Systems of Differential Equations that are Competitive or Cooperative II: Convergence Almost Everywhere

Morris W. Hirsch

Type: Article

Publication Date: 1985-05-01

Citations: 546

DOI: https://doi.org/10.1137/0516030

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.

Locations

eScholarship (California Digital Library) - View - PDF
SIAM Journal on Mathematical Analysis - View

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