Intrinsic noise and discrete-time processes

TL;DR

This paper develops a formalism for modeling intrinsic noise in discrete-time processes using Markov chains, providing a Gaussian approximation for finite populations and applying it to the logistic map.

Contribution

It introduces a new Markov chain framework that converges to a deterministic map and offers an approximate Gaussian scheme for stochastic fluctuations in finite populations.

Findings

The formalism accurately models intrinsic noise in discrete systems.

The Gaussian approximation effectively describes stochastic fluctuations.

Application to the logistic map demonstrates the method's utility.

Abstract

A general formalism is developed to construct a Markov chain model that converges to a one-dimensional map in the infinite population limit. Stochastic fluctuations are therefore internal to the system and not externally specified. For finite populations an approximate Gaussian scheme is devised to describe the stochastic fluctuations in the non-chaotic regime. More generally, the stochastic dynamics can be captured using a stochastic difference equation, derived through an approximation to the Markov chain. The scheme is demonstrated using the logistic map as a case study.