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

Persistent trajectories of the n-dimensional system x(over dot)(i) = x(i)N(i)(x(1),..., x(n)), x(i) >= 0, are studied under the assumptions that the system is competitive and dissipative with irreducible community matrices [partial derivative N-i/partial derivative x(j)]. The main result is that there is a canonically defined countable (generically finite) family of disjoint invariant open (n-1) cells which attract all non-convergent persistent trajectories. These cells are Lipschitz submanifolds and are transverse to 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 tech cohomology group and containing an equilibrium. Thus the existence of a persistent trajectory 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.