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Complex Turing patterns in chaotic dynamics of autocatalytic reactions with the Caputo fractional derivative

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Abstract

Many chemical systems exhibit a range of patterns, a noticeable and interesting class of numerical patterns that arise in autocatalytic reactions which changes with increasing spatial domains. In this paper, autocatalytic spatiotemporal patterns were demonstrated using the system of chemical species modeled with the time-fractional Caputo derivatives of subdiffusive orders. It is not new that a spectral algorithm with its entire nature is more accurate when compared with a finite-difference scheme to solve a range of integer and non-integer order time partial differential equations. This is because the Fourier spectral techniques have the upper hand on high-order spectral accuracy, and are computationally efficient. Hence, it is regarded as the best approach to existing lower-order methods for integrating the second-order partial derivatives in space. This motivates the present study to explore the usefulness of Fourier spectral methods in resolving and obtaining complex Turing patterns arising from nonlinear fractional autocatalytic reaction-diffusion problems in high dimensions. The autocatalysis model was examines for linear stability in an attempt to obtain the correct choice of parameters that are likely to lead to the formation of new complex turing-like patterns. Numerical experiments in the 2D lead to a striking range of patterns arising from catalytic reactions of fractional-order labyrinthine pattern-like structures. Analysis of pattern formation was also extended to 3D dynamics to obtain a new set of patterns like a star-, cyclic-, diamond-like, and the emergence of apple-shaped structures, which are greatly influenced by either the choice parameters that are involved or that of the initial conditions.

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References

  1. Alqhtani M, Owolabi KM, Saad KM, Pindza E (2022) Efficient numerical techniques for computing the Riesz fractional-order reaction-diffusion models arising in biology. Chaos Solitons Fractals 161:112394

    MATH  MathSciNet  Google Scholar 

  2. Alqhtani M, Owolabi KM, Saad KM (2022) Spatiotemporal (target) patterns in sub-diffusive predator-prey system with the Caputo operator. Chaos Solitons Fractals 160:112267

    MATH  MathSciNet  Google Scholar 

  3. Ana X, Liu F, Zheng M, Anh VV, Turner IW (2021) A space-time spectral method for time-fractional Black-Scholes equation. Appl Numer Math 165:152–166

    MATH  MathSciNet  Google Scholar 

  4. Animasaun IL (2016) 47nm alumina-water nanofluid flow within boundary layer formed on upper horizontal surface of paraboloid of revolution in the presence of quartic autocatalysis chemical reaction. Alex Eng J 55:2375–2389

    Google Scholar 

  5. Ascher UM, Ruth SJ, Wetton BTR (1995) Implicit-explicit methods for time-dependent partial differential equations. SIAM J Numer Anal 32:797–823

    MATH  MathSciNet  Google Scholar 

  6. Ascher UM, Ruth SJ, Spiteri RJ (1997) Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Appl Numer Math 25:151–167

    MATH  MathSciNet  Google Scholar 

  7. Atangana A (2016) Derivative with a New Parameter: Theory. Academic Press, NY

    MATH  Google Scholar 

  8. Atangana A (2017) Fractional operators with constant and variable order with application to geo-hydrology. Academic Press, NY

    MATH  Google Scholar 

  9. Atangana A, Owolabi KM (2018) New numerical approach for fractional differential equations. Math Modell Nat Phenom 13(3):21. https://doi.org/10.1051/mmnp/2018010

    Article  MATH  MathSciNet  Google Scholar 

  10. Barrio RA, Varea C, Aragon JL, Maini PK (1999) A two-dimensional numerical study of spatial pPattern formation in interacting Turing systems. Bull Math Biol 61:483–505

    MATH  Google Scholar 

  11. Blackmond D (2009) An examination of the role of autocatalytic cycles in the chemistry of proposed primordial reactions. Angew Chem 48:386–390

    Google Scholar 

  12. Boyd JP (2001) Chebyshev and fourier spectral methods. Dover, NY

    MATH  Google Scholar 

  13. Calvo M, Palencia C (2006) A class of explicit multi-step exponential integrators for semi-linear problems. Numer Math 102:367–381

    MATH  MathSciNet  Google Scholar 

  14. Chen W-C (2008) Nonlinear dynamics and chaos in a fractional-order financial system. Chaos Solitons Fractals 36:1305–1314

