Skip to main content
eScholarship
Open Access Publications from the University of California

UC Berkeley

UC Berkeley Previously Published Works bannerUC Berkeley

Systems of differential equations which arecompetitive or cooperative:III. Competingspecies

Abstract

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.

Many UC-authored scholarly publications are freely available on this site because of the UC's open access policies. Let us know how this access is important for you.

Main Content
For improved accessibility of PDF content, download the file to your device.
Current View