Geodesic
Functioning modes of asynchronous cellular automata simulating nonlinear spatial dynamics
Prikladnaâ diskretnaâ matematika, no. 1 (2015), pp. 105-119.

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The shift of scientific interest from physical phenomena obeying laws of thermodynamics towards nonlinear dissipative processes containing chemical and biological transformations stimulates a similar turn in mathematical modeling: from differential equation solution to direct and stochastic simulation. A foundation for discrete simulation is the asynchronous cellular automaton – a stochastic analogue of von-Neumann's cellular automaton. For the time being, there is no systematic methodology for constructing asynchronous cellular automata simulating processes composed of many actions transforming a common discrete space. It is not known, how different are simulation results obtained by different ways of composing simple operations for organizing a complex computational process. In the paper, an attempt is made to answer this question by means of performing a series of simulation of three typical reaction-diffusion processes with different asynchronous modes of functioning, and comparative analysis of their evolutions and invariants. The obtained result shows that qualitative character of the process under simulation does not depend on the composition mode, and quantitative differences may be corrected.
Keywords: discrete mathematical modeling, modes of functioning, spacial self organization.
Mots-clés : asynchronous cellular automaton, reaction-diffusion processes 
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O. L. Bandman. Functioning modes of asynchronous cellular automata simulating nonlinear spatial dynamics. Prikladnaâ diskretnaâ matematika, no. 1 (2015), pp. 105-119. http://geodesic.mathdoc.fr/item/PDM_2015_1_a11/

[1] Von Neumann J., Teoriya samovosproizvodjashhihsja avtomatov, Mir Publ., Moscow, 1971, 384 pp. (in Russian)

[2] Metropolis N., Ulam S., “The Monte Carlo method”, Amer. Statist. Assoc., 44:247 (1949), 335–341 | DOI | MR | Zbl

[3] Chatterjee A., Vlaches D. G., “An overview of spatial microscopic and accelerated kinetic Monte-Carlo methods”, J. Computer-Aided Mater. Des., 14 (2007), 253–308 | DOI

[4] Nurminen L., Kuonen A., Kaski K., “Kinetic Monte-Carlo simulation on patterned substrates”, Phys. Rev. B, 63, 29 December (2000), 035407 | DOI

[5] Matveev A. V., Latkin E. I., Elokhin V. I, Gorodetskii V. V., “Turbulent and stripes wave patterns caused by limited $\mathrm{CO}_{ads}$ diffusion during CO oxidation over Pd(110) surface: kinetic Monte Carlo studies”, Chem. Eng. J., 107 (2005), 181–189 | DOI

[6] Bandman O. L., “Diskretnoe modelirovanie fiziko-himicheskih processov.”, Prikladnaya Diskretnaya Matematika, 2009, no. 3, 33–49 (in Russian)

[7] Kireeva A., “Parallel implementation of totalistic cellular automata model of stable patterns formation”, 12th Int. Conf. “Parallel Computing Technologies” (St.-Petersbourg, 2013), LNCS, 7979, 2013, 347–360

[8] Bandman O. L., “Metody kompozicii kletochnyh avtomatov dlja modelirovanija prostranstvennoj dinamiki”, Vestnik Tomskogo gosudarstvennogo universiteta, 2004, Prilozhenie no. 9(1), 183–193 (in Russian)

[9] Achasova S., Bandman O., Markova V., Piskunov S., Parallel Substitution Algorithm. Theory and Application, World Scientific, Singapore, 1994, 180 pp. | Zbl

[10] Kolmogorov A. N., Petrovskij I. G., Piskunov I. S., “Issledovanie uravnenija diffuzii, soedinennoj s vozrastaniem kolichestva veshhestva, i ego primenenie k odnoj biologicheskoj probleme”, MSU Bull., Sec. A, 1937, no. 6, 1–25 (in Russian)

[11] Fisher R. A., The genetical theory of natural selection, Univ. Press, Oxford, 1930, 58 pp. | MR | Zbl

[12] Szakàly T., Lagzi I., Izsàk F., et al., “Stochastic cellular automata modelling excitable systems”, Central Eur. J. Phys., 5:4 (2007), 471–486 | DOI

[13] Van Saarloos W., “Front propagation into unstable states”, Phys. Rep., 386 (2003), 29–222 | DOI | Zbl

[14] Witten T. A. (Jr.), Sander L. M., “Diffusion-limited aggregation, a kinetic critical phenomenon”, Phys. Rev. Lett., 47:19 (1981), 1400–1403 | DOI

[15] Ackland G. J., Tweedie E. S., “Microscopic model of diffusion limited aggregation and electrodeposition in the presence of leveling molecules”, Phys. Rev. E, 73, 26 January (2006), 011606 | DOI

[16] Bogoyavlenskiy V. A., Chernova N. A., “Diffusion-limited aggregation: a relationship between surface thermodynamics and crystal morphology”, Phys. Rev. E, 61:2 (2000), 1629–1633 | DOI

[17] Batty M., Longley P., “Urban growth and form: scaling, fractal geometry, and diffusion-limited aggregation”, Environment and Planning, 21:11 (1989), 1447–1472 | DOI

[18] Svirezhev Ju. M., Nelinejnye volny, dissipativnye struktury i katastrofy v jekologii, Nauka Publ., Moscow, 1987, 368 pp. (in Russian) | MR

[19] Nicolis G., Prigogine I., Self-organization in Nonequilibrium Systems: From Dissipative Structures to Order through Fluctuations, Wiley, N.Y., 1977 | MR | Zbl

[20] Kutson J. D., A survey of the use of cellular automata and cellular automata-like models for simulating a population of biological cells, Graduate Thesis and Dissertations. Paper 10133, Iowa State University, 2011 http://lib.dr.iastate.edu/etd/10133

[21] Chen Q., Mao J., Li W., “Stability analysis of harvesting strategies in a cellular automata based predator – prey model”, Cellular Automata, LNCS, 4173, 2006, 268–376