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

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, Germany
2. 

Department of Mechanics and Mathematics, Kharkov University, 4 Svobody Square, 310077 Kharkov, Ukraine

Received  August 2000 Revised  October 2000 Published  November 2000

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