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January  2015, 20(1): 173-187. doi: 10.3934/dcdsb.2015.20.173

Spatial pattern of discrete and ultradiscrete Gray-Scott model

Keisuke Matsuya 1, and  Mikio Murata 2,

1. 

Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan
2. 

Institute of Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Nakacho Koganei-shi, Tokyo 184-8588, Japan

Received  April 2013 Revised  December 2013 Published  November 2014

Ultradiscretization is a limiting procedure transforming a given difference equation into a cellular automaton. In addition the cellular automaton constructed by this procedure preserves the essential properties of the original equation, such as the structure of exact solutions for integrable equations. In this article, we propose a discretization and an ultradiscretization of Gray-Scott model which is not an integrable system and which gives various spatial patterns with appropriate initial data and parameters. The resulting systems give a traveling pulse and a self-replication pattern with appropriate initial data and parameters. The ultradiscrete system is directly related to the elementary cellular automaton Rule 90 which gives a Sierpinski gasket pattern. A $(2+1)$D ultradiscrete Gray-Scott model that gives a ring pattern and a self-replication pattern are also constructed.
Keywords: Gray-Scott model, cellular automaton, ultradiscretization, pattern formation., discretization.
Mathematics Subject Classification: Primary: 39A12; Secondary: 35K5.
Citation: Keisuke Matsuya, Mikio Murata. Spatial pattern of discrete and ultradiscrete Gray-Scott model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 173-187. doi: 10.3934/dcdsb.2015.20.173
References:
[1]

M. J. Ablowitz, J. M. Keiser and L. A. Takhtajan, Stable, multi-state, time-reversible cellular automata with rich particle content,, Quaestiones Math., 15 (1992), 325.  doi: 10.1080/16073606.1992.9631695.  Google Scholar

[2]

M. E. Alexander and S. M. Moghadas, $\mathcalO(l)$ shift in Hopf bifurcations for a class of non-standard numerical schemes,, in Proceedings of the 2004 Conference on Differential Equations and Applications in Mathematical Biology, (2004).   Google Scholar

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A. S. Fokas, E. Papadopoulou and Y. Saridakis, Soliton cellular automata,, Physica D, 41 (1990), 297.  doi: 10.1016/0167-2789(90)90001-6.  Google Scholar

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A. S. Fokas, E. Papadopoulou, Y. Saridakis and M. J. Ablowitz, Interaction of simple particles in soliton cellular automata,, Stud. Appl. Math., 81 (1989), 153.   Google Scholar

[5]

P. Gray and S. K. Scott, Sustained oscillations and other exotic patterns of behaviour in isothermal reactions,, J. Phys. Chem., 89 (1985), 22.  doi: 10.1021/j100247a009.  Google Scholar

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W. Kunishima, A. Nishiyama, H. Tanaka and T. Tokihiro, Differential equations for creating complex cellular automaton patterns,, J. Phys. Soc. Japan, 73 (2004), 2033.  doi: 10.1143/JPSJ.73.2033.  Google Scholar

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J. Matsukidaira, J. Satsuma, D. Takahashi, T. Tokihiro and M. Torii, Toda-type cellular automaton and its N-soliton solution,, Phys. Lett. A, 225 (1997), 287.  doi: 10.1016/S0375-9601(96)00899-7.  Google Scholar

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W. Mazin, K. E. Rasmussen, E. Mosekilde, P. Borckmans and G. Dewel, Pattern formation in the bistable Gray-Scott model,, Math. Comput. Simul., 40 (1996), 371.  doi: 10.1016/0378-4754(95)00044-5.  Google Scholar

[9]

M. Murata, Exact solutions with two parameters for an ultradiscrete Painlevé equation of type $A_6^{(1)}$,, SIGMA, 7 (2011).   Google Scholar

[10]

M. Murata, Tropical discretization: Ultradiscrete Fisher-KPP equation and ultradiscrete Allen-Cahn equation,, J. Difference. Equ. Appl., 19 (2013), 1008.  doi: 10.1080/10236198.2012.705834.  Google Scholar

[11]

M. Murata, S. Isojima, A. Nobe and J. Satsuma, Exact solutions for discrete and ultradiscrete modified KdV equations and their relation to box-ball systems,, J. Phys. A Math. Gen., 39 (2006).  doi: 10.1088/0305-4470/39/1/L04.  Google Scholar

