A 3D competitive Lotka–Volterra system with three limit cycles: A falsification of a conjecture by Hofbauer and So

For three-dimensional competitive Lotka–Volterra systems, Zeeman [M.L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka–Volterra systems, Dyn. Stab. Syst. 8 (1993) 189–217] identified 33 stable nullcline equivalence classes. Among these, only classes 26–31 may have limit cycles. Hofbauer and So [J. Hofbauer, J.W.-H. So, Multiple limit cycles for three dimensional Lotka–Volterra equations, Appl. Math. Lett. 7 (1994) 65–70] conjectured that the number of limit cycles is at most two for these systems. In this paper, we construct three limit cycles for class 29 without a heteroclinic polycycle in Zeeman’s classification.