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

Abstract

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.