Stability analysis of fractional two-dimensional reaction-diffusion model with applications in biological processes

Ishtiaq Ali; Saeed Islam; Ishtiaq Ali; Saeed Islam

doi:10.3934/math.2025531

AIMS Mathematics

2025, Volume 10, Issue 5: 11732-11756. doi: 10.3934/math.2025531

Previous Article Next Article

Research article Special Issues

Stability analysis of fractional two-dimensional reaction-diffusion model with applications in biological processes

Ishtiaq Ali ^{1
,
,},
Saeed Islam ^{2
,
,}

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa, 31982, Saudi Arabia
2.
Department of Mechanical Engineering, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia

Received: 20 March 2025 Revised: 03 May 2025 Accepted: 19 May 2025 Published: 21 May 2025
MSC : 35K57, 35R11, 35B35, 37M05

We study the stability of a two-dimensional fractional reaction-diffusion system under the Caputo differential operator in time. Our model is based on the Grey-Scott model, a well-known coupled reaction-diffusion system that describes the interaction between two chemical species. The diffusion term captures the species special spread, while the nonlinear term in the system describes the chemical reaction, resulting in a wide range of difficult, self-organizing patterns, including spots, stripes, or spirals, depending on the parameter values. We derive conditions for local stability of the homogeneous equilibrium by linearizing the system and analyzing the eigenvalues of the Jacobian. Furthermore, we construct appropriate Lyapunov functionals to establish global asymptotic stability of the discrete model under suitable conditions. This approach seeks to provide a robust framework for analyzing complex dynamical behaviors in systems governed by fractional-order in-time reaction-diffusion systems. The numerical simulations employ the Chebyshev spectral method for spatial discretization and the $ L1 $ scheme for fractional time derivatives. These simulations validate the theoretical findings, demonstrating the model's ability to replicate intricate patterns often observed in reaction-diffusion systems. The results suggest that the fractional-order framework enhances the understanding of pattern formation in such systems, making this model a valuable tool for studying anomalous diffusion and non-local dynamics in biological and chemical processes.
- fraction reaction-diffusion system in 2D,
- biological processes,
- local and global stability,
- spectral method,
- Lyapunov method,
- numerical simulations
Citation: Ishtiaq Ali, Saeed Islam. Stability analysis of fractional two-dimensional reaction-diffusion model with applications in biological processes[J]. AIMS Mathematics, 2025, 10(5): 11732-11756. doi: 10.3934/math.2025531

Related Papers:

Abstract

We study the stability of a two-dimensional fractional reaction-diffusion system under the Caputo differential operator in time. Our model is based on the Grey-Scott model, a well-known coupled reaction-diffusion system that describes the interaction between two chemical species. The diffusion term captures the species special spread, while the nonlinear term in the system describes the chemical reaction, resulting in a wide range of difficult, self-organizing patterns, including spots, stripes, or spirals, depending on the parameter values. We derive conditions for local stability of the homogeneous equilibrium by linearizing the system and analyzing the eigenvalues of the Jacobian. Furthermore, we construct appropriate Lyapunov functionals to establish global asymptotic stability of the discrete model under suitable conditions. This approach seeks to provide a robust framework for analyzing complex dynamical behaviors in systems governed by fractional-order in-time reaction-diffusion systems. The numerical simulations employ the Chebyshev spectral method for spatial discretization and the $ L1 $ scheme for fractional time derivatives. These simulations validate the theoretical findings, demonstrating the model's ability to replicate intricate patterns often observed in reaction-diffusion systems. The results suggest that the fractional-order framework enhances the understanding of pattern formation in such systems, making this model a valuable tool for studying anomalous diffusion and non-local dynamics in biological and chemical processes.

