Four small limit cycles around a Hopf singular point in 3-dimensional competitive Lotka-Volterra systems

The 3-dimensional competitive Lotka-Volterra (LV) systems have been studied for more than two decades, and particular attention has been focused on bifurcation of limit cycles. For such a system, Zeeman (1993) identified 33 stable equivalence classes on a carrying simplex, among which only classes 26-31 may have limit cycles. It has been shown that all these 6 classes may possess two limit cycles, and the existence of three limit cycles was claimed in some of these classes. Recently, Gyllenberg and Yan (2009) studied the existence of four limit cycles, three of them are small-amplitude limit cycles due to Hopf bifurcation and one additional limit cycle, enclosing all the three small-amplitude limit cycles, is due to the existence of a heteroclinic cycle, and proposed a new conjecture including: (i) There exists a 3-d competitive LV system with at least 5 limit cycles. (ii) In the case of a heteroclinic cycle on the boundary of the carrying simplex of a 3-d competitive LV system, the vanishing of the first four focus values (the vanishing of the zero-order focus value means that there is a pair of purely imaginary eigenvalues at the positive equilibrium) does not imply that the heteroclinic cycle is neutrally stable, and hence it does not imply that the positive equilibrium is a center. (iii) In the case of a heteroclinic cycle on the boundary of the carrying simplex of a 3-d competitive LV system, the vanishing of the first three focus values and that the heteroclinic cycle is neutrally stable do not imply the vanishing of the third-order focus value, and hence they do not imply that the positive equilibrium is a center. In this paper, we will present two examples belonging to class 27 and another two examples belonging to class 26, which exhibit at least four small-amplitude limit cycles in the vicinity of the positive equilibrium due to Hopf bifurcations, and prove that the items (ii) and (iii) in the conjecture are true. Moreover, showing the existence of four small-amplitude limit cycles is a necessary step towards proving item (i) of the conjecture.