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SYSTEMS OF DIFFERENTIAL EQUATIONSTHAT ARE COMPETITIVE OR COOPERATIVE II:CONVERGENCE ALMOST EVERYWHERE

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

 A vector field in n-space determines a competitive (or cooperative) system of differentialequations provided all of the off-diagonal terms of its Jacobian matrix are nonpositive (or nonnegative). Themain results in this article are the following. A cooperative system cannot have nonconstant attractingperiodic solutions. In a cooperative system whose Jacobian matrices are irreducible the forward orbitconverges for almost every point having compact forward orbit closure. In a cooperative system in 2dimensions, every solution is eventually monotone. Applications are made to generalizations of positivefeedback loops.

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