Minimal Models for Pattern Formation in Stochastic Reaction-Diffusion Systems

Abstract

Complex patterns arise in the world around us on a vast range of scales. In biological examples, at the tissue, organism, and population levels, the formation of regular heterogeneity -- as one example: the striped, spotted, or otherwise patchy pigmentation distribution on an animal's skin -- is often an emergent property of processes on a smaller scale, where stochasticity is an important factor. In order to understand the driving forces behind these emergent patterns, it is thus necessary to construct models which can account directly for both small scale dynamics and their inherent randomness.

In this thesis we analyse a mesoscopic compartment-based framework, in which particles move in space according to stochastic position jump processes and interact with each other via prescribed schemes of reactions. The larger-scale behaviour of the system then manifests through the concentrations of particles in different spatial compartments. This framework demonstrates, and enables the analysis of, how emergent macro-scale phenomena are driven by, and thus directly tied to, the micro-scale particle dynamics. We show that, under the parsimonious assumptions of mass-action kinetics and isotropic diffusion, in the simplest systems that have the potential to produce patterns there are certain types of micro-scale reaction that are necessary to facilitate this. In particular, patterning requires a multi-particulate auto-catalytic reaction and a multi-particulate interspecific inhibitory reaction. Guided by this new understanding of necessary reactions, we then construct a classification of all `minimal' reaction schemes (i.e. those with the fewest number of reactions) that are sufficient to produce patterns. We analyse the patterns produced by such schemes using both linear and weakly nonlinear analysis in the regime of large numbers of particles, and we also begin to explore their relation to intrinsic system stochasticity.

The exploration of these minimal models furthers our understanding of the principal micro-scale interactions and associated parameter groups that determine the spatiotemporal characteristics of an emergent pattern, as well as its dynamical properties. Our exploration of the processes linking two distinct analytical scales can perhaps help to contextualise the potential origins of observed patterning processes and could (with further elaboration) even guide system designs where patterning is a desired outcome.

Date of Award	10 Dec 2025
Original language	English
Awarding Institution	University of Bath
Supervisor	Kit Yates (Supervisor) & Jonathan Dawes (Supervisor)