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

Systems of Differential Equations that are Competitive or Cooperative I: Limit Sets

  • Author(s): Hirsch, Morris W
  • et al.

Abstract. A vector field in n-space determines a competitive (or cooperative) system of differential equations provided all the off-diagonal terms of its Jacobian matrix are nonpositive (or nonnegative). The principal result is that limit sets of such systems cannot be more complicated than invariant sets of systems of one lower dimension. In fact orthogonal projection along any positive direction maps a limit set homeomorphically and equivariantly onto an invariant set of a Lipschitz vector field in a hyperplane. Limit sets are nowhere dense, unknotted and unlinked. In dimension 2 every trajectory is eventually monotone.In dimension 3 a compact limit set which does not contain an equilibrium is a closed orbit or a cylinder of closed orbits.

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
Current View