SYSTEMS OF DIFFERENTIAL EQUATIONSTHAT ARE COMPETITIVE OR COOPERATIVE II:CONVERGENCE ALMOST EVERYWHERE*MORRIS W. HIRSCH
- Authors
- Publication Date
- May 01, 1885
- Source
- eScholarship - University of California
- Keywords
- License
- Unknown
- External links
Abstract
A vector field in n-space determines a competitive (or cooperative) system of differential equations provided all of the off-diagonal terms of its Jacobian matrix are nonpositive (or nonnegative). The main results in this article are the following. A cooperative system cannot have nonconstant attracting periodic solutions. In a cooperative system whose Jacobian matrices are irreducible the forward orbit converges for almost every point having compact forward orbit closure. In a cooperative system in 2 dimensions, every solution is eventually monotone. Applications are made to generalizations of positive feedback loops.
