# On a New Version of Grierer-Meinhardt Model Using Fractional Discrete Calculus

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## Abstract

As mathematical models of biological pattern generation, this study studies the dynamics of fractional discrete Gierer-Meinhardt reaction-diffusion system. After deriving the discrete non-integer fractional variant of the Gierer-Meinhardt system and establishing that the system has a unique equilibrium, we analyze the system’s local asymptotic behavior with both the presence and absence of the diffusion. The requirements for the steady-state solution’s global stability are found with the help of relevant approaches and the Lyapunov method. Throughout the study, two large biological models and simulations are employed to validate the usefulness of the considered approach.

## References

H. Wilhelmsson, E. Lazzaro, Reaction–diffusion problems in the physics of hot plasmas. Bristol and Philadelphia:

Institute of Physics Publishing; 2001.

N. F. Britton, Reaction-diffusion equations and their applications to biology. Academic Press; 1986.

I. Lengyel, I. R. Epstein, A chemical approach to designing Turing patterns in reaction-diffusion systems. Proceedings

of the National Academy of Sciences, vol. 89, pp. 3977–3979, 1992.

V.M. Kenkre, M.N. Kuperman, “Applicability of the Fisher equation to bacterial population dynamics”. Physical

Review E, vol. 67, pp. 051921, 2003.

M. Bar, N. Gottschalk, M. Eiswirth, G. Ertl, “Spiral waves in a surface reaction: model calculations”. The Journal of

chemical physics, vol. 100, pp. 1202–1214, 1994.

Bataihah A, Shatanawi W, Qawasmeh T, Hatamleh R. On H -Simulation Functions and Fixed Point Results in the

Setting of ωt-Distance Mappings with Application on Matrix Equations. Mathematics. 2020; 8(5):837.

Ahmed Salem Heilat, Belal Batiha, Tariq Qawasmeh, Raed Hatamleh, Hybrid Cubic B-spline Method for Solving

a Class of Singular Boundary Value Problems. (2023). European Journal of Pure and Applied Mathematics, 16(2),

–762.

Qawasmeh T, Tallafha A, Shatanawi W. Fixed Point Theorems through Modified ω-Distance and Application to

Nontrivial Equations. Axioms. 2019; 8(2):57.

H. Weitzner, G.M. Zaslavsky, “Some applications of fractional equations”. Communications in nonlinear science and

numerical simulation, vol. 8, pp. 273–281, 2003.

Ramzi B. Albadarneh; Abderrahmane Abbes; Adel Ouannas; Iqbal M. Batiha; Taki-Eddine Oussaeif, On chaos in the

fractional-order discrete-time macroeconomic systems, AIP Conf. Proc. 2023; 2849(1):030014.

Shatnawi MT, Khennaoui AA, Ouannas A, Grassi G, Radogna AV, Bataihah A, Batiha IM. A Multistable Discrete

Memristor and Its Application to Discrete-Time FitzHugh–Nagumo Model. Electronics. 2023; 12(13):2929.

I. M. Batiha, N. Djenina, A. Ouannas, T. -E. Oussaeif, L. B. Aoua and S. Momani, “Control of chaos in incommensurate

fractional order discrete system,” 2023 International Conference on Fractional Differentiation and Its Applications

(ICFDA), Ajman, United Arab Emirates, 2023, pp. 1–4.

S. B. Ahmed, A. Ouannas, M. A. Horani, A. A. Khennaoui and I. M. Batiha, “Chaotic Attractors in Quadratic Discrete

Tinkerbell System With Non-Commensurate Fractional Variable-Orders: Complexity, Chaos and Entropy*,” 2023

International Conference on Fractional Differentiation and Its Applications (ICFDA), Ajman, United Arab Emirates,

, pp. 1–5.

R. C. Karoun, A. Ouannas, M. A. Horani, T. Ziar, I. M. Batiha and Z. Dibi, “Chaos in The Fractional Variable Order

Discrete-Time Neural Networks*,” 2023 International Conference on Fractional Differentiation and Its Applications

(ICFDA), Ajman, United Arab Emirates, 2023, pp. 1–5.

Mohd Taib Shatnawi, Abderrahmane Abbes, Adel Ouannas, Iqbal M Batiha, Hidden multistability of fractional discrete non-equilibrium point memristor based map, Physica Scripta. 2023; 98(3):035213.

Mohd Taib Shatnawi, Abderrahmane Abbes, Adel Ouannas, Iqbal M Batiha, A new two-dimensional fractional discrete rational map: chaos and complexity, Physica Scripta. 2022; 98(1): 015208.

I. H. Jebril, N. Biswas and S. K. Datta, “On the growth analysis of meromorphic solutions of finite logarithmic order

of linear difference equations in the unit disc,” 2021 International Conference on Information Technology (ICIT),

Amman, Jordan, 2021, pp. 582–588.

