推广的Volterra方程的定性分析

Qualitative Analysis of Extended Volterra's Equation

运用Ｌｉａｐｕｎｏｖ第二方法研究系统ｘ·＝ ｘ〔ａ（ｘ）－ ｙ〕ｙ·＝ ｄｙ（ｘ－ ｍ ）正平衡点的稳定性． 采用Ｐｏｉｎｃａｒｅ－ Ｂｅｎｄｉｘｏｎ 环域定理论证明了当ａ′（ｍ ）≥０ 时，针对ｍ 的不同数值，系统在Ｒ＋ ＝ ｛（ｘ，ｙ）：ｘ＞ ０，ｙ＞ ０｝内极限环的存在性稳定性，并结合几个具体方程进行了定性分析． 推广了文献［１］的结果．

In this paper,we utilize Liapunov's second method for follwoing system x·=x[a(x)-y] y·=dy[x-m]. We study the steadiness of positive balance points in this system.The main result is that limit rings of the system are existing and steady in R +=｛(x,y):x>0,y>0｝ when a′(m)≥0. We also carry on a qualitalive analysis of some conerate equations.