    Google Scholar 

  15. Cox SM, Matthews PC (2002) Exponential time differencing for stiff systems. J Comput Phys 176:430–455

    MATH  MathSciNet  Google Scholar 

  16. Diethelm K, Freed AD (1999) On the solution of nonlinear fractional order differential equations used in the modelling of viscoplasticity. In: Scientific computing in chemical engineering II: computational fluid dynamics, reaction engineering and molecular properties. Springer Verlag, Heidelberg, pp 217–224

  17. Ervin VJ, Roop JP (2006) Variational formulation for the stationary fractional advection dispersion equation. Numer Methods Partial Differ Eq 22:558–576

    MATH  MathSciNet  Google Scholar 

  18. Ervin VJ, Heuer N, Roop JP (2018) Regularity of the solution to 1-D fractional order diffusion equations. Math Comput 87:2273–2294

    MATH  MathSciNet  Google Scholar 

  19. Fornberg B, Driscoll TA (1999) A fast spectral algorithm for nonlinear wave equations with linear dispersion. J Comput Phys 155:456–467

    MATH  MathSciNet  Google Scholar 

  20. Gao G, Sun Z (2011) A compact finite difference scheme for the fractional sub-diffusion equations. J Comput Phys 230:586–595

    MATH  MathSciNet  Google Scholar 

  21. Gorenflo R, Mainardi F, Scalas E, Raberto M (2001) Fractional calculus and continuous-time finance. III. The diffusion limit, Math. Finance (Konstanz, 2000) 171–180

  22. Hochbruck M, Ostermann A (2005) Exponential Runge-Kutta methods for parabolic problems. Appl Numer Math 53:323–339

    MATH  MathSciNet  Google Scholar 

  23. Hochbruck M, Ostermann A (2005) Explicit exponential Runge-Kutta methods for semilinear parabolic problems. SIAM J Numer Anal 43:1069–1090

    MATH  MathSciNet  Google Scholar 

  24. Hochbruck, Ostermann A (2010) Exponential integrators. Acta Numer 19:209–286

    MATH  MathSciNet  Google Scholar 

  25. Hochbruck M, Ostermann A (2011) Exponential multistep methods of Adams-type. BIT Numer Math 51:889–908

    MATH  MathSciNet  Google Scholar 

  26. Karig D, Martini KM, Lu T, Weiss R (2018) Stochastic Turing patterns in a synthetic bacterial population. PNAS 115:6572–6577

    Google Scholar 

  27. Kassam AK, Trefethen LN (2005) Fourth-order time-stepping for stiff PDEs. SIAM J Sci Comput 26:1214–1233

    MATH  MathSciNet  Google Scholar 

  28. Lacitignola D, Sgura I, Bozzini B (2021) Turing-Hopf patterns in a morphochemical model for electrodeposition with cross-diffusion. Appl Eng Sci 5:100034

    Google Scholar 

  29. Landge AN, Jordan BM, Diego X, Müller P (2020) Pattern formation mechanisms of self-organizing reaction-diffusion systems. Dev Biol 460:2–11

    Google Scholar 

  30. Li X, Xu C (2009) A space-time spectral method for the time fractional diffusion equation. SIAM J Numer Anal 47(3):2108–2131

    MATH  MathSciNet  Google Scholar 

  31. Li X, Xu C (2010) Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput Phys 8:1016–1051

    MATH  MathSciNet  Google Scholar 

  32. Liu F, Anh V, Turner I (2004) Numerical solution of the space fractional Fokker-Planck equation. J Comput Appl Math 166:209–219

    MATH  MathSciNet  Google Scholar 

  33. Makinde OD, Animasaun IL (2016) Bioconvection in MHD nanofluid flow with nonlinear thermal radiation and quartic autocatalysis chemical reaction past an upper surface of a paraboloid of revolution. Int J Thermal Sci 109:159–171

    Google Scholar 

  34. Makinde OD, Animasaun IL (2016) Thermophoresis and Brownian motion effects on MHD bioconvection of nanofluid with nonlinear thermal radiation and quartic chemical reaction past an upper horizontal surface of a paraboloid of revolution. J Mol Liquids 221:733–743

