Cooperative random and stochastic differential equations

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January 2001, 7(1): 1-33. doi: 10.3934/dcds.2001.7.1

Cooperative random and stochastic differential equations

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

Received August 2000 Revised October 2000 Published November 2000

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

Mathematics Subject Classification: 37H10, 60H10, 34F05, 47B60, 93E1.

Citation: Ludwig Arnold, Igor Chueshov. Cooperative random and stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 1-33. doi: 10.3934/dcds.2001.7.1

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