Quasimonotone random and stochastic functional differential equations with applications.

In this paper, we study monotone properties of random and stochastic functional differential equations and their global dynamics. First, we show that random functional differential equations (RFDEs) generate the random dynamical system (RDS) if and only if all the solutions are globally defined, and establish the comparison theorem for RFDEs and the random Riesz representation theorem. These three results lead to the Borel measurability of coefficient functions in the Riesz representation of variational equations for quasimonotone RFDEs, which paves the way following the Smith line to establish eventual strong monotonicity for the RDS under cooperative and irreducible conditions. Then strong comparison principles, strong sublinearity theorems and the existence of random attractors for RFDEs are proved. Finally, criteria are presented for the existence of a unique random equilibrium and its global stability in the universe of all the tempered random closed sets of the positive cone. Applications to typical random or stochastic delay models in monotone dynamical systems, such as biochemical control circuits, cyclic gene models and Hopfield-type neural networks, are given. [ABSTRACT FROM AUTHOR]