Pattern selection in the Schnakenberg equations: From normal to anomalous diffusion

TL;DR

This paper investigates how anomalous diffusion influences pattern formation in the Schnakenberg equations, revealing that fractional diffusion leads to more chaotic, multiscale patterns with larger unstable wave numbers.

Contribution

It provides a comprehensive analysis of pattern formation under fractional diffusion, combining linear stability, weakly nonlinear analysis, and numerical simulations to reveal new behaviors.

Findings

Fractional diffusion widens the unstable wave number band.

Superdiffusion results in multiscale, chaotic patterns.

Smaller diffusion powers increase pattern complexity.

Abstract

Pattern formation in the classical and fractional Schnakenberg equations is studied to understand the nonlocal effects of anomalous diffusion. Starting with linear stability analysis, we find that if the activator and inhibitor have the same diffusion power, the Turing instability space depends only on the ratio of diffusion coefficients $\kappa_1/\kappa_2$. However, the smaller diffusive powers might introduce larger unstable wave numbers with wider band, implying that the patterns may be more chaotic in the fractional cases. We then apply a weakly nonlinear analysis to predict the parameter regimes for spot, stripe, and mixed patterns in the Turing space. Our numerical simulations confirm the analytical results and demonstrate the differences of normal and anomalous diffusion on pattern formation. We find that in the presence of superdiffusion the patterns exhibit multiscale structures. The smaller the diffusion powers, the larger the unstable wave numbers and the smaller the pattern scales.