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Traveling waves for the Lotka-Volterra predator-prey system without diffusion of the predator
Spatial pattern of discrete and ultradiscrete Gray-Scott model
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 |
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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