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Novel patterns in the space variable fractional order Gray–Scott model

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Abstract

In this paper, we apply Fourier spectral method to simulate the process of pattern formations of the space variable fractional order Gray–Scott model. This spatially extended reaction–diffusion model is obtained from the standard Gray–Scott model by using the Riesz variable fractional derivative in space. To validate the high efficiency and low computational complexity of the present numerical method, some numerical examples are provided as well as the numerical results are compared with those obtained by other numerical methods. The effects of order with different forms on pattern formation are discussed including a trigonometric function and a polynomial function. Furthermore, the temporal evolution of patterns in the space variable fractional order three-dimensional Gray–Scott model is given. Some interesting process of pattern formations are observed in numerical study which are different from any that have been previously reported. The Fourier spectral method is applicable for the Gray–Scott model with space variable fractional order, and it will be applied to some other space variable fractional order reaction–diffusion models in the future.

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All data generated or analyzed during this study are included in this paper.

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Funding

The authors thank the reviewers and editors for their valuable suggestions, which greatly improved the quality of the paper. This work is supported by the Fundamental Research Funds for the Central Universities (2024JBZX003), by the National Natural Science Foundation of China under Grant No. 12275017, by the State Key Laboratory of Integrated Services Networks (Contract No. ISN25-16), **dian University and by the Bei**g Laboratory of National Economic Security Early-warning Engineering, Bei**g Jiaotong University.

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  1. Department of Mathematics, Bei**g Jiaotong University, Bei**g, 100044, China

    Che Han & **ng Lü

  2. State Key Laboratory of Integrated Services Networks, **dian University, **’an, 710071, China

    **ng Lü

  3. Bei**g Laboratory of National Economic Security Early-warning Engineering, Bei**g Jiaotong University, Bei**g, 100044, China

    **ng Lü

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  1. Che Han
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Correspondence to **ng Lü.

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Han, C., Lü, X. Novel patterns in the space variable fractional order Gray–Scott model. Nonlinear Dyn (2024). https://doi.org/10.1007/s11071-024-09857-5

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