Exploring fractional dynamics in the fractional-in-time Gierer-Meinhardt reaction-diffusion model with periodic boundary condition

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

This paper presents a new numerical method for simulating the dynamic behavior of the fractional-in-time Gierer-Meinhardt reaction-diffusion model with periodic boundary conditions. A recursive algorithm for binomial coefficients is introduced, avoiding numerical instabilities associated with Gamma functions. High-precision polynomial expansions and the short-memory principle are employed to enhance efficiency and accuracy. Numerical simulations reveal diverse pattern formation.

Keywords

fractional-in-time Gierer-Meinhardt model / pattern formation / numerical method / short-memory principle

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] K. B. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , Academic Press: New York, NY, USA, 1974. https://doi.org/10.1016/s0076-5392(09)x6012-1

[2] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations , John Wiley & Sons: New York, NY, USA, 1993.

[3] I. Podlubny, Fractional Differential Equations , Academic Press: San Diego, CA, USA, 1999. https://doi.org/10.1016/s0076-5392(99)x8001-5

[4] I. Petráš, Fractional-Order Nonlinear Systems, Modeling, Analysis and Simulation , Springer: Heidelberg, Germany, 2011. https://doi.org/10.1007/978-3-642-18101-6

[5] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , Springer-Verlag, 2010. https://doi.org/10.1007/978-3-642-14574-2

[6] C. P. Li, F. H. Zeng, Numerical Methods for Fractional Calculus , CRC Press, Boca Raton, 2015. https://doi.org/10.1201/b18503

[7] C. P. Li, M. Cai, Theory and Numerical Approximations of Fractional Integrals and Derivatives , Society for Industrial and Applied Mathematics, Philadelphia, 2020. https://doi.org/10.1137/1.9781611975888

[8] M. B. Almatrafi, Abundant traveling wave and numerical solutions for Novikov-Veselov system with their stability and accuracy, Appl. Anal., 102(2023), 2389-2402. https://doi.org/10.1080/00036811.2022.2027381

[9] H. L. Zhang, Z. Y. Li, X. Y Li . Numerical simulation of pattern and chaos dynamic behaviour in the fractional-order-in-time Lengyel-Epstein reaction-diffusion system, Int. J. Comput. Math., https://doi.org/10.1080/00207160.2025.2612196

[10] X. H. Wang, H. L. Zhang, Y. L. Wang, Z. Y. Li, Dynamic properties and numerical simulations of the fractional Hastings-Powell model with the Grunwald-Letnikov differential derivative, Int. J. Bifurcat. Chaos, 35(2025), 1250145. https://doi.org/10.1142/S0218127425501457

[11] X. L. Gao, Z. Y. Li, Y. L. Wang, Chaotic dynamic behavior of a fractional-order financial system with constant inelastic demand, Int. J. Bifurcat. Chaos, 34(2024), 2450111. https://doi.org/10.1142/S0218127424501116

[12] H. Che, Y. L. Wang, Z. Y. Li, Novel patterns in a class of fractional reaction-diffusion models with the Riesz fractional derivative, Math. Comput. Simul., 202(2022), 149-163. https://doi.org/10.1016/j.matcom.2022.05.037

[13] C. Han, Y. L. Wang, Z. Y. Li, A high-precision numerical approach to solving space fractional Gray-Scott model, Appl. Math. Lett., 125(2022), 107759. https://doi.org/10.1016/j.aml.2021.107759

[14] C. Han, Y. L. Wang, Z. Y. Li, Numerical solutions of space fractional variable-coefficient KdV-modified KdV equation by Fourier spectral method, Fractals, 29(2021), 2150246. https://doi.org/10.1142/S0218348X21502467

[15] L. Zhu, T. Zheng, Pattern dynamics analysis and application of West Nile virus spatiotemporal models based on higher-order network topology, Bull. Math. Biol., 87(2025), 121. https://doi.org/10.1007/s11538-025-01501-6

[16] L. Zhu, Y. Ding, S. Shen, Green behavior propagation analysis based on statistical theory and intelligent algorithm in data-driven environment, Math. Biosci., 379(2025), 109340. https://doi.org/10.1016/j.mbs.2024.109340

[17] T. Yang, L. Zhu, S. Shen, L. He, Pattern dynamics analysis and parameter identification of spatiotemporal infectious disease models on complex networks, Math. Biosci., 387(2025), 109502. https://doi.org/10.1016/j.mbs.2025.109502

[18] H. L. Zhang, Y. L. Wang, J. X. Bi, S. H. Bao, Novel pattern dynamics in a vegetation-water reaction-diffusion model, Math. Comput. Simul., 241(2026), 97-116. https://doi.org/10.1016/j.matcom.2025.09.020