References

[1]	B. Baeumer, M. Kovács, M. M. Meerschaert, Numerical solutions for fractional reaction-diffusion equations, Comput. Math. Appl., 55 (2008), 2212–2226. https://doi.org/10.1016/j.camwa.2007.11.012 doi: 10.1016/j.camwa.2007.11.012
[2]	A. Ghafoor, M. Fiaz, M. Hussain, A. Ullah, E. A. A. Ismail, F. A. Awwad, Dynamics of the time-fractional reaction-diffusion coupled equations in biological and chemical processes, Sci. Rep., 14 (2024), 7549. https://doi.org/10.1038/s41598-024-58073-z doi: 10.1038/s41598-024-58073-z
[3]	A. El Hassani, K. Hattaf, N. Achtaich, Global stability of reaction-diffusion equations with fractional Laplacian operator and applications in biology, Commun. Math. Biol. Neurosci., 2022 (2022), 56. https://doi.org/10.28919/cmbn/7485 doi: 10.28919/cmbn/7485
[4]	T. Wang, F. Song, H. Wang, G. E. Karniadakis, Fractional Gray–Scott model: Well-posedness, discretization, and simulations, Comput. Methods Appl. Mech. Eng., 347 (2019), 1030–1049. https://doi.org/10.1016/j.cma.2019.01.002 doi: 10.1016/j.cma.2019.01.002
[5]	K. J. Lee, W. D. McCormick, J. E. Pearson, H. L. Swinney, Experimental observation of self-replicating spots in a reaction–diffusion system, Nature, 369 (1994), 215–218. https://doi.org/10.1038/369215a0 doi: 10.1038/369215a0
[6]	A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. B Biol. Sci., 237 (1952), 37–72. https://doi.org/10.1098/rstb.1952.0012 doi: 10.1098/rstb.1952.0012
[7]	V. Castets, E. Dulos, J. Boissonade, P. De Kepper, Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern, Phys. Rev. Lett., 64 (1990), 2953–2965. https://doi.org/10.1103/PhysRevLett.64.2953 doi: 10.1103/PhysRevLett.64.2953
[8]	Q. Ouyang, H. L. Swinney, Transition from a uniform state to chemical patterns in a reaction-diffusion system, Nature, 352 (1991), 610–612. https://doi.org/10.1038/352610a0 doi: 10.1038/352610a0
[9]	L. Xu, L. J. Zhao, Z. X. Chang, J. T. Feng, Turing instability and pattern formation in a semi-discrete Brusselator model, Mod. Phys. Lett. B, 27 (2013), 1350006. https://doi.org/10.1142/S0217984913500061 doi: 10.1142/S0217984913500061
[10]	W. Mazin, K. E. Rasmussen, E. Mosekilde, P. Borckmans, Pattern formation in the bistable Gray–Scott model, Math. Comput. Simul., 40 (1996), 371–396. https://doi.org/10.1016/0378-4754(95)00044-5 doi: 10.1016/0378-4754(95)00044-5
[11]	O. Ersoy, I. Dağ, Numerical solutions of the reaction-diffusion system by using exponential cubic B-spline collocation algorithms, Open Phys., 13 (2015), 153–158. https://doi.org/10.1515/phys-2015-0047 doi: 10.1515/phys-2015-0047
[12]	A. T. Onarcan, N. Adar, I. Dağ, Trigonometric cubic B-spline collocation algorithm for numerical solutions of reaction-diffusion equation systems, Comput. Appl. Math., 37 (2018), 6848–6869. https://doi.org/10.1007/s40314-018-0713-4 doi: 10.1007/s40314-018-0713-4
[13]	C. S. Chou, Y. T. Zhang, R. Zhao, Q. Nie, Numerical methods for stiff reaction-diffusion systems, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 515–525. https://doi.org/10.3934/dcdsb.2007.7.515 doi: 10.3934/dcdsb.2007.7.515
[14]	E. Özuğurlu, A note on the numerical approach for the reaction-diffusion problem to model the density of the tumor growth dynamics, Comput. Math. Appl., 69 (2015), 1504–1517. https://doi.org/10.1016/j.camwa.2015.04.018 doi: 10.1016/j.camwa.2015.04.018
[15]	A. Madzvamuse, A. H. Chung, The bulk-surface finite element method for reaction-diffusion systems on stationary volumes, Finite Elem. Anal. Des., 108 (2016), 9–21. https://doi.org/10.1016/j.finel.2015.09.002 doi: 10.1016/j.finel.2015.09.002
[16]	R. Mittal, R. Jiwari, Numerical study of two-dimensional reaction-diffusion Brusselator system by differential quadrature method, Int. J. Comput. Methods Eng. Sci. Mech., 12 (2011), 14–25. https://doi.org/10.1080/15502287.2010.540300 doi: 10.1080/15502287.2010.540300
[17]	P. Gray, S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system $A + 2B \rightarrow 3B$, $B \rightarrow C$, Chem. Eng. Sci., 39 (1984), 1087–1097. https://doi.org/10.1016/0009-2509(84)87017-7 doi: 10.1016/0009-2509(84)87017-7
