Fractional diffusion-reaction stochastic simulations.

A novel method is presented for the simulation of a discrete state space, continuous time Markov process subject to fractional diffusion. The method is based on Lie-Trotter operator splitting of the diffusion and reaction terms in the master equation. The diffusion term follows a multinomial distribution governed by a kernel that is the discretized solution of the fractional diffusion equation. The algorithm is validated and simulations are provided for the Fisher-KPP wavefront. It is shown that the wave speed is dictated by the order of the fractional derivative, where lower values result in a faster wave than in the case of classical diffusion. Since many physical processes deviate from classical diffusion, fractional diffusion methods are necessary for accurate simulations.

[1]  David Bernstein,et al.  Simulating mesoscopic reaction-diffusion systems using the Gillespie algorithm. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Sotiria Lampoudi,et al.  The multinomial simulation algorithm for discrete stochastic simulation of reaction-diffusion systems. , 2009, The Journal of chemical physics.

[3]  Daniel T Gillespie,et al.  Stochastic simulation of chemical kinetics. , 2007, Annual review of physical chemistry.

[4]  José Canosa,et al.  Diffusion in Nonlinear Multiplicative Media , 1969 .

[5]  S. Ritchie,et al.  Successful establishment of Wolbachia in Aedes populations to suppress dengue transmission , 2011, Nature.

[6]  Linda R Petzold,et al.  Efficient step size selection for the tau-leaping simulation method. , 2006, The Journal of chemical physics.

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

[8]  Anne Auger,et al.  R-leaping: accelerating the stochastic simulation algorithm by reaction leaps. , 2006, The Journal of chemical physics.

[9]  W. Huisinga,et al.  Solving the chemical master equation for monomolecular reaction systems analytically , 2006, Journal of mathematical biology.

[10]  Gianni De Fabritiis,et al.  Discrete random walk models for symmetric Lévy–Feller diffusion processes , 1999 .

[11]  Solomon,et al.  Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow. , 1993, Physical review letters.

[12]  Stanley,et al.  Stochastic process with ultraslow convergence to a Gaussian: The truncated Lévy flight. , 1994, Physical review letters.

[13]  Andreas Hellander,et al.  Coupled Mesoscopic and Microscopic Simulation of Stochastic Reaction-Diffusion Processes in Mixed Dimensions , 2012, Multiscale Model. Simul..

[14]  D. Gillespie.  A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions , 1976 .

[15]  V E Lynch,et al.  Front dynamics in reaction-diffusion systems with Levy flights: a fractional diffusion approach. , 2002, Physical review letters.

[16]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[17]  R. Gorenflo,et al.  Discrete random walk models for space-time fractional diffusion , 2002, cond-mat/0702072.

[18]  P Grigolini,et al.  Lévy diffusion as an effect of sporadic randomness. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[19]  J. Elf,et al.  Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases. , 2004, Systems biology.

[20]  R. Fisher.  THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .

[21]  Brian Drawert,et al.  The diffusive finite state projection algorithm for efficient simulation of the stochastic reaction-diffusion master equation. , 2010, The Journal of chemical physics.

[22]  Petros Koumoutsakos,et al.  Adaptive mesh refinement for stochastic reaction-diffusion processes , 2011, J. Comput. Phys..

[23]  Ott,et al.  Anomalous diffusion in "living polymers": A genuine Levy flight? , 1990, Physical review letters.

[24]  Klaus Dietz,et al.  Epidemics and Rumours: A Survey , 1967 .

[25]  T. Geisel,et al.  The scaling laws of human travel , 2006, Nature.

[26]  Eric Deleersnijder,et al.  Front dynamics in fractional-order epidemic models. , 2011, Journal of theoretical biology.

[27]  D. Gillespie.  Exact Stochastic Simulation of Coupled Chemical Reactions , 1977 .

[28]  Tobias Jahnke,et al.  Efficient simulation of discrete stochastic reaction systems with a splitting method , 2010 .

[29]  D. Gillespie.  Approximate accelerated stochastic simulation of chemically reacting systems , 2001 .

[30]  D. Helbing.  Traffic and related self-driven many-particle systems , 2000, cond-mat/0012229.

[31]  O. Bychuk,et al.  Anomalous diffusion of surface-active species at liquid-fluid and liquid-solid interfaces , 1994 .

[32]  Hans Engler,et al.  On the Speed of Spread for Fractional Reaction-Diffusion Equations , 2009, 0908.0024.

[33]  Reuter,et al.  Surfactants in epitaxial growth. , 1989, Physical review letters.

[34]  Francesco Petruccione,et al.  Fluctuation effects on wave propagation in a reaction-diffusion process , 1994 .

[35]  P. A. Prince,et al.  Lévy flight search patterns of wandering albatrosses , 1996, Nature.

[36]  Andreas Hellander,et al.  Hybrid method for the chemical master equation , 2007, J. Comput. Phys..