Abstract: A prey-predator system with Holling’s Ⅰ functional response is discussed. When the response function is φ(x), the behaviour of equilibrium points, the boundedness of the solutions and 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.

Key words: boundedness, Liapunov function, limit cycle, globally asymptotic stability