    Google Scholar 

  35. Carpinteri A, Mainardi F (1997) Fractals Fract Calc Contin Mech. Springer-Verlag, Wien

    Google Scholar 

  36. Meerschaert MM, Tadjeran C (2004) Finite difference approximations for fractional advection-dispersion flow equations. J Comput Appl Math 172:65–77

    MATH  MathSciNet  Google Scholar 

  37. Metzler R, Klafter J (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339(1):1–77

    MATH  MathSciNet  Google Scholar 

  38. Metler R, Klafter J (2004) The restaurant at the end of random walk: recent developments in the description of anomalous transport by fractional dynamics. J Phys A 37:R161–R208

    MATH  MathSciNet  Google Scholar 

  39. Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, Hoboken

    MATH  Google Scholar 

  40. Moskon M, Komac R, Zimic N et al (2021) Distributed biological computation: from oscillators, logic gates and switches to a multicellular processor and neural computing applications. Neural Comput Appl 33:8923–8938

    Google Scholar 

  41. Müller P, El-Sherif E (2020) A systems-level view of pattern formation mechanisms in development. Dev Biol 460:1. https://doi.org/10.1016/j.ydbio.2019.10.034

    Article  Google Scholar 

  42. Murray JD (2002) Mathematical biology I: an introduction. Springer-Verlag, NY

    MATH  Google Scholar 

  43. Murray JD (2003) Mathematical biology II: spatial models and biomedical applications. Springer-Verlag, Berlin

    MATH  Google Scholar 

  44. Oldham KB, Spanier J (1974) The fractional calculus. Academic Press, NY

    MATH  Google Scholar 

  45. Ostwald W (1912) Outlines of general chemistry (trad. Taylor, W.W.), chap. XI.1. Macmillan and co, NY, p. 301

  46. Owolabi KM, Atangana A (2016) Numerical solution of fractional-in-space nonlinear Schrödinger equation with the Riesz fractional derivative. Eur Phys J Plus 131:335

    Google Scholar 

  47. Owolabi KM (2017) Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order. Commun Nonlinear Sci Numer Simul 44:304–317

    MATH  MathSciNet  Google Scholar 

  48. Owolabi KM (2018) Mathematical analysis and numerical simulation of chaotic non-integer order differential systems with Riemann-Liouville derivative. Numer Methods Partial Differ Eq 34:274–95

    MATH  Google Scholar 

  49. Owolabi KM, Atangana A (2019) Higher-order solvers for space-fractional differential equations with Riesz derivative. Discret Contin Dynamic Syst Ser S 12:567–590

    MATH  Google Scholar 

  50. Owolabi KM (2020) Numerical simulation of fractional-order reaction-diffusion equations with the Riesz and Caputo derivatives. Neural Comput Appl 32:4093–4104

    Google Scholar 

  51. Owolabi KM, Baleanu D (2021) Emergent patterns in diffusive Turing-like systems with fractional-order operator. Neural Comput Appl 33:12703–12720

    Google Scholar 

  52. Owolabi KM (2021) Computational analysis of different Pseudoplatystoma species patterns the Caputo-Fabrizio derivative. Chaos Solitons Fractals 144:110675

    MATH  MathSciNet  Google Scholar 

  53. Alqhtani M, Owolabi KM, Saad KM (2022) Spatiotemporal (target) patterns in sub-diffusive predator-prey system with the Caputo operator. Chaos Solitons Fractals 160:112267

    MATH  MathSciNet  Google Scholar 

  54. Alqhtani M, Owolabi KM, Saad KM, Pindza E (2022) Efficient numerical techniques for computing the Riesz fractional-order reaction-diffusion models arising in biology. Chaos Solitons Fractals 161:112394

    MATH  MathSciNet  Google Scholar 

  55. Pindza E, Owolabi KM (2016) Fourier spectral method for higher order space fractional reaction-diffusion equations. Commun Nonlinear Sci Numer Simul 40:112–128

    MATH  MathSciNet  Google Scholar 

  56. Plasson R, Kondepudi DK, Bersini H, Commeyras A, Asakura K (2007) Emergence of homochirality in far-from-equilibrium systems: mechanisms and role in prebiotic chemistry. Chirality 19:589–600

    Google Scholar 

  57. Plasson R (2008) Comment on re-examination of reversibility in reaction models for the spontaneous emergence of homochirality. J Phys Chem B 112:9550–9552

