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Turing proposed a model for the development of patterns found in nature in 1952. Turing instability is known as diffusion-driven instability, which states that a stable spatially homogeneous equilibrium may lose its stability due to the unequal spatial diffusion coefficients. The Gierer-Mainhardt model is an activator and inhibitor system to model the generating mechanism of biological patterns. The reaction-diffusion system is often used to describe the pattern formation model arising in biology. In this paper, the mechanism of the pattern formation of the Gierer-Meinhardt model is deduced from the reactive diffusion model. It is explained that the steady equilibrium state of the nonlinear ordinary differential equation system will be unstable after adding of the diffusion term and produce the Turing pattern. The parameters of the Turing pattern are obtained by calculating the model. There are a variety of numerical methods including finite difference method and finite element method. Compared with the finite difference method and finite element method, which have low order precision, the spectral method can achieve the convergence of the exponential order with only a small number of nodes and the discretization of the suitable orthogonal polynomials. In the present work, an efficient high-precision numerical scheme is used in the numerical simulation of the reaction-diffusion equations. In spatial discretization, we construct Chebyshev differentiation matrices based on the Chebyshev points and use these matrices to differentiate the second derivative in the reaction-diffusion equation. After the spatial discretization, we obtain the nonlinear ordinary differential equations. Since the spectral differential matrix obtained by the spectral collocation method is full and cannot use the fast solution of algebraic linear equations, we choose the compact implicit integration factor method to solve the nonlinear ordinary differential equations. By introducing a compact representation for the spectral differential matrix, the compact implicit integration factor method uses matrix exponential operations sequentially in every spatial direction. As a result, exponential matrices which are calculated and stored have small sizes, as those in the one-dimensional problem. This method decouples the exact evaluation of the linear part from the implicit treatment of the nonlinear reaction terms. We only solve a local nonlinear system at each spatial grid point. This method combines with the advantages of the spectral method and the compact implicit integration factor method, i.e., high precision, good stability, and small storage and so on. Numerical simulations show that it can have a great influence on the generation of patterns that the system control parameters take different values under otherwise identical conditions. The numerical results verify the theoretical results.
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Keywords:
- reaction-diffusion equation /
- Gierer-Meinhardt model /
- Turing pattern /
- Chebyshev spectral method
[1] Turing A M 1952 Philos. Trans. R. Soc. Lond. B 2 37
[2] Li X Z, Bai Z G, Li Y, Zhao K, He Y F 2013 Acta Phys. Sin. 62 220503 (in Chinese) [李新政, 白占国, 李燕, 赵昆, 贺亚峰 2013 物理学报 62 220503]
[3] Zhang L, Liu S Y 2007 Appl. Math. Mec. 28 1102 (in Chinese) [张丽, 刘三阳 2007 应用数学和力学 28 1102]
