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The Center Conditions and Hopf Cyclicity for a 3DLotka-Volterra System
  
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KeyWord:3D Lotka-Volterra system; Hopf bifurcation; Center problem; Singular point quantities
Author NameAffiliation
Qinlong Wang School of Computing Science and Mathematics, Guangxi Key Laborato-ry of Trusted Software, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China 
Jingping Lu School of Computing Science and Mathematics, Guangxi Key Laborato-ry of Trusted Software, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China 
Wentao Huang School of Mathematics and Statistics, Guangxi Normal University, Guilin, Guangxi 541004, China 
Bo Sang Liaocheng University, School of Mathematical Sciences, Liaocheng, Shandong 252059, China 
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Abstract:
      The main objective of this paper is not only to find necessary and The main objective of this paper is not only to find necessary and sufficient conditions for the existence of a center on a local center manifold for a three dimensional Lotka-Volterra system, but also to determine the maximum number of limit cycles that can bifurcate from the positive equilibrium as a fine focus. Firstly, the singular point quantities are computed and simplified to obtain necessary conditions for local integrability, and Darboux method is applied to show the sufficiency. Then, the Hopf bifurcation on the center manifold is investigated, from this, the conclusion of at most five small limit cycles generated in the vicinity of the equilibrium is obtained. To the best of our knowledge, this is the first case with five possible limit cycles for the cyclicity of 3D Lotka-Volterra systems.