一类食饵种群具有投放率的捕食系统的定性分析

Qualitative Analysis of a Kind of Predator-Prey System Exploited

关键词:  捕食系统  极限环  平衡点

Key words:  predator-prey system  limit circle  equilibrium

摘要:

讨论一类食饵种群被开发的两种群捕食系统
{_{(dy/dt)=y(x-1),}^{(dx/dt)=x(a₀+a₁-a₂x2-a₃y3)+s₀,}
(其中a₀>0,a₂>0,a₃>0,s₀>0,a₁不定号)、平衡点的行为以及系统的全局稳定性.用Bendixson环域定理证明极限环的存在性,由张芷芬唯一性定理证明极限环的唯一性,同时对相应结论的生态学意义给予说明.

Abstract:

The system
{_{(dy/dt)=y(x-1),}^{(dx/dt)=x(a₀+a₁-a₂x2-a₃y3)+s₀,}
(where a₀>0,a₂>0,a₃>0,s₀>0,a₁ can be positive or negative),the behaviors of the equilibria and the global stability of the system are studied. The existence of the limit circle is proved by Bendixson Theroem. The uniqueness of the limit circle around the neighborhood of the positive equilibrium exploited by Uniqueness Theorem of Zhang Zifen. The significance of the conclusions are interpreted.