Population explosion suppressed by noise: Stationary distributions and how to simulate them

TL;DR

This paper demonstrates that adding Gaussian white noise to different dynamical systems can lead to similar stationary distributions and discusses how to effectively simulate these systems, identifying an optimal noise level for stability.

Contribution

It reveals the unifying effect of noise on diverse systems' stationary distributions and provides practical methods for their numerical simulation.

Findings

Different systems share similar stationary distributions under noise

An optimal noise level maximizes the physical simulation interval

Numerical examples validate analytical results

Abstract

We show that two dynamical systems exhibiting very different deterministic behaviours possess very similar stationary distributions when stabilized by a multiplicative Gaussian white noise. We also discuss practical aspects of numerically simulating these systems. We show that there exists a noise level that is optimal in the sense that the interval during which discrete-time versions of the systems remain physical is maximized. Analytical results are illustrated by numerical examples.