一类Kolmogorov捕食系统的定性分析

The qualitative analysis of a Kolmogrov's predator-prey system

研究一类Kolmogorov捕食系统:ddxt=x(a0-a1x+a2xn-1-a3xn+a4xφ(y)),ddyt=y(b1xn-b2),其中φ(0)=0,φ′(y)>ε>0,(y>0).首先运用等式gf((uu))′=Δlui→m0f(u+Δu)g(u+Δu)-gf((uu))Δu将张芷芬唯一性定理和微分不等式定理中需要的两个不等式联系起来,再配合运用环域定理、ΦИЛИППОВ变换及Dulac函数法得到了该系统存在唯一极限环和不存在极限环的充要条件,从而对其参数范围就其极限环存在性与不存在性讨论完全,推广了前人相关的结果.

The author studies the Kolmogorov＇ s predator -prey system x =x（a0-a1x＋a2x^n-1-a3x^n＋a4xφ（y））, y=y（b1x^n-b2）, where φ（0）=0,φ＇（y）〉ε〉0,（y〉0）. And obtains the surfficient and necessary condition for the existence and uniqueness of limit cycle surrounding the positive equilibrium point while the sufficient and necessary condition for nonexistence of limit cycle is also considered and derived, by using important equation （f（u）/g（u））＇=lim dx-0 f（u＋△u）/g（u＋△u）-f（u）/g（u）/△u to link two inequalities in Zhang＇ s theorem and differential inequality theorem and employing annular region theorem, ФИЛИППОВ transformation and Dulac function method.