Cooperative random and stochastic differential equations

Ludwig Arnold; Igor Chueshov

doi:10.3934/dcds.2001.7.1

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2001, Volume 7, Issue 1: 1-33. Doi: 10.3934/dcds.2001.7.1

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Cooperative random and stochastic differential equations

Ludwig Arnold^1, and
Igor Chueshov^2,

1.
Institut für Dynamische Systeme, Fachbereich 3, Universität, Postfach 33 04 40, 28334 Bremen
2.
Department of Mechanics and Mathematics, Kharkov University, 4 Svobody Square, 310077 Kharkov

Received: July 31, 2000

Revised: September 30, 2000

Published: December 2000

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Abstract

This is a systematic study of order-preserving (or monotone) random dynamical systems which are generated by cooperative random or stochastic differential equations. Our main results concern the long-term behavior of these systems, in particular the existence of equilibria and attractors and a limit set trichotomy theorem. Several applications (models of the control of the protein synthesis in a cell, of gonorrhea infection and of symbiotic interaction in a random environment) are treated in detail.

Keywords:

Cooperative differential equation,
order-preserving or monotone random dynamical system,
random equilibrium,
random attractor,
long-term behavior,
limit set trichotomy.

Mathematics Subject Classification: 37H10, 60H10, 34F05, 47B60, 93E15.

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