# American Institute of Mathematical Sciences

January  2012, 11(1): 189-207. doi: 10.3934/cpaa.2012.11.189

## An effective design method to produce stationary chemical reaction-diffusion patterns

 1 Centre De Recherche Paul Pascal, CNRS, Av. Schweitzer, 33600 Pessac, France 2 Institute of Chemistry, Laboratory of Nonlinear Chemical Dyanmics, Eötvös L. University, P.O. Box H-1518 Budapest 112, Hungary

Received  March 2010 Revised  July 2010 Published  September 2010

We present a semi-empirical experimental design method to produce nontrivial chemical reaction-diffusion patterns in open reactors. We specially focus on the development of stationary patterns. The method is based on autoactivated reactions that produces spatial bistability, the addition of an independent antagonist reaction to produce spatio-temporal oscillations, and the introduction of a low mobility complexing agent that rapidly and reversibly binds the main autoactivatory species. The method is presented in formal way. Actual experimental results are used for illustration. We point out the open problems of the mathematical description: they relate to the boundary conditions, to the dimensionality of the system, and to the coupled time- and space-scale changes induced by the complexing agent.
Citation: Patrick De Kepper, István Szalai. An effective design method to produce stationary chemical reaction-diffusion patterns. Communications on Pure & Applied Analysis, 2012, 11 (1) : 189-207. doi: 10.3934/cpaa.2012.11.189
##### References:
 [1] A. M. Turing, The chemical basis of morphogenesis,, Phil. Trans. Roy. Soc. Ser B, 237 (1952), 37. doi: 10.1098/rstb.1952.0012. Google Scholar [2] J. D. Murray, "Mathematical Biology I-II,'', 3rd edition, (2002). doi: 10.1007/b98868. Google Scholar [3] H. Ikeda, M. Mimura and Y. Nishiura, Global bifurcation phenomena of travelling wave solutions for some bistable reaction-diffusion systems,, Nonl. Anal. TMA, 13 (1989). doi: 10.1016/0362-546X(89)90061-8. Google Scholar [4] A. Hagberg and E. Meron, Complex patterns in reaction-diffusion systems: a tale of two front instabilities,, Chaos, 4 (1994). doi: 10.1063/1.166047. Google Scholar [5] G. Nicolis and I. Prigogine, "Self Organization in Nonequilibrium Systems,'', Wiley, (1977). Google Scholar [6] H. Meinhardt, "Models of Biological Pattern Formation,'', Academic Press, (1982). Google Scholar [7] E. Ammelt, Y. A. Astrov and H. G. Purwins, Stripe Turing structures in a two-dimensional gas discharge system,, Physical Review E, 55 (2001), 6731. doi: 10.1103/PhysRevE.55.6731. Google Scholar [8] L. A. Lugiato, C. Oldano and L. M. Narducci, Cooperative frequency locking and stationary spatial structures in lasers,, Journal of the Optical Society of America B -Optical Physics, 5 (1988), 879. doi: 10.1364/JOSAB.5.000879. Google Scholar [9] S. A. Levin, The problem of pattern and scale in ecology,, Ecology, 73 (1992), 1943. doi: 10.2307/1941447. Google Scholar [10] F. Borgogno, P. D'Odorico and F. Laio, et al, Mathematical models of vegetation pattern formation in ecohydrology,, Reviews of Geophysics, 47 (2009). doi: 10.1029/2007RG000256. Google Scholar [11] P. De Kepper, J. Boissonade and I. Szalai, From sustained oscillations to stationnary reaction-diffusion patterns,, in, (2009), 978. doi: 10.1007/978-90-481-2993-5\_1. Google Scholar [12] V. Castets, E. Dulos, J. Boissonade and P. De Kepper, Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern,, Phys. Rev. Lett., 64 (1990), 2953. doi: 10.1103/PhysRevLett.64.2953. Google Scholar [13] K. J. Lee, W. D. McCormick, Q. Ouyang and H. L. Swinney, Pattern formation by interacting chemical fronts,, Science, 261 (1993), 192. doi: 10.1126/science.261.5118.192. Google Scholar [14] E. Dulos, P. W. Davies, B. Rudovics and P. De Kepper, From quasi-2D to 3D Turing patterns in ramped systems,, Physica D, 98 (1996), 53. doi: 10.1016/0167-2789(96)00072-3. Google Scholar [15] P. De Kepper, J.-J. Perraud, B. Rudovics and E. Dulos, Experimental study of stationary Turing patterns and their interaction with traveling waves in a chemical system,, Int. J. Bifurcation and Chaos, 6 (1994), 1077. doi: 10.1142/S0218127494000915. Google Scholar [16] K. J. Lee, W. D. McCormick, H. L. Swinney and J. E. Pearson, Experimental observation of self-replicating spots in a reaction-diffusion system,, Nature, 369 (1994), 215. doi: 10.1038/369215a0. Google Scholar [17] J. Horváth, I. Szalai and P. De Kepper, An experimental design method leading to chemical turing patterns,, Science, 324 (2009), 772. doi: 10.1126/science.1169973. Google Scholar [18] V. K. Vanag and I. R. Epstein, Pattern formation in a tunable