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The linear noise approximation for reaction-diffusion systems on networks

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Abstract

Stochastic reaction-diffusion models can be analytically studied on complex networks using the linear noise approximation. This is illustrated through the use of a specific stochastic model, which displays travelling waves in its deterministic limit. The role of stochastic fluctuations is investigated and shown to drive the emergence of stochastic waves beyond the region of the instability predicted from the deterministic theory. Simulations are performed to test the theoretical results and are analyzed via a generalized Fourier transform algorithm. This transform is defined using the eigenvectors of the discrete Laplacian defined on the network. A peak in the numerical power spectrum of the fluctuations is observed in quantitative agreement with the theoretical predictions.

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References

  1. M. Mimura, J.D. Murray, J. Theor. Biol. 75, 249 (1978)

    Article  MathSciNet  Google Scholar 

  2. J.L. Maron, S. Harrison, Science 278, 1619 (1997)

    Article  ADS  Google Scholar 

  3. M. Baurmann, T. Gross, U. Feudel, J. Theor. Biol. 245, 220 (2007)

    Article  MathSciNet  Google Scholar 

  4. M. Rietkerk, J. van de Koppel, Trends Ecol. Evol. 23, 169 (2008)

    Article  Google Scholar 

  5. H. Meinhardt, A. Gierer, BioEssays 22, 753 (2000)

    Article  Google Scholar 

  6. M.P. Harris, S. Williamson, J.F. Fallon, H. Meinhardt, R.O. Prum, Proc. Natl. Acad. Sci. USA 102, 11734 (2005)

    Article  ADS  Google Scholar 

  7. P.K. Maini, R.E. Baker, C.-M. Chuong, Science 314, 1397 (2006)

    Article  Google Scholar 

  8. S.A. Newman, R. Bhat, Birth Defects Res. C 81, 305 (2007)

    Article  Google Scholar 

  9. T. Miura, K. Shiota, Dev. Dyn. 217, 241 (2000)

    Article  Google Scholar 

  10. I. Prigogine, R. Lefever, J. Chem. Phys. 48, 1695 (1968)

    Article  ADS  Google Scholar 

  11. V. Castets, E. Dulos, J. Boissonade, P. De Kepper, Phys. Rev. Lett. 64, 2953 (1990)

    Article  ADS  Google Scholar 

  12. Q. Ouyang, H.L. Swinney, Nature 352, 610 (1991)

    Article  ADS  Google Scholar 

  13. A.M. Turing, Phil. Trans. R. Soc. Lond. B 237, 37 (1952)

    Article  ADS  Google Scholar 

  14. H. Nakao, A.S. Mikhailov, Nat. Phys. 6, 544 (2010)

    Article  Google Scholar 

  15. R. Pastor-Satorras, A. Vespignani, Nat. Phys. 6, 480 (2010)

    Article  Google Scholar 

  16. V. Colizza, A. Barrat, M. Barthelemy, A. Vespignani, Proc. Natl. Acad. Sci. USA 103, 2015 (2006)

    Article  ADS  Google Scholar 

  17. R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001)

    Article  ADS  Google Scholar 

  18. V. Colizza, R. Pastor-Satorras, A. Vespignani, Nat. Phys. 3, 276 (2007)

    Article  Google Scholar 

  19. M. Asslani, F. Di Patti, D. Fanelli, Phys. Rev. E 86, 046105 (2102)

    Article  Google Scholar 

  20. A.J. McKane, T.J. Newman, Phys. Rev. Lett. 94, 218102 (2005)

    Article  ADS  Google Scholar 

  21. T. Dauxois, F. Di Patti, D. Fanelli, A.J. McKane, Phys. Rev. E 79, 036112 (2009)

    Article  ADS  Google Scholar 

  22. T. Butler, N. Goldenfeld, Phys. Rev. E 80, 030902(R) (2009)

    Article  ADS  Google Scholar 

  23. T. Biancalani, D. Fanelli, F. Di Patti, Phys. Rev. E 81, 046215 (2010)

    Article  ADS  Google Scholar 

  24. T.E. Woolley, R.E. Baker, E.A. Gaffney, P.K. Maini, Phys. Rev. E 84, 046216 (2011)

    Article  ADS  Google Scholar 

  25. T. Biancalani, T. Galla, A.J. McKane, Phys. Rev. E 84, 026201 (2011)

    Article  ADS  Google Scholar 

  26. I. Hanski, Metapopulation Ecology (Oxford University Press, Oxford, 1999)

  27. T. Maruyama, Stochastic Problems in Population Genetics, Lectures in Biomathematics 17 (Springer, Berlin, 1977)

  28. G. Rozhnova, A. Nunes, A.J. McKane, Phys. Rev. E 84, 051919 (2011)

    Article  ADS  Google Scholar 

  29. J.D. Challenger, A.J. McKane, Phys. Rev. E 88, 012107 (2013)

    Article  ADS  Google Scholar 

  30. A.M. Zhabotinsky, M. Dolnik, I.R. Epstein, J. Chem. Phys. 103, 10306 (1995)

    Article  ADS  Google Scholar 

  31. A-L. Barabási, R. Albert, Science 286, 509 (1999)

    Article  MathSciNet  ADS  Google Scholar 

  32. D.T. Gillespie, J. Comp. Phys. 22, 403 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  33. N.E. Kouvaris, H. Kori, A.S. Mikhailov, PLoS ONE 7, e45029 (2012)

    Article  ADS  Google Scholar 

  34. N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North Holland Pubs., Amsterdam, 1992)

  35. R. Burioni, S. Chibbaro, D. Vergni, A. Vulpiani, Phys. Rev. E 86, 055101(R) (2012)

    Article  ADS  Google Scholar 

  36. I. Simonsen, K.A. Eriksen, S. Maslov, K. Sneppen, Physica A 336, 163 (2004)

    Article  ADS  Google Scholar 

  37. M. Cross, H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems (Cambridge University Press, Cambridge, 2009)

  38. P.N. McGraw, M. Menzinger, Phys. Rev. E 77, 031102 (2008)

    Article  ADS  Google Scholar 

  39. C.A. Lugo, A.J. McKane, Phys. Rev. E 78, 051911 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  40. P. de Anna, F. Di Patti, D. Fanelli, A.J. McKane, T. Dauxois, Phys. Rev. E 81, 056110 (2010)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, via Valleggio 11, 22100, Como, Italy

    Malbor Asllani

  2. Theoretical Physics, School of Physics and Astronomy, University of Manchester, M13 9, PL Manchester, UK

    Tommaso Biancalani & Alan J. McKane

  3. Dipartimento di Fisica e Astronomia, Università di Firenze and INFN, via Sansone 1, 50019 Sesto Fiorentino, Florence, Italy

    Duccio Fanelli

Authors
  1. Malbor Asllani
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  2. Tommaso Biancalani
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  3. Duccio Fanelli
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  4. Alan J. McKane
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Correspondence to Duccio Fanelli.

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Asllani, M., Biancalani, T., Fanelli, D. et al. The linear noise approximation for reaction-diffusion systems on networks. Eur. Phys. J. B 86, 476 (2013). https://doi.org/10.1140/epjb/e2013-40570-8

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  • DOI: https://doi.org/10.1140/epjb/e2013-40570-8

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