有密度制约的HollingI类捕食系统的定性分析

The Qualitative Analysis of a Prey-predator System with Holling’s Ⅰ Functional Response

讨论一类被捕食者种群有密度制约的HollingI类捕食系统,当功能反应函数为φ(x)=cx0,x>x0cx,0≤x≤x0时,采用定性分析方法,研究了系统平衡点的性态、解的有界性和正平衡点的全局渐近稳定性,得到了极限环不存在的条件,并利用Poincare-Bendixson环域定理和构造Liapunov函数方法,得到了极限环存在的充分条件.

A prey-predator system with Holling＇ s Ⅰ functional response is discussed. When the response function is φ（x）={cx0,x〉x0 cx,0≤x≤x0, the behaviour of equilibrium points, the boundedness of the solutionsand the globally asymptotic stability of positive balance point were studied by means of the qualitative analysis method, conditions of nonexistence of the limit cycles were obtained. Utilizing the method of Liapunov function and the Poincare-Bendixson theorem and forming Liapunov function method, the sufficient conditions of the existence of the limit cycles were also obtained.