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

Abstract

For three-dimensional competitiveLotka-Volterra systems, Zeeman identified 33 stable equivalenceclasses. Among these, only classes 26-31 may have limit cycles. Weconstruct two limit cycles without a heteroclinic cycle (classes 30and 31 in Zeeman's classification). Our construction together withHofbauer and So [9] and Lu and Luo [10] gives acomplete answer to Hofbauer's and So's problem [9] concerningtwo limit cycles for three-dimensional competitive Lotka-Volterrasystems.