Turing patterns on networks

TL;DR

This paper investigates Turing pattern formation on networks, revealing how diffusion rates influence instability, and demonstrates complex pattern dynamics through simulations and mean-field theory.

Contribution

It introduces a mean-field theory for nonlinear Turing patterns on networks and explores the effects of diffusion rates and network topology on pattern formation.

Findings

Turing instability occurs when inhibitor diffusion exceeds that of activator.

Final patterns can differ significantly from initial unstable modes.

Multistability and hysteresis are observed in pattern dynamics.

Abstract

Turing patterns formed by activator-inhibitor systems on networks are considered. The linear stability analysis shows that the Turing instability generally occurs when the inhibitor diffuses sufficiently faster than the activator. Numerical simulations, using a prey-predator model on a scale-free random network, demonstrate that the final, asymptotically reached Turing patterns can be largely different from the critical modes at the onset of instability, and multistability and hysteresis are typically observed. An approximate mean-field theory of nonlinear Turing patterns on the networks is constructed.