On the number of limit cycles for three dimensional Lotka-Volterra systems

For three-dimensional competitive Lotka-Volterra systems, Zee-man identified 33 stable equivalence classes. Among these, only classes 26-31 may have limit cycles. We construct two limit cycles without a heteroclinic cycle (classes 30 and 31 in Zeeman's classification). Our construction together with Hofbauer and So [9] and Lu and Luo [10] gives a complete answer to Hofbauer's and So's problem [9] concerning two limit cycles for three-dimensional competitive Lotka-Volterra systems.