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 (Formula presented.). However, 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 super diffusion the patterns exhibit multiscale structures. The smaller the diffusion powers, the larger the unstable wave numbers, and the smaller the pattern scales.


Mathematics and Statistics

Second Department

Business and Information Technology


National Science Foundation, Grant DMS‐1620465

Keywords and Phrases

anomalous diffusion; fractional Laplacian; pattern formation; Schnakenberg equations; Turing instability

International Standard Serial Number (ISSN)

1098-2426; 0749-159X

Document Type

Article - Journal

Document Version

Final Version

File Type





© 2023 Wiley, All rights reserved.

Publication Date

01 Nov 2022