Fractal Dimension of Turing Instability in the Fractional Lengyel–Epstein Model

We investigate pattern formation driven by Turing instability in the Lengyel–Epstein model incorporating the fractional Laplacian operator. This approach simulates anomalous superdiffusion, where particle clouds spread more rapidly than predicted by the classical diffusion models. Such phase phenomena are often observed in viruses or biochemical substances that require unusually rapid diffusion. Our study allows for the estimation of superdiffusion effects in biochemistry or thermodynamics by analyzing temporal changes in fractional dimensions that exhibit abnormal patterns. Specifically, we employ the fractal dimension to quantify the complexity of these patterns using the box-counting method. Since this method is based on examining neighboring patterns, that is self-similarity, we observe that the formation of patterns varies with the bifurcations induced by Turing instability. Interestingly, despite these instabilities and variations, the fractal dimension stabilizes to a certain value once the patterns are formed. We also investigate how fractional diffusion affects the fractal dimension. Numerical experiments were performed with various initial conditions, including random values and circles centered with a fixed distance. The fractal dimension was calculated for Turing instability of the Lengyel–Epstein model, and is analyzed for both classical and fractional diffusion following a Lévy process. These tests revealed the formation of different patterns under the Turing instability by varying parameters, including a time variable t, and mass-conserving the chemical concentrations of the activator iodide u and the inhibitor chlorite v. Then, we analyzed these results in terms of the fractal dimension of the patterns. Furthermore, we analyze the consistency with the mesh refinement numerically and examine parameter sensitivity. Our numerical simulation of the fractional-Lengyel–Epstein model examined the relationship between the fractal dimension and the parameters b, c, and the fractional order s, while holding the scaling parameter a as a constant. In general, in regions of instability, higher values of parameter b are associated with lower fractal dimensions, while higher values of the parameter c lead to an increase in fractal dimension. In addition, as the fractional order s increases, the fractal dimension tends to decrease. Although specific parameter values around the bifurcation point influence the fractal dimension, the overall evolutionary trend follows the aforementioned patterns.