[12]

M. Murata, J. Satsuma, A. Ramani and B. Grammaticos, How to discretize differential systems in a systematic way,, J. Phys. A: Math. Theor., 43 (2010).  doi: 10.1088/1751-8113/43/31/315203.  Google Scholar

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A. Nagai, D. Takahashi and T. Tokihiro, Soliton cellular automaton, toda molecule equation and sorting algorithm,, Phys. Lett. A, 255 (1999), 265.  doi: 10.1016/S0375-9601(99)00162-0.  Google Scholar

[14]

Y. Nishiura and D. Ueyama, A skeleton structure of self-replicating dynamics,, Physica D, 130 (1999), 73.  doi: 10.1016/S0167-2789(99)00010-X.  Google Scholar

[15]

Y. Nishiura and D. Ueyama, Spatio-temporal chaos for the Gray-Scott model,, Physica D, 150 (2001), 137.  doi: 10.1016/S0167-2789(00)00214-1.  Google Scholar

[16]

J. K. Park, K. Steiglitz and W. P. Thurston, Soliton-like behavior in automata,, Physica D, 19 (1986), 423.  doi: 10.1016/0167-2789(86)90068-0.  Google Scholar

[17]

J. E. Pearson, Complex patterns in a simple system,, Science, 261 (1993), 189.  doi: 10.1126/science.261.5118.189.  Google Scholar

[18]

D. Takahashi and J. Satsuma, A soliton cellular automaton,, J. Phys. Soc. Japan, 59 (1990), 3514.  doi: 10.1143/JPSJ.59.3514.  Google Scholar

[19]

D. Takahashi, A. Shida and M. Usami, On the pattern formation mechanism of (2+1)D max-plus models,, J. Phys. A: Math. Gen., 34 (2001), 10715.  doi: 10.1088/0305-4470/34/48/333.  Google Scholar

[20]

H. Tanaka, A. Nakajima, A. Nishiyama and T. Tokihiro, Derivation of a differential equation exhibiting replicative time-evolution patterns by inverse ultra-discretization,, J. Phys. Soc. Japan, 78 (2009).  doi: 10.1143/JPSJ.78.034002.  Google Scholar

[21]

T. Tokihiro, D. Takahashi, J. Matsukidaira and J. Satsuma, From soliton equations to integrable cellular automata through a limiting procedure,, Phys. Rev. Lett., 76 (1996), 3247.  doi: 10.1103/PhysRevLett.76.3247.  Google Scholar

[22]

S. Wolfram, Twenty Problems in the Theory of Cellular Automata,, Physica Scripta., T9 (1985), 170.  doi: 10.1088/0031-8949/1985/T9/029.  Google Scholar

show all references

References:
[1]

M. J. Ablowitz, J. M. Keiser and L. A. Takhtajan, Stable, multi-state, time-reversible cellular automata with rich particle content,, Quaestiones Math., 15 (1992), 325.  doi: 10.1080/16073606.1992.9631695.  Google Scholar

[2]

M. E. Alexander and S. M. Moghadas, $\mathcalO(l)$ shift in Hopf bifurcations for a class of non-standard numerical schemes,, in Proceedings of the 2004 Conference on Differential Equations and Applications in Mathematical Biology, (2004).   Google Scholar

[3]

A. S. Fokas, E. Papadopoulou and Y. Saridakis, Soliton cellular automata,, Physica D, 41 (1990), 297.  doi: 10.1016/0167-2789(90)90001-6.  Google Scholar

[4]

A. S. Fokas, E. Papadopoulou, Y. Saridakis and M. J. Ablowitz, Interaction of simple particles in soliton cellular automata,, Stud. Appl. Math., 81 (1989), 153.   Google Scholar

[5]

P. Gray and S. K. Scott, Sustained oscillations and other exotic patterns of behaviour in isothermal reactions,, J. Phys. Chem., 89 (1985), 22.  doi: 10.1021/j100247a009.  Google Scholar

[6]

