Existence of Two Limit Cycles in Zeeman’s Class 30 for 3D Lotka-Volterra Competitive System  

作　　者：Yaoqi Li 

机构地区：[1]Mathematics and Science College,Shanghai Normal University,Shanghai,200234,China

出　　处：《Communications on Applied Mathematics and Computation》2023年第4期1584-1590,共7页应用数学与计算数学学报（英文）

基　　金：the National Natural Science Foundation of China(NSFC)under Grant No.12171321.

摘　　要：Gyllenberg and Yan(Discrete Contin Dyn Syst Ser B 11(2):347–352,2009)presented a system in Zeeman’s class 30 of 3-dimensional Lotka-Volterra(3D LV)competitive systems to admit at least two limit cycles,one of which is generated by the Hopf bifurcation and the other is obtained by the Poincaré-Bendixson theorem.Yu et al.(J Math Anal Appl 436:521–555,2016,Sect.3.4)recalculated the first Liapunov coefficient of Gyllenberg and Yan’s system to be positive,rather than negative as in Gyllenberg and Yan(2009),and pointed out that the Poincaré-Bendixson theorem is not applicable for that system.Jiang et al.(J Differ Equ 284:183–218,2021,p.213)proposed an open question:“whether Zeeman’s class 30 can be rigorously proved to admit at least two limit cycles by the Hopf theorem and the Poincaré-Bendixson theorem?”This paper provides four systems in Zeeman’s class 30 to admit at least two limit cycles by the Hopf theorem and the Poincaré-Bendixson theorem and gives an answer to the above question.

关 键 词：3-dimensional Lotka-Volterra(3D LV)competitive system Zeeman’s class 30 Fine focus Hopf bifurcation Poincaré-Bendixson theorem Limit cycle 

分 类 号：O17[理学—数学]