Qualitative Analysis Of A Kind Of Cubic Kolmogorov Differential System

In this paper,we study the cubic Kolmogorov differential system (*)whereandcan be positive or negative.Assume that.In this paper,we will discuss the existence,the non-existence and the uniqueness of limit cycle system(*).The methods employed are Poincare-Bendixsion theorem,method of Dulac function and Zhang zhifen's uniqueness theorem.Our main results are as follows:Theorem 1. When,there exists at least one limit cycle to the system (*) in the first quadrant;Theorem 2. When,there exsits at most one limit cycle to the system (*) in the first quadrant;Theorem 3. When, there exactly exists one limit cycle to the system (*) in the first quadrant. In the cases of and ,there has no closed orbit to the system (*) in the first quadrant. We find that there are some mistakes in paper [2].In particular,the conditions for the uniqueness of limit cycle in [2] can not be realized.All of these mistakes have been corrected in this paper.We give an example in the end of this paper to illustrate that our conditions on the uniqueness of limit cycle can be realized.