Stochastic dynamics on slow manifolds

TL;DR

This paper extends the concept of slow manifolds from deterministic to stochastic systems, demonstrating how trajectories conditioned on staying near the manifold can simplify models in ecology and epidemiology.

Contribution

It introduces a methodology for applying slow manifold theory to stochastic dynamical systems, enabling model reduction and analytical approximation in stochastic contexts.

Findings

Successful reduction of model dimensions in ecology and epidemiology models

High accuracy of analytical approximations for stochastic trajectories

Demonstration of the method's applicability to real-world stochastic systems

Abstract

The theory of slow manifolds is an important tool in the study of deterministic dynamical systems, giving a practical method by which to reduce the number of relevant degrees of freedom in a model, thereby often resulting in a considerable simplification. In this article we demonstrate how the same basic methodology may also be applied to stochastic dynamical systems, by examining the behaviour of trajectories conditioned on the event that they do not depart the slow manifold. We apply the method to two models: one from ecology and one from epidemiology, achieving a reduction in model dimension and illustrating the high quality of the analytical approximations.