Limit cycles for the competitive three dimensional Lotka-Volterra system

In the first part of this paper, it is proved that the number of limit cycl es of the competitive three-dimensional Lotka-Volterra system in R-+(3) is finite if this system has not ally heteroclinic polycycles in R-+(3). In th e second part of this paper, a 3D competitive Lotka-Volterra system with tw o small parameters is discussed. This system always has a heteroclinic poly cycle with three saddles. It is proved that there exists one parameter rang e in which the system is persistence and has at least two limit cycles, and there exists other parameter ranges in which the system is not persistence and has at least one limit cycle. (C) 2000 Academic Press.