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.
H. K. Khudhair et al., "Pattern Selection in the Schnakenberg Equations: from Normal to Anomalous Diffusion," Numerical Methods for Partial Differential Equations, vol. 38, no. 6, pp. 1843 - 1860, Wiley, Nov 2022.
The definitive version is available at https://doi.org/10.1002/num.22842
Mathematics and Statistics
Business and Information Technology
Keywords and Phrases
anomalous diffusion; fractional Laplacian; pattern formation; Schnakenberg equations; Turing instability
International Standard Serial Number (ISSN)
Article - Journal
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01 Nov 2022
National Science Foundation, Grant DMS‐1620465