Four limit cycles for a three-dimensional competitive Lotka–Volterra system with a heteroclinic cycle

For three-dimensional competitive Lotka–Volterra systems, Zeeman [M.L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka–Volterra systems, Dynamics and Stability of Systems 8 (1993) 189–217] identified 33 stable equivalence classes. Among these, only classes 26–31 may have limit cycles. It is known that all these classes may possess two limit cycles and in classes 27 and 29 three limit cycles have been constructed. It has been conjectured that the maximum number of limit cycles is three. In this paper, we disprove the conjecture by constructing four limit cycles in the three-dimensional competitive Lotka–Volterra system with a heteroclinic cycle (class 27 in Zeeman’s classification). Furthermore, in the case of a heteroclinic cycle on the boundary of the carrying simplex of three-dimensional competitive Lotka–Volterra systems, we show that: (i) the conditions (a) there is a pair of purely imaginary eigenvalues at a positive equilibrium, (b) the first focal values vanishes, and (c′) the second focal values vanishes do not imply that the heteroclinic cycle is neutrally stable, and hence they do not imply that the interior equilibrium is a center; (ii) the conditions (a) there is a pair of purely imaginary eigenvalues at a positive equilibrium, (b) the first focal value vanishes, and (c) the heteroclinic cycle is neutrally stable do not imply the second focal value vanishes, and hence they do not imply that the interior equilibrium is a center. This refutes a conjecture by 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].