Stochastic pattern formation and spontaneous polarisation: the linear noise approximation and beyond

Abstract

We review the mathematical formalism underlying the modelling of stochasticity in biological systems. Beginning with a description of the system in terms of its basic constituents, we derive the mesoscopic equations governing the dynamics which generalise the more familiar macroscopic equations. We apply this formalism to the analysis of two specific noise-induced phenomena observed in biologically-inspired models. In the first example, we show how the stochastic amplification of a Turing instability gives rise to spatial and temporal patterns which may be understood within the linear noise approximation. The second example concerns the spontaneous emergence of cell polarity, where we make analytic progress by exploiting a separation of time-scales.

arXiv comment27 pages, 6 figures

References in corpus (8)

• Stochastic Turing patterns in the Brusselator model
• Robust ecological pattern formation induced by demographic noise
• Fluctuation-driven Turing patterns
• Quasi-cycles in a spatial predator-prey model
• A jump-growth model for predator-prey dynamics: derivation and application to marine ecosystems
• Stochastic waves in a Brusselator model with nonlocal interaction
• Noise-induced metastability in biochemical networks
• Spontaneous genetic clustering in populations of competing organisms