[4] Li B, Wang M X 2008 Appl. Math. Mec. 29 749 (in Chinese) [李波, 王明新 2008 应用数学和力学 29 749]
[5] Hu W Y, Shao Y Z 2014 Acta Phys. Sin. 63 238202 (in Chinese) [胡文勇, 邵元智 2014 物理学报 63 238202]
[6] Peng R Wang M 2007 Sci. China A 50 377
[7] Copie F, Conforti M, Kudlinski A, Mussot A, Trillo S 2016 Phys. Rev. Lett. 116 143901
[8] Tompkins N, Li N, Girabawe C, Heymann M, Ermentrout G B, Epstein I R, Fraden S 2014 Proc. Natl. Acad. Sci. USA 111 4397
[9] Lacitignola D, Bozzini B, Frittelli M, Sgura I 2017 Commun. Nonlinear Sci. Numer. Simul. 48 484
[10] Gaskins D K, Pruc E E, Epstein I R, Dolnik M 2016 Phys. Rev. Lett. 117 056001
[11] Zhang R P, Yu X J, Zhu J, Loula A 2014 Appl. Math. Model. 38 1612
[12] Zhang R P, Zhu J, Loula A, Yu X J 2016 J. Comput. Appl. Math. 302 312
[13] Bai Z G, Dong L F, Li Y H, Fan W L 2011 Acta Phys. Sin. 60 118201 (in Chinese) [白占国, 董丽芳, 李永辉, 范伟丽 2011 物理学报 60 118201]
[14] Zhang R, Zhu J, Yu X, Li M, Loula A F D 2017 Appl. Math. Comput. 310 194
[15] Lv Z Q, Zhang L M, Wang Y S 2014 Chin. Phys. B 23 120203
[16] Wang H 2010 Comput. Phys. Commun. 181 325
[17] Hoz F D L, Vadillo F 2013 Commun. Comput. Phys. 14 1001
[18] Nie Q, Zhang Y T, Zhao R 2006 J. Comput. Phys. 214 521
[19] Nie Q, Wan F Y M, Zhang Y T, Liu X F 2008 J. Comput. Phys. 227 5238
[20] Gierer A, Meinhardt H 1972 Kybernetik 12 30
[21] Ward M J, Wei J 2003 J. Nonlinear Sci. 13 209
[22] Wei J, Winter M 2004 J. Math. Pures Appl. 83 433
[23] Li H X 2015 J. Northeast Normal University 3 26 (in Chinese) [李海侠 2015 东北师大学报 3 26]
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[1] Turing A M 1952 Philos. Trans. R. Soc. Lond. B 2 37
[2] Li X Z, Bai Z G, Li Y, Zhao K, He Y F 2013 Acta Phys. Sin. 62 220503 (in Chinese) [李新政, 白占国, 李燕, 赵昆, 贺亚峰 2013 物理学报 62 220503]
[3] Zhang L, Liu S Y 2007 Appl. Math. Mec. 28 1102 (in Chinese) [张丽, 刘三阳 2007 应用数学和力学 28 1102]
[4] Li B, Wang M X 2008 Appl. Math. Mec. 29 749 (in Chinese) [李波, 王明新 2008 应用数学和力学 29 749]
[5] Hu W Y, Shao Y Z 2014 Acta Phys. Sin. 63 238202 (in Chinese) [胡文勇, 邵元智 2014 物理学报 63 238202]
[6] Peng R Wang M 2007 Sci. China A 50 377
[7] Copie F, Conforti M, Kudlinski A, Mussot A, Trillo S 2016 Phys. Rev. Lett. 116 143901
[8] Tompkins N, Li N, Girabawe C, Heymann M, Ermentrout G B, Epstein I R, Fraden S 2014 Proc. Natl. Acad. Sci. USA 111 4397
[9] Lacitignola D, Bozzini B, Frittelli M, Sgura I 2017 Commun. Nonlinear Sci. Numer. Simul. 48 484
[10] Gaskins D K, Pruc E E, Epstein I R, Dolnik M 2016 Phys. Rev. Lett. 117 056001
[11] Zhang R P, Yu X J, Zhu J, Loula A 2014 Appl. Math. Model. 38 1612
[12] Zhang R P, Zhu J, Loula A, Yu X J 2016 J. Comput. Appl. Math. 302 312
[13] Bai Z G, Dong L F, Li Y H, Fan W L 2011 Acta Phys. Sin. 60 118201 (in Chinese) [白占国, 董丽芳, 李永辉, 范伟丽 2011 物理学报 60 118201]
[14] Zhang R, Zhu J, Yu X, Li M, Loula A F D 2017 Appl. Math. Comput. 310 194
[15] Lv Z Q, Zhang L M, Wang Y S 2014 Chin. Phys. B 23 120203
[16] Wang H 2010 Comput. Phys. Commun. 181 325
[17] Hoz F D L, Vadillo F 2013 Commun. Comput. Phys. 14 1001
[18] Nie Q, Zhang Y T, Zhao R 2006 J. Comput. Phys. 214 521
[19] Nie Q, Wan F Y M, Zhang Y T, Liu X F 2008 J. Comput. Phys. 227 5238
[20] Gierer A, Meinhardt H 1972 Kybernetik 12 30
[21] Ward M J, Wei J 2003 J. Nonlinear Sci. 13 209
[22] Wei J, Winter M 2004 J. Math. Pures Appl. 83 433
[23] Li H X 2015 J. Northeast Normal University 3 26 (in Chinese) [李海侠 2015 东北师大学报 3 26]
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