[18]	P. Gray, S. K. Scott, Chemical oscillations and instabilities, Oxford University Press, Oxford, 1990.
[19]	E. E. Selkov, Self-oscillations in glycolysis, Eur. J. Biochem., 4 (1968), 79–86. https://doi.org/10.1111/j.1432-1033.1968.tb00175.x doi: 10.1111/j.1432-1033.1968.tb00175.x
[20]	J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189–192. https://doi.org/10.1126/science.261.5118.189 doi: 10.1126/science.261.5118.189
[21]	S. Aljhani, M. S. Md Noorani, K. M. Saad, A. K. Alomari, Numerical solutions of certain new models of the time-fractional Gray–Scott, J. Funct. Spaces, 2021 (2021), 2544688. https://doi.org/10.1155/2021/2544688 doi: 10.1155/2021/2544688
[22]	J. S. McGough, K. Riley, Pattern formation in the Gray–Scott model, Nonlinear Anal. Real World Appl., 5 (2004), 105–121. https://doi.org/10.1016/S1468-1218(03)00020-8 doi: 10.1016/S1468-1218(03)00020-8
[23]	I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, San Diego, 1998.
[24]	R. Magin, Fractional calculus in bioengineering, part 1, Crit. Rev. Biomed. Eng., 32 (2004), 1–104. https://doi.org/10.1615/CritRevBiomedEng.v32.i1.10 doi: 10.1615/CritRevBiomedEng.v32.i1.10
[25]	K. M. Owolabi, R. P. Agarwal, E. Pindza, S. Bernstein, M. S. Osman, Complex Turing patterns in chaotic dynamics of autocatalytic reactions with the Caputo fractional derivative, Neural Comput. Appl., 2023, 1–27. https://doi.org/10.1007/s00521-023-08298-2
[26]	M. Alqhtani, K. M. Owolabi, K. M. Saad, E. Pindza, Spatiotemporal chaos in spatially extended fractional dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 119 (2023), 107118. https://doi.org/10.1016/j.cnsns.2023.107118 doi: 10.1016/j.cnsns.2023.107118
[27]	M. Alqhtani, K. M. Owolabi, K. M. Saad, Spatiotemporal (target) patterns in sub-diffusive predator-prey system with the Caputo operator, Chaos Soliton. Fract., 160 (2022), 112267. https://doi.org/10.1016/j.chaos.2022.112267 doi: 10.1016/j.chaos.2022.112267
[28]	K. M. Owolabi, E. Pindza, A. Atangana, Analysis and pattern formation scenarios in the superdiffusive system of predation described with Caputo operator, Chaos Soliton. Fract., 152 (2021), 111468. https://doi.org/10.1016/j.chaos.2021.111468 doi: 10.1016/j.chaos.2021.111468
[29]	K. M. Owolabi, B. Karaagac, D. Baleanu, Dynamics of pattern formation process in fractional-order super-diffusive processes: A computational approach, Soft Comput., 25 (2021), 11191–11208. https://doi.org/10.1007/s00500-021-05885-0 doi: 10.1007/s00500-021-05885-0
[30]	K. M. Owolabi, D. Baleanu, Emergent patterns in diffusive Turing-like systems with fractional-order operator, Neural Comput. Appl., 33 (2021), 12703–12720. https://doi.org/10.1007/s00521-021-05917-8 doi: 10.1007/s00521-021-05917-8
[31]	K. M. Owolabi, S. Jain, Spatial patterns through diffusion-driven instability in modified predator-prey models with chaotic behaviors, Chaos Soliton. Fract., 174 (2023), 113839. https://doi.org/10.1016/j.chaos.2023.113839 doi: 10.1016/j.chaos.2023.113839
[32]	K. M. Owolabi, K. C. Patidar, Higher-order time-stepping methods for time-dependent reaction–diffusion equations arising in biology, Appl. Math. Comput., 240 (2014), 30–50. https://doi.org/10.1016/j.amc.2014.04.055 doi: 10.1016/j.amc.2014.04.055
[33]	K. M. Owolabi, Mathematical analysis and numerical simulation of patterns in fractional and classical reaction–diffusion systems, Chaos Soliton. Fract., 93 (2016), 89–98. https://doi.org/10.1016/j.chaos.2016.10.005 doi: 10.1016/j.chaos.2016.10.005
[34]	E. Pindza, K. M. Owolabi, Fourier spectral method for higher order space fractional reaction–diffusion equations, Commun. Nonlinear Sci. Numer. Simul., 40 (2016), 112–128. https://doi.org/10.1016/j.cnsns.2016.04.020 doi: 10.1016/j.cnsns.2016.04.020
[35]	K. J. Lee, H. L. Swinney, Lamellar structures and self-replicating spots in a reaction-diffusion system, Phys. Rev. E, 51 (1995), 1899–1915. https://doi.org/10.1103/PhysRevE.51.1899 doi: 10.1103/PhysRevE.51.1899
