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

Abstract: 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.

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