Systems of differential equations which are competitive or cooperative: III. Competing species

Persistent trojectories of an n-dimensional system are studied under the assumptions that the system is competitive and dissipativewith irreducible community matrices. The main result is that there is acanonically defined countable (generically finite) family of disjoint invariant open (n - 1)-cells which attract all non-convergent persistenttrajectories. These cells are Lipschiiz submanifolds and are transverseto positive rays. In dimension 3 this implies that an omega-limit set of a persistent orbit is either an equilibrium, a cycle bounding an invariant disc, or a one-dimensional continuum having a non-trivial first Cech cohomology group and containing an equilibrium. Thus the existence of a persistenttrajectory in the three-dimensional case implies the existence of a positive equilibrium. In any dimension it is shown that if the community matrices are strictly negative then there is a closed invariant (n - 1) cell which attracts every persistent trajectory. This shows that a seemingly special construction by Smale of certain competitive systems is in fact close to the general case.