Turing pattern in the fractional Gierer-Meinhardt model

doi:10.1088/1674-1056/28/5/050503

Abstract It is well-known that reaction-diffusion systems are used to describe the pattern formation models. In this paper, we will investigate the pattern formation generated by the fractional reaction-diffusion systems. We first explore the mathematical mechanism of the pattern by applying the linear stability analysis for the fractional Gierer-Meinhardt system. Then, an efficient high-precision numerical scheme is used in the numerical simulation. The proposed method is based on an exponential time differencing Runge-Kutta method in temporal direction and a Fourier spectral method in spatial direction. This method has the advantages of high precision, better stability, and less storage. Numerical simulations show that the system control parameters and fractional order exponent have decisive influence on the generation of patterns. Our numerical results verify our theoretical results.

Keywords: Turing patterns fractional Gierer-Meinhardt model Fourier spectral method

Received: 22 December 2018
Revised: 04 March 2019
Accepted manuscript online:

PACS:	05.30.Pr	(Fractional statistics systems)
	05.65.+b	(Self-organized systems)
	82.40.Ck	(Pattern formation in reactions with diffusion, flow and heat transfer)
	02.70.Hm	(Spectral methods)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 61573008 and 61703290) and Natural Science Foundation of Liaoning Province, China (Grant No. 20180550996).

Corresponding Authors: Rongpei Zhang
E-mail: rongpeizhang@163.com

Cite this article:

Yu Wang(王语), Rongpei Zhang(张荣培), Zhen Wang(王震), Zijian Han(韩子健) Turing pattern in the fractional Gierer-Meinhardt model 2019 Chin. Phys. B 28 050503

[1]	Turing A M 1952 Philos. Trans. Royal Soc. B 237 37
[2]	Baurmann M, Gross T and Feudel U 2007 J. Theor. Biol. 245 220
[3]	Ouyang Q and Swinney H L 1991 Nat. 352 610
[4]	Liang P 1995 Phys. Rev. Lett. 75 1863
[5]	Staliunas K, Sánchez-Morcillo V J 2000 Opt. Commun. 177 389
[6]	Epstein I R 1984 Nat. 307 692
[7]	L'Heureux I 2013 Philos. Trans. 371 20120356
[8]	Iron D, Wei J and Winter M 2004 J. Math. Biol. 49 358
[9]	Li Q, Zheng C G, Wang N C and Shi B C 2001 J. Sci. Comput. 16 121
[10]	Wang T, Song F, Wang H and Karniadakis G 2019 Comput. Meth. Appl. Mech. Engrg. 347 1030
[11]	Yang R and Song Y 2016 Nonlinear Anal. Real World Appl. 31 356
[12]	Gambino G, Lombardo M C, Sammartino M and Sciacca V 2013 Phys. Rev. E 88 042925
[13]	Golovin A A, Matkowsky B J and Volpert V A 2008 SIAM J. Appl. Math. 69 251
[14]	Metzler R and Klafter J 2004 J. Phys. A: Math. Gen. 37 R161
[15]	Li B and Wang J 2003 Phys. Rev. Lett. 91 044301
[16]	Nec Y, Nepomnyashchy A A and Golovin A A 2008 Physica D 237 3237
[17]	Feng F, Yan J, He Y F and Liu F C 2016 Chin. Phys. B 25 104702
[18]	Li X and Xu C 2010 Commun. Comput. Phys. 8 1016
[19]	Khader M M 2011 Commun. Nonlinear. Sci. Numer. Simul. 16 2535
[20]	Khader M M and Sweilam N H 2014 Comput. Appl. Math. 33 739
[21]	Hanert E 2010 Environ Fluid Mech. 10 7
[22]	Zayernouri M and Karniadakis G E 2014 SIAM J. Sci. Comput. 36 A40
[23]	Cox S M and Matthews P C 2002 J. Comput. Phys. 176 430
[24]	Kassam A K and Trefethen L N 2005 Soc. Ind. Appl. Mech. 26 1214
[25]	Gierer A and Meinhardt H 1972 Kybernetik 12 30
[26]	Zhang R P, Wang Z, Wang Y and Han Z J 2018 Acta Phys. Sin. 67 050503 (in Chinese)
[27]	Bueno-Orovio A, Kay D and Burrage K 2014 BIT Numer. Math. 54 937

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