    Google Scholar 

  58. Podlubny I (1999) Fractional Differential Equations. Academic Press, NY

    MATH  Google Scholar 

  59. Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives: theory and applications. Gordon and Breach, Amsterdam

    MATH  Google Scholar 

  60. Scalas E, Gorenflo R, Mainardi F (2000) Fractional calculus and continuous-time finance. Phys A Stat Mech Appl 284(1–4):376–384

    MathSciNet  Google Scholar 

  61. Schiesser WE (1991) Numerical method of lines integration of partial differential equations. Academic Press, San Diego

    MATH  Google Scholar 

  62. Schisser WE, Griffiths GW (2009) A compendium of partial differential equation models: method of lines analysis with Matlab. Cambridge University Press, Cambridge

    Google Scholar 

  63. Soai K, Shibata T, Morioka H, Choji K (1995) Asymmetric autocatalysis and amplification of enantiomeric excess of a chiral molecule. Nature 378:767–768

    Google Scholar 

  64. Trefethen LN (1996) Finite difference and spectral methods for ordinary and partial differential equations. Upson Hall Cornell University Ithaca, NY

    Google Scholar 

  65. Trefethen LN (2000) Spectral methods in MATLAB. SIAM, Philadelphia

    MATH  Google Scholar 

  66. Trefethen LN, Embere M (2005) Spectra and pseudospectra: the behavior of nonnormal matrices and operators. Princeton University Press, New Jersey

    Google Scholar 

  67. Vittadello ST, Leyshon T, Schnoerr D, Stumpf MPH (2021) Turing pattern design principles and their robustness. Phil Trans R Soc A 379:20200272. https://doi.org/10.1098/rsta.2020.0272

    Article  Google Scholar 

  68. Yuste SB, Acedo L, Lindenberg K (2004) Reaction front in an \(A + B\rightarrow C\) reaction-subdiffusion process. Phys Rev E 69(3):036126

    Google Scholar 

  69. Zaslavsky GM (2002) Chaos, fractional kinetics, and anomalous transport. Phys Rep 371(6):461–580

    MATH  MathSciNet  Google Scholar 

  70. Zhang H, Jiang X, Wang C, Fan W (2018) Galerkin-Legendre spectral schemes for nonlinear space fractional Schrödinger equation. Numer. Algorithms 79:337–356

    MATH  MathSciNet  Google Scholar 

  71. Zhang H, Jiang X, Yang X (2018) A time-space spectral method for the time-space fractional Fokker-Planck equation and its inverse problem. Appl Math Comput 320:302–318

    MATH  MathSciNet  Google Scholar 

  72. Zhao M, Wang H (2019) Fast finite difference methods for space-time fractional partial differential equations in three space dimensions with nonlocal boundary conditions. Appl Numer Math 145:411–428

    MATH  MathSciNet  Google Scholar 

  73. Zheng M, Liu F, Turner I, Anh V (2015) A novel high order space-time spectral method for the time fractional Fokker-Planck equation. SIAM J Comput 37(2):A701–A724

    MATH  MathSciNet  Google Scholar 

  74. Zheng M, Liu F, Anh V, Turner I (2016) A high-order spectral method for the multi-term time-fractional diffusion equations. Appl Math Model 40(7–8):4970–4985

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Federal University of Technology Akure, PMB 704, Akure, Ondo State, Nigeria

    Kolade M. Owolabi

  2. Department of Mathematics, Texas A &M University-Kingsville, Kingsville, TX, 78363-8202, USA

    Ravi P. Agarwal

  3. Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 002, South Africa

    Edson Pindza

  4. Department of Mathematics and Informatics, Institute of Applied Analysis, Technische Universität Bergakademie Freiberg, 09599, Freiberg, Germany

    Swanhild Bernstein

  5. Department of Mathematics, Faculty of Applied Science, Umm Al-Qura University, Makkah, 21955, Saudi Arabia

    Mohamed S. Osman

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  1. Kolade M. Owolabi
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Owolabi, K.M., Agarwal, R.P., Pindza, E. et al. Complex Turing patterns in chaotic dynamics of autocatalytic reactions with the Caputo fractional derivative. Neural Comput & Applic 35, 11309–11335 (2023). https://doi.org/10.1007/s00521-023-08298-2

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