W. Kunishima, A. Nishiyama, H. Tanaka and T. Tokihiro, Differential equations for creating complex cellular automaton patterns,, J. Phys. Soc. Japan, 73 (2004), 2033.  doi: 10.1143/JPSJ.73.2033.  Google Scholar

[7]

J. Matsukidaira, J. Satsuma, D. Takahashi, T. Tokihiro and M. Torii, Toda-type cellular automaton and its N-soliton solution,, Phys. Lett. A, 225 (1997), 287.  doi: 10.1016/S0375-9601(96)00899-7.  Google Scholar

[8]

W. Mazin, K. E. Rasmussen, E. Mosekilde, P. Borckmans and G. Dewel, Pattern formation in the bistable Gray-Scott model,, Math. Comput. Simul., 40 (1996), 371.  doi: 10.1016/0378-4754(95)00044-5.  Google Scholar

[9]

M. Murata, Exact solutions with two parameters for an ultradiscrete Painlevé equation of type $A_6^{(1)}$,, SIGMA, 7 (2011).   Google Scholar

[10]

M. Murata, Tropical discretization: Ultradiscrete Fisher-KPP equation and ultradiscrete Allen-Cahn equation,, J. Difference. Equ. Appl., 19 (2013), 1008.  doi: 10.1080/10236198.2012.705834.  Google Scholar

[11]

M. Murata, S. Isojima, A. Nobe and J. Satsuma, Exact solutions for discrete and ultradiscrete modified KdV equations and their relation to box-ball systems,, J. Phys. A Math. Gen., 39 (2006).  doi: 10.1088/0305-4470/39/1/L04.  Google Scholar

[12]

M. Murata, J. Satsuma, A. Ramani and B. Grammaticos, How to discretize differential systems in a systematic way,, J. Phys. A: Math. Theor., 43 (2010).  doi: 10.1088/1751-8113/43/31/315203.  Google Scholar

[13]

A. Nagai, D. Takahashi and T. Tokihiro, Soliton cellular automaton, toda molecule equation and sorting algorithm,, Phys. Lett. A, 255 (1999), 265.  doi: 10.1016/S0375-9601(99)00162-0.  Google Scholar

[14]

Y. Nishiura and D. Ueyama, A skeleton structure of self-replicating dynamics,, Physica D, 130 (1999), 73.  doi: 10.1016/S0167-2789(99)00010-X.  Google Scholar

[15]

Y. Nishiura and D. Ueyama, Spatio-temporal chaos for the Gray-Scott model,, Physica D, 150 (2001), 137.  doi: 10.1016/S0167-2789(00)00214-1.  Google Scholar

[16]

J. K. Park, K. Steiglitz and W. P. Thurston, Soliton-like behavior in automata,, Physica D, 19 (1986), 423.  doi: 10.1016/0167-2789(86)90068-0.  Google Scholar

[17]

J. E. Pearson, Complex patterns in a simple system,, Science, 261 (1993), 189.  doi: 10.1126/science.261.5118.189.  Google Scholar

[18]

D. Takahashi and J. Satsuma, A soliton cellular automaton,, J. Phys. Soc. Japan, 59 (1990), 3514.  doi: 10.1143/JPSJ.59.3514.  Google Scholar

[19]

D. Takahashi, A. Shida and M. Usami, On the pattern formation mechanism of (2+1)D max-plus models,, J. Phys. A: Math. Gen., 34 (2001), 10715.  doi: 10.1088/0305-4470/34/48/333.  Google Scholar

[20]

H. Tanaka, A. Nakajima, A. Nishiyama and T. Tokihiro, Derivation of a differential equation exhibiting replicative time-evolution patterns by inverse ultra-discretization,, J. Phys. Soc. Japan, 78 (2009).  doi: 10.1143/JPSJ.78.034002.  Google Scholar

[21]

T. Tokihiro, D. Takahashi, J. Matsukidaira and J. Satsuma, From soliton equations to integrable cellular automata through a limiting procedure,, Phys. Rev. Lett., 76 (1996), 3247.  doi: 10.1103/PhysRevLett.76.3247.  Google Scholar

[22]

S. Wolfram, Twenty Problems in the Theory of Cellular Automata,, Physica Scripta., T9 (1985), 170.  doi: 10.1088/0031-8949/1985/T9/029.  Google Scholar

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