捕食者有常数投入的n+1次系统的极限环

The limit cycle of a kind of n+1-degreesystem which put into some predator

主要对一类n+1次系统dxdt=x(a0+a1x-a2x2+a3yn)+hdydt=y(b1x2-b2)(a0,a1,a2,b1,b2,h>0,a3为不定号,n≥1,n∈N)进行研究.利用微分方程定性理论,给出该系统平衡点的渐近性态,得到局部性质.找出闭轨线不存在性的充分条件,并给出极限环的存在性和唯一性的条件.

A kind of n＋1 degree system dxdt=x（a0＋a1x-a2x-a2x2＋a3yn）＋hdydt=y（b1x2-b2）（a0,a1,a2,b1,b2,h＆gt;0,a3≠0,n≥1 and n∈N） was mainly studied.By using qualitative theory of ordinary differential equation,the quality of the equilibrium point was discussed in this system.The sufficient conditions of nonexistence of the closed orbit were found and the conditions of existence of the unique and stable limit cycle were presented in the first quadrant of the system.