our construction together with Hofbauer and So [9] and Lu and Luo [10] gives a complete answer to Hofbauer's and So's problem [9] concerning two limit cycles for three-dimensional competitive Lotka-Volterra systems.
Keywords: Hopf Bifurcation, Limit cycles, carrying simplex, Competitive, Lotka-Volterra system.
Mathematics Subject Classification: Primary: 34C12, 34C23; Secondary: 95D25.
Citation: Mats Gyllenberg, Ping Yan. On the number of limit cycles for three dimensional Lotka-Volterra systems. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 347-352. doi: 10.3934/dcdsb.2009.11.347
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