Systems of Differential Equations Which Are Competitive or Cooperative: I. Limit Sets

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.