A concrete example with three limit cycles in Zeeman’s class 29 for three dimensional Lotka–Volterra competitive systems

• Published in: Mathematical biosciences Vol. 308; pp. 38 - 41
• Main Author: Murakami, Kouichi
• Format: Journal Article
• Language: English
• Published: United States Elsevier Inc 01.02.2019; Elsevier Science Ltd
• Subjects: 3D Lotka–Volterra competitive systems; Competitive advantage; Economic models; Ecosystem; Hopf bifurcation; Mathematical models; Models, Biological; Multiple limit cycles; Perturbation; Hopf bifurcation; Multiple limit cycles; 3D Lotka–Volterra competitive systems
• ISSN: 0025-5564; 1879-3134
• DOI: 10.1016/j.mbs.2018.12.006

•A concrete example with three limit cycles for 3D Lotka–Volterra competitive systems is presented.•The critical parameter values such that the interior equilibrium is an exact unstable weak focus are given explicitly.•The phase portrait that we can see three limit cycles is drawn by numerical simulation. The number of limit cycles for three dimensional Lotka–Volterra competitive systems is an open problem. Recently, we have presented a concrete example with three limit cycles in Zeeman’s class 27 [6]. In this paper, we present a concrete example with three limit cycles which belongs to Zeeman’s class 29. We explicitly give the critical parameter values such that the interior equilibrium is an exact unstable weak focus of order two. Also we verify that the system is permanent. This implies that there can exist three limit cycles around the interior equilibrium under suitable perturbations. We actually generate multiple limit cycles, and confirm them by numerical simulation. In addition, we present some other examples with three limit cycles in Zeeman’s class 27.