Limit Cycles for the Competitive Three Dimensional Lotka–Volterra System

Abstract

In the first part of this paper, it is proved that the number of limit cycles of the competitive three-dimensional Lotka–Volterra system in R3+ is finite if this system has not any heteroclinic polycycles in R3+. In the second part of this paper, a 3D competitive Lotka–Volterra system with two small parameters is discussed. This system always has a heteroclinic polycycle with three saddles. It is proved that there exists one parameter range 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.