Qualitative Analysis of a Class of Competitive and Cooperative Systems

Hirsch[1, 2] studied the limiting behavior of solutions of competitive or cooperative systems, and showed that if L is an .-limit set of a three-dimensional cooperative system, which contains no equilibrium, then L is a nonattracting closed orbit. Smith considered a three-dimensional irreducible competitive system and showed that an co-limit set containing no equilibrium must be a closed orbit which has a simple Floquet multiplier . < 1, and may be attracting. In this paper we carry out the qualitative analysis of a class of competitive and cooperative systems, and a generalization of the result of Levine is given. The stability problem of closed orbits raised in [5] and [6] is resolved.