Minimal reaction schemes for Turing instabilities #

Fraser Waters, Kit Yates, Jonathan Dawes

12:50 Monday in 2Q49.

Part of the Dynamics of reaction-transport systems session.

Abstract #

Particle-based systems can form emergent diffusion-driven patterns on a much larger length scale than that of the reacting particles themselves. Under the assumption of mass-action kinetics for two interacting particle species, and by considering the infinite-particle limit, we deduce the catalogue of particle reactions that are necessary in order for the corresponding continuum models to exhibit macro-scale patterning instability via the Turing mechanism. We thus derive the list of `minimal’ reaction schemes which generate Turing patterns.

Our catalogue reveals that two-species Turing patterns in which the concentration peaks are spatially in-phase can result from sets of only three reactions, while, surprisingly, four reactions are required to generate patterns in which the concentration peaks are spatially in anti-phase. We establish the precise conditions under which Turing patterns form which thereby links particle-scale and continuum models directly, while also informing the design of sensible stochastic simulations for pattern formation.