T. Abdeljawad, “On Riemann and Caputo fractional differences”. Computers & Mathematics with Applications, vol.

, pp. 1602–1611, 2011.

F. M. Atici, P. W. Eloe, “A transform method in discrete fractional calculus”. International Journal of Difference

Equations, vol. 2, 2007.

A. C. Peterson, C. Goodrich, Discrete fractional calculus. Cham: Springer. Vol. 10, pp, 2015, 978–3,

W. G. Kelley, A. C. Peterson, Difference equations: an introduction with applications. Academic press. 2001,

D. Baleanu, G. C. Wu, Y. R. Bai, F. L. Chen, “Stability analysis of Caputo–like discrete fractional systems”.

Communications in Nonlinear Science and Numerical Simulation, vol. 48, pp. 520–530, 2017.

F. Atici, P. Eloe, Initial value problems in discrete fractional calculus. Proceedings of the American Mathematical

Society, 137, 2009, pp. 981–989.

N. Anakira, A. Hioual, A. Ouannas, T. E. Oussaeif, I. M. Batiha, Global Asymptotic Stability for Discrete-Time SEI

Reaction-Diffusion Model. In International Conference on Mathematics and Computations, Singapore: Springer

Nature Singapore, pp. 345–357, 2022.

P. Ostalczyk, Discrete fractional calculus: applications in control and image processing, World scientific, Vol. 4, 2015.

G. I. Marchuk, A. A. Brown, Methods of numerical mathematics, New York: Springer-verlag, Vol. 2, 1982.

T. Hamadneh, A. Hioual, O. Alsayyed, Y. A. AL-Khassawneh, A. Al-Husban, A. Ouannas, “Local Stability, Global

Stability, and Simulations in a Fractional Discrete Glycolysis Reaction–Diffusion Model”. Fractal and Fractional,

vol. 7, pp. 587, 2023.

I. Abu Falahah, A. Hioual, M.O. Al-Qadri, Y.A. AL-Khassawneh, A. Al-Husban, T. Hamadneh, A. Ouannas

“Synchronization of Fractional Partial Difference Equations via Linear Methods”. Axioms. vol. 12, pp. 728, 2023.

O. A. Almatroud, A. Hioual, A. Ouannas, G. Grassi, “On Fractional-Order Discrete-Time Reaction Diffusion Systems”.

Mathematics, vol. 11, pp. 2447, 2023.

T. Hamadneh, A. Hioual, O. Alsayyed, Y.A. Al-Khassawneh, A. Al-Husban, A. Ouannas, A. “The FitzHugh–Nagumo

Model Described by Fractional Difference Equations: Stability and Numerical Simulation”. Axioms, vol. 12, pp. 806,

O. Alsayyed, A. Hioual, G.H. Gharib, M. Abualhomos, H. Al-Tarawneh, M.S. Alsauodi, N. Abu-Alkishik, A.A.

Al-Husban, A. Ouannas, “On Stability of a Fractional Discrete Reaction–Diffusion Epidemic Model”. Fractal and

Fractional, vol. 7, pp. 729, 2023.

A. Gierer, H. Meinhardt, “A theory of biological pattern formation”. Kybernetik, vol. 12, pp. 30-39, 1972.

H. Meinhardt, A. J. Koch, G. Bernasconi, ” Models of pattern formation applied to plant development”. Symmetry in

plants, vol. 4, pp. 723-758, 1998.

F. Rothe, Global solutions of reaction–diffusion equations, Lecture Notes in Mathematics, Springer–Verlag, Berlin,

, 1984.

J. Wu, Y. Li, “Global classical solution for the activator–inhibitor model”, Acta Mathematicae Applicatae Sinica (in

Chinese), vol. 13, pp. 501–505, 1990.

H. Jiang, “Global existence of Solution of an Activator-Inhibitor System”, Disc. Cont. Dyn. Sys., vol. 14, pp. 737–751,

M. Medeiros, J. Wei, W. Yang, “Existence and stability of symmetric and asymmetric patterns for the half-Laplacian

Gierer–Meinhardt system in one-dimensional domain”. Math. Models Methods Appl. Sci. 2022.

A. M. Turing, “The chemical basis of morphogenesis, Philosophical Trans”. Roy. Soc. (B), vol. 237, pp. 37–72, 1952.

S. Elaydi, An Introduction to Difference Equations. Springer, San Antonio, Texas, 2015.

J. Cermák, L. Nechvátal, “On a problem of linearized stability for fractional difference equations”, Nonlinear

Dynamics, vol. 104, pp. 1253–1267, 2021.

L. Xu, L.J. Zhao, Z.X. Chang, J.T. Feng, G. Zhang, “Turing instability and pattern formation in a semi-discrete

Brusselator model”. Mod. Phys. Lett. vol. 27, pp. 1350006, 2013.

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