Spatial pattern of discrete and ultradiscrete Gray-Scott model

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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

Abstract
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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 and 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), 325-343. doi: 10.1080/16073606.1992.9631695.
[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, Electron. J. Differ. Equ. Conf., 12, Texas State Univ.San Marcos, Dept. Math., San Marcos, TX, 2005, 12 pp. (electronic).
[3]	A. S. Fokas, E. Papadopoulou and Y. Saridakis, Soliton cellular automata, Physica D, 41 (1990), 297-321. doi: 10.1016/0167-2789(90)90001-6.
[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), 153-180.
[5]	P. Gray and S. K. Scott, Sustained oscillations and other exotic patterns of behaviour in isothermal reactions, J. Phys. Chem., 89 (1985), 22-32. doi: 10.1021/j100247a009.
[6]	W. Kunishima, A. Nishiyama, H. Tanaka and T. Tokihiro, Differential equations for creating complex cellular automaton patterns, J. Phys. Soc. Japan, 73 (2004), 2033-2036. doi: 10.1143/JPSJ.73.2033.
[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), 287-295. doi: 10.1016/S0375-9601(96)00899-7.
[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), 371-396. doi: 10.1016/0378-4754(95)00044-5.
[9]	M. Murata, Exact solutions with two parameters for an ultradiscrete Painlevé equation of type $A_6^{(1)}$, SIGMA, 7 (2011), Paper059, 15 pp.
[10]	M. Murata, Tropical discretization: Ultradiscrete Fisher-KPP equation and ultradiscrete Allen-Cahn equation, J. Difference. Equ. Appl., 19 (2013), 1008-1021. doi: 10.1080/10236198.2012.705834.
[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), L27-L34. doi: 10.1088/0305-4470/39/1/L04.
[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), 315203, 15 pp. doi: 10.1088/1751-8113/43/31/315203.
[13]	A. Nagai, D. Takahashi and T. Tokihiro, Soliton cellular automaton, toda molecule equation and sorting algorithm, Phys. Lett. A, 255 (1999), 265-271. doi: 10.1016/S0375-9601(99)00162-0.
[14]	Y. Nishiura and D. Ueyama, A skeleton structure of self-replicating dynamics, Physica D, 130 (1999), 73-104. doi: 10.1016/S0167-2789(99)00010-X.
[15]	Y. Nishiura and D. Ueyama, Spatio-temporal chaos for the Gray-Scott model, Physica D, 150 (2001), 137-162. doi: 10.1016/S0167-2789(00)00214-1.
[16]	J. K. Park, K. Steiglitz and W. P. Thurston, Soliton-like behavior in automata, Physica D, 19 (1986), 423-432. doi: 10.1016/0167-2789(86)90068-0.
[17]	J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. doi: 10.1126/science.261.5118.189.
[18]	D. Takahashi and J. Satsuma, A soliton cellular automaton, J. Phys. Soc. Japan, 59 (1990), 3514-3519. doi: 10.1143/JPSJ.59.3514.
[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), 10715-10726. doi: 10.1088/0305-4470/34/48/333.
[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), 034002, 5 pp. doi: 10.1143/JPSJ.78.034002.
[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), 3247-3250. doi: 10.1103/PhysRevLett.76.3247.
[22]	S. Wolfram, Twenty Problems in the Theory of Cellular Automata, Physica Scripta., T9 (1985), 170-183. doi: 10.1088/0031-8949/1985/T9/029.

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), 325-343. doi: 10.1080/16073606.1992.9631695.
[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, Electron. J. Differ. Equ. Conf., 12, Texas State Univ.San Marcos, Dept. Math., San Marcos, TX, 2005, 12 pp. (electronic).
[3]	A. S. Fokas, E. Papadopoulou and Y. Saridakis, Soliton cellular automata, Physica D, 41 (1990), 297-321. doi: 10.1016/0167-2789(90)90001-6.
[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), 153-180.
[5]	P. Gray and S. K. Scott, Sustained oscillations and other exotic patterns of behaviour in isothermal reactions, J. Phys. Chem., 89 (1985), 22-32. doi: 10.1021/j100247a009.
[6]	W. Kunishima, A. Nishiyama, H. Tanaka and T. Tokihiro, Differential equations for creating complex cellular automaton patterns, J. Phys. Soc. Japan, 73 (2004), 2033-2036. doi: 10.1143/JPSJ.73.2033.
[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), 287-295. doi: 10.1016/S0375-9601(96)00899-7.
[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), 371-396. doi: 10.1016/0378-4754(95)00044-5.
[9]	M. Murata, Exact solutions with two parameters for an ultradiscrete Painlevé equation of type $A_6^{(1)}$, SIGMA, 7 (2011), Paper059, 15 pp.
[10]	M. Murata, Tropical discretization: Ultradiscrete Fisher-KPP equation and ultradiscrete Allen-Cahn equation, J. Difference. Equ. Appl., 19 (2013), 1008-1021. doi: 10.1080/10236198.2012.705834.
[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), L27-L34. doi: 10.1088/0305-4470/39/1/L04.
[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), 315203, 15 pp. doi: 10.1088/1751-8113/43/31/315203.
[13]	A. Nagai, D. Takahashi and T. Tokihiro, Soliton cellular automaton, toda molecule equation and sorting algorithm, Phys. Lett. A, 255 (1999), 265-271. doi: 10.1016/S0375-9601(99)00162-0.
[14]	Y. Nishiura and D. Ueyama, A skeleton structure of self-replicating dynamics, Physica D, 130 (1999), 73-104. doi: 10.1016/S0167-2789(99)00010-X.
[15]	Y. Nishiura and D. Ueyama, Spatio-temporal chaos for the Gray-Scott model, Physica D, 150 (2001), 137-162. doi: 10.1016/S0167-2789(00)00214-1.
[16]	J. K. Park, K. Steiglitz and W. P. Thurston, Soliton-like behavior in automata, Physica D, 19 (1986), 423-432. doi: 10.1016/0167-2789(86)90068-0.
[17]	J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. doi: 10.1126/science.261.5118.189.
[18]	D. Takahashi and J. Satsuma, A soliton cellular automaton, J. Phys. Soc. Japan, 59 (1990), 3514-3519. doi: 10.1143/JPSJ.59.3514.
[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), 10715-10726. doi: 10.1088/0305-4470/34/48/333.
[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), 034002, 5 pp. doi: 10.1143/JPSJ.78.034002.
[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), 3247-3250. doi: 10.1103/PhysRevLett.76.3247.
[22]	S. Wolfram, Twenty Problems in the Theory of Cellular Automata, Physica Scripta., T9 (1985), 170-183. doi: 10.1088/0031-8949/1985/T9/029.

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