Dimension reduction for stochastic dynamical systems forced onto a manifold by large drift: a constructive approach with examples from theoretical biology

TL;DR

This paper develops a constructive method for reducing the dimensionality of stochastic dynamical systems constrained near a manifold, providing explicit formulas and examples relevant to biological and ecological models.

Contribution

It introduces a general, explicit formula for dimension reduction in stochastic systems with slow-fast dynamics, applicable to systems with conserved quantities and non-Gaussian noise.

Findings

Derived explicit reduction formulas for small noise limits

Applied method to enzyme reactions and population models

Extended approach to infinite-dimensional and non-Gaussian systems

Abstract

Systems composed of large numbers of interacting agents often admit an effective coarse-grained description in terms of a multidimensional stochastic dynamical system, driven by small-amplitude intrinsic noise. In applications to biological, ecological, chemical and social dynamics it is common for these models to posses quantities that are approximately conserved on short timescales, in which case system trajectories are observed to remain close to some lower-dimensional subspace. Here, we derive explicit and general formulae for a reduced-dimension description of such processes that is exact in the limit of small noise and well-separated slow and fast dynamics. The Michaelis-Menten law of enzyme-catalyzed reactions, and the link between the Lotka-Voltera and Wright-Fisher processes are explored as a simple worked examples. Extensions of the method are presented for infinite dimensional systems and processes coupled to non-Gaussian noise sources.