[19] H. Sha, L. Zhu, Dynamic analysis of pattern and optimal control research of rumor propagation model on different networks, Inf. Process. Manage., 62(2025), 104016. https://doi.org/10.1016/j.ipm.2024.104016

[20] J. Shi, L. Zhu, Turing pattern theory on homogeneous and heterogeneous higher-order temporal network system, J. Math. Phys., 66(2025), 042706. https://doi.org/10.1063/5.0211728

[21] X. L. Gao, H. L. Zhang, Y. L. Wang, Z. Y. Li, Research on pattern dynamics behavior of a fractional vegetation-water model in arid flat environment, Fractal Fract., 8(2024), 264. https://doi.org/10.3390/fractalfract8050264

[22] B. Li, L. Zhu, Turing instability analysis of a reaction-diffusion system for rumor propagation in continuous space and complex networks, Inf. Process. Manage., 61(2024), 103621. https://doi.org/10.1016/j.ipm.2023.103621

[23] J. Y. Yang, Y. L. Wang, Z. Y. Li, Exploring dynamics and pattern formation of a fractional-order three-variable Oregonator model, Netw. Heterog. Media, 20(2025), 1201-1229. https://doi.org/10.3934/nhm.2025052

[24] J. Ning, Y. L. Wang, Fourier spectral method for solving fractional-in-space variable coefficient KdV-Burgers equation, Indian J. Phys., 98(2024), 1727-1744. https://doi.org/10.1007/s12648-023-02934-2

[25] X. L. Gao, H. L. Zhang, X. Y. Li, Research on pattern dynamics of a class of predator-prey model with interval biological coefficients for capture, AIMS Math., 9(2024), 18506-18527. https://doi.org/10.3934/math.2024901

[26] S. Zhang, H. L. Zhang, Y. L. Wang, Z. Y. Li, Dynamic properties and numerical simulations of a fractional phytoplankton-zooplankton ecological model, Netw. Heterog. Media, 20(2025), 648-669. https://doi.org/10.3934/nhm.2025028

[27] M. McCourt, N. Dovidio, M. Gilbert, Spectral methods for resolving spike dynamics in the Geirer-Meinhardt model, Commun. Comput. Phys., 3(2008), 659-678.

[28] G. Q. Sun, C. H. Wang, Z. Y. Wu, Pattern dynamics of a Gierer-Meinhardt model with spatial effects, Nonlinear Dyn., 88(2017), 1385-1396. https://doi.org/10.1007/s11071-016-3317-9

[29] G. Karali, T. Suzuki, Y. Yamada, Global-in-time behavior of the solution to a Gierer-Meinhardt system, Discrete Contin. Dyn. Syst., 33(2013), 2885-2900. https://doi.org/10.3934/dcds.2013.33.2885

[30] S. Chen, Y. Salmaniw, R. Xu, Global existence for a singular Gierer-Meinhardt system, J. Differ. Equations, 262(2017), 2940-2960. https://doi.org/10.1016/j.jde.2016.11.022

[31] J. Wang, Y. Li, X. Hou, Supercritical Hopf bifurcation and Turing patterns for an activator and inhibitor model with different sources, Adv. Differ. Equ., 2018(2018), 241. https://doi.org/10.1186/s13662-018-1697-5

[32] Y. Song, R. Yang, G. Sun, Pattern dynamics in a Gierer-Meinhardt model with a saturating term, Appl. Math. Model., 46(2017), 476-491. https://doi.org/10.1016/j.apm.2017.01.081

[33] L. Guo, X. Shi, J. Cao, Turing patterns of Gierer-Meinhardt model on complex networks, Nonlinear Dyn., 105(2021), 899-909. https://doi.org/10.1007/s11071-021-06618-6

[34] X. P. Yan, Y. J. Ding, C. H. Zhang, Dynamics analysis in a Gierer-Meinhardt reaction-diffusion model with homogeneous Neumann boundary condition, Int. J. Bifurcat. Chaos, 29(2019), 1930025. https://doi.org/10.1142/S0218127419300258

[35] Z. Qiao, Numerical investigations of the dynamical behaviors and instabilities for the Gierer-Meinhardt system, Commun. Comput. Phys., 3(2008), 406-426.

[36] Y. S. Choi, P. J. McKenna, A singular Gierer-Meinhardt system of elliptic equations: The classical case, Nonlinear Anal., 55(2003), 521-541. https://doi.org/10.1016/j.na.2003.07.003

[37] R. Yang, X. Q. Yu, Turing-Hopf bifurcation in diffusive Gierer-Meinhardt model, Int. J. Bifurcat. Chaos, 32(2022), 2250046. https://doi.org/10.1142/S0218127422500468

[38] D. Y. Xue, L. Bai, Numerical algorithms for Caputo fractional-order differential equations, Int. J. Control, 90(2016), 1201-1211. https://doi.org/10.1080/00207179.2016.1158419