[36]	K. J. Lin, W. D. McCormick, J. E. Pearson, H. L. Swinney, Experimental observation of self-replicating spots in a reaction-diffusion system, Nature, 369 (1994), 215–218. https://doi.org/10.1038/369215a0 doi: 10.1038/369215a0
[37]	B. Liu, R. Wu, N. Iqbal, L. Chen, Turing patterns in the Lengyel–Epstein system with super diffusion, Int. J. Bifurcat. Chaos, 27 (2017), 1730026. https://doi.org/10.1142/S0218127417300269 doi: 10.1142/S0218127417300269
[38]	F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation, Appl. Math. Comput., 191 (2007), 12–20. https://doi.org/10.1016/j.amc.2006.08.162 doi: 10.1016/j.amc.2006.08.162
[39]	W. Mazin, K. E. Rasmussen, E. Mosekilde, P. Borckmans, G. Dewel, Pattern formation in the bistable Gray–Scott model, Math. Comput. Simul., 40 (1996), 371–396. https://doi.org/10.1016/0378-4754(95)00044-5 doi: 10.1016/0378-4754(95)00044-5
[40]	J. S. McGough, K. Riley, Pattern formation in the Gray–Scott model, Nonlinear Anal. Real World Appl., 5 (2004), 105–121. https://doi.org/10.1016/S1468-1218(03)00020-8 doi: 10.1016/S1468-1218(03)00020-8
[41]	M. M. Meerschaert, C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56 (2006), 80–90. https://doi.org/10.1016/j.apnum.2005.02.008 doi: 10.1016/j.apnum.2005.02.008
[42]	Z. Zhang, Q. Ouyang, Global existence, blow-up and optimal decay for a nonlinear viscoelastic equation with nonlinear damping and source term, Discrete Contin. Dyn. Syst. Ser. B, 28 (2023), 4735–4760. https://doi.org/10.3934/dcdsb.2023038 doi: 10.3934/dcdsb.2023038
[43]	Z. Y. Zhang, Z. H. Liu, Y. J. Deng, J. H. Huang, C. X. Huang, Long time behavior of solutions to the damped forced generalized Ostrovsky equation below the energy space, Proc. Amer. Math. Soc., 149 (2021), 1527–1542. https://doi.org/10.1090/proc/15322 doi: 10.1090/proc/15322
[44]	Y. Wen, X. F. Zhou, Z. Zhang, S. Liu, Lyapunov method for nonlinear fractional differential systems with delay, Nonlinear Dyn., 82 (2015), 1015–1025. https://doi.org/10.1007/s11071-015-2214-y doi: 10.1007/s11071-015-2214-y
[45]	Y. Li, Y. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput. Math. Appl., 59 (2010), 1810–1821. https://doi.org/10.1016/j.camwa.2009.08.019 doi: 10.1016/j.camwa.2009.08.019
[46]	S. Liu, W. Jiang, X. Li, X. F. Zhou, Lyapunov stability analysis of fractional nonlinear systems, Appl. Math. Lett., 51 (2016), 13–19. https://doi.org/10.1016/j.aml.2015.06.018 doi: 10.1016/j.aml.2015.06.018
[47]	I. Ali, M. T. Saleem, Spatiotemporal dynamics of reaction–diffusion system and its application to Turing pattern formation in a Gray–Scott model, Mathematics, 11 (2023), 1459. https://doi.org/10.3390/math11061459 doi: 10.3390/math11061459
[48]	I. Ali, S. U. Khan, Threshold of stochastic SIRS epidemic model from infectious to susceptible class with saturated incidence rate using spectral method, Symmetry, 14 (2022), 1838. https://doi.org/10.3390/sym14091838 doi: 10.3390/sym14091838
[49]	I. Ali, S. U. Khan, A dynamic competition analysis of stochastic fractional differential equation arising in finance via pseudospectral method, Mathematics, 11 (2023), 1328. https://doi.org/10.3390/math11061328 doi: 10.3390/math11061328
[50]	I. Ali, S. U. Khan, Dynamics and simulations of stochastic COVID-19 epidemic model using Legendre spectral collocation method, AIMS Math., 8 (2023), 4220–4236. https://doi.org/10.3934/math.2023210 doi: 10.3934/math.2023210
[51]	I. Ali, Dynamical analysis of two-dimensional fractional-order-in-time biological population model using Chebyshev spectral method, Fractals Fract., 8 (2024), 325. https://doi.org/10.3390/fractalfract8060325 doi: 10.3390/fractalfract8060325
[52]	D. Gottlieb, S. A. Orszag, Numerical analysis of spectral methods: Theory and applications, SIAM, Philadelphia, PA, USA, 1977. https://doi.org/10.1137/1.9781611970425
[53]	J. C. Mason, D. C. Handscomb, Chebyshev Polynomials, CRC Press LLC, New York, NY, USA, 2003. https://doi.org/10.1201/9781420036114
[54]	C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer Series in Scientific Computation, Springer, New York, NY, USA, 2006. https://doi.org/10.1007/978-3-540-30726-6

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)