Construction, mathematical description and coding of reactive lattice-gas cellular automaton

Abstract We construct a reactive lattice-gas cellular automaton (LGCA) for reaction–diffusion systems and provide extensive discussion of its software coding aspects. The software coding aspects provide rationale for some choices in the construction of LGCA which has been inspired by molecular dynamics. Portability of the C language source code, of the data structures, and of the data formats is discussed and explained. We illustrate the ideas behind the development of LGCA and its code by considering a particular reacting system, the Sel'kov model with immobile complexing species. We demonstrate usefulness of LGCA modelling of reactive systems by presenting various simulation results. We compare these results with the standard numerical simulations of reaction–diffusion equations. We conclude the paper by discussing how LGCA methodology can be applied and extended to other contexts.

[1]  Rovinsky,et al.  Chemical instability induced by a differential flow. , 1992, Physical review letters.

[2]  Anna T. Lawniczak,et al.  Lattice gas automata for reactive systems , 1995, comp-gas/9512001.

[3]  I. Epstein,et al.  Modeling of Turing Structures in the Chlorite—Iodide—Malonic Acid—Starch Reaction System , 1991, Science.

[4]  A. Deutsch,et al.  Probabilistic lattice models of collective motion and aggregation: from individual to collective dynamics. , 1999, Mathematical biosciences.

[5]  Tommaso Toffoli,et al.  Cellular automata machines - a new environment for modeling , 1987, MIT Press series in scientific computation.

[6]  Pierre Lallemand,et al.  Lattice Gas Hydrodynamics in Two and Three Dimensions , 1987, Complex Syst..

[7]  B. Hess,et al.  Self-organization in living cells. , 1994, Science.

[8]  Kapral,et al.  Oscillations and waves in a reactive lattice-gas automaton. , 1991, Physical review letters.

[9]  Bastien Chopard,et al.  Cellular automata model for the diffusion equation , 1991 .

[10]  Anna T. Lawniczak,et al.  Pattern Formation and Lattice Gas Automata , 1995 .

[11]  H. Swinney,et al.  Transition from a uniform state to hexagonal and striped Turing patterns , 1991, Nature.

[12]  Peter H. Richter,et al.  Control and Dissipation in Oscillatory Chemical Engines , 1981 .

[13]  Tommaso Toffoli,et al.  Cellular Automata Machines , 1987, Complex Syst..

[14]  DAWSCMV.,et al.  LATTICE METHODS AND THEIR APPLICATIONS TO REACTING SYSTEMS , 1994 .

[15]  E. Sel'kov,et al.  Self-oscillations in glycolysis. 1. A simple kinetic model. , 1968, European journal of biochemistry.

[16]  Frisch,et al.  Lattice gas automata for the Navier-Stokes equations. a new approach to hydrodynamics and turbulence , 1989 .

[17]  Andreas Deutsch.  ORIENTATION-INDUCED PATTERN FORMATION: SWARM DYNAMICS IN A LATTICE-GAS AUTOMATON MODEL , 1996 .

[18]  Anna T. Lawniczak,et al.  Reactive dynamics in a multispecies lattice‐gas automaton , 1992 .

[19]  A. M. Turing,et al.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[20]  Anna T. Lawniczak,et al.  Turing pattern formation in heterogeneous media , 1996 .

[21]  Y. Pomeau,et al.  Lattice-gas automata for the Navier-Stokes equation. , 1986, Physical review letters.

[22]  George Marsaglia,et al.  A random number generator for PC's , 1990 .

[23]  ScienceDirect.  Simulation practice and theory , 2002 .

[24]  P. Lallemand,et al.  Lattice-gas cellular automata, simple models of complex hydrodynamics , 1998 .

[25]  W. J. Freeman,et al.  Alan Turing: The Chemical Basis of Morphogenesis , 1986 .

[26]  Tommaso Toffoli,et al.  Cellular Automata as an Alternative to (Rather than an Approximation of) Differential Equations in M , 1984 .

[27]  Dulos,et al.  Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. , 1990, Physical review letters.

[28]  B. Hasslacher,et al.  Molecular Turing structures in the biochemistry of the cell. , 1993, Chaos.

[29]  Gary D. Doolen.  Lattice Gas Methods For Partial Differential Equations , 1990 .

[30]  G. Doolen,et al.  Lattice gas methods for partial differential equations : a volume of lattice gas reprints and articles, including selected papers from the Workshop on Large Nonlinear Systems, held August, 1987, in Los Alamos, New Mexico , 1990 .

[31]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.