medium: The Belousov-Zhabotinsky reaction in an aerosol ot microemulsion,, Phys. Rev. Lett., 87 (2001). doi: 10.1103/PhysRevLett.87.228301. Google Scholar [19] P. Érdi and J. Tóth, "Mathematical Models of Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models,'', Princeton University Press, (1989). Google Scholar [20] J. Boissonade and P. De Kepper, Transitions from bistability to limit cycle oscillations. Theoretical analysis and experimental evidence in an open chemical system,, J. Phys. Chem., 84 (1980), 501. doi: 10.1021/j100442a009. Google Scholar [21] G. Dangelmayr and J. Guckenheimer, On a four parameter family of planar vector fields,, Archive for Rational Mechanics and Analysis, 97 (1987), 321. doi: 10.1007/BF00280410. Google Scholar [22] I. R. Epstein and J. A. Pojman, "An Introduction to Nonlinear Chemical Dynamics Oscillations, Waves, Patterns, and Chaos,'', Oxford University Press, (1998). Google Scholar [23] P. Gray and S. K. Scott, "Chemical Oscillations and Instabilities,'', Clarendon Press, (1990). Google Scholar [24] P. Blanchedeau and J. Boissonade, Resolving an experimental paradox in open spatial reactors: The role of spatial bistability,, Phys. Rev. Lett., 81 (1998), 5007. doi: 10.1103/PhysRevLett.81.5007. Google Scholar [25] K. Benyaich, T. Erneux T, S. Metens, S, S. Villain and P. Borckmans, Spatio-temporal behaviors of a clock reaction in an open gel reactor,, Chaos, 16 (2006). doi: 10.1063/1.2219703 . Google Scholar [26] J. Boissonade, E. Dulos, F. Gauffre, M. N. Kuperman and P. De Kepper, Spatial bistability and waves in a reaction with acid autocatalysis,, Faraday Discuss., 120 (2001), 353. doi: 10.1039/b103240m. Google Scholar [27] P. Blanchedeau, J. Boissonade and P. De Kepper, Theoretical and experimental studies of spatial bistability in the chlorine-dioxide-iodide reaction,, Physica D, 147 (2000), 283. doi: 10.1016/S0167-2789(00)00169-X. Google Scholar [28] Z. Virányi, I. Szalai, J. Boissonade and P. De Kepper, Sustained Spatiotemporal Patterns in the Bromate-Sulfite Reaction,, J. Phys. Chem. A, 111 (2007), 8090. doi: 10.1021/jp0723721. Google Scholar [29] I. Szalai and P. De Kepper, Spatial bistability, oscillations and excitability in the Landolt reaction,, Phys. Chem. Chem. Phys., 8 (2006), 1105. doi: 10.1039/b515620c. Google Scholar [30] I. Szalai and P. De Kepper, Pattern formation in the ferrocyanide-iodate-sulfite reaction: The control of space scale separation,, Chaos, 18 (2008). doi: 10.1063/1.2912719. Google Scholar [31] J. Horváth, I. Szalai and P. De Kepper, Pattern formation in the Thiourea-Iodate-Sulfite system: spatial bistability, waves, and stationary patterns,, Physica D, 239 (2010), 776. doi: 10.1016/j.physd.2009.07.005. Google Scholar [32] I. Szalai and P. De Kepper, Patterns of the Ferrocyanide-Iodate-Sulfite reaction revisited: the role of immobilized carboxylic functions,, J. Phys. Chem. A, 112 (2008), 783. doi: 10.1021/jp711849m. Google Scholar [33] S. Ponce Dawson, M. V. D'Angelo and J. E. Pearson, Towards a global classification of excitable reaction-diffusion systems,, Phys. Lett. A, 265 (2000), 346. doi: 10.1016/S0375-9601(00)00008-6. Google Scholar [34] I. Lengyel and I. R. Epstein, A chemical approach to design Turing patterns in reaction-diffusion systems,, Proc. Natl. Acad. Sci. USA, 89 (1992), 3977. doi: 10.1073/pnas.89.9.3977. Google Scholar [35] J. E. Pearson and W. Bruno, Pattern formation in an N+Q component reaction-diffusion system,, Chaos, 2 (1992), 513. doi: 10.1063/1.165893. Google Scholar [36] D. E. Strier and S. P. Dawson, Turing patterns inside cells,, PLoS ONE, 2 (2007). doi: 10.1371/journal.pone.0001053. Google Scholar [37] D. Horváth and Á. Tóth, Diffusion-driven front instabilities in the chlorite-tetrathionate reaction,, J. Chem. Phys., 108 (1998). doi: 10.1063/1.475355. Google Scholar [38] D. Horváth and Á. Tóth, Turing patterns in a single-step autocatalytic reaction,, J. Chem. Soc. Farad. Trans., 93 (1997). doi: 10.1039/a705895k. Google Scholar [39] I. Szalai and P. De Kepper, Turing patterns, spatial bistability, and front instabilities in a reaction-diffusion system,, J. Phys. Chem. A, 108 (2004), 5315. doi: 10.1021/jp049168n. Google Scholar [40] I. Szalai N. Takács, J. Horváth and P. De Kepper, Sustained self-organizing pH patterns in hydrogen peroxide driven aqueous redox systems,, preprint, (2011). Google Scholar [41] B. Rudovics, E. Barillot, P. W. Davies, E. Dulos, J. Boissonade and P. De Kepper, Experimental Studies and Quantitative Modeling of Turing Patterns in the (Chlorine Dioxide, Iodine, Malonic Acid) Reaction,, J. Phys. Chem. A, 103 (1999), 1790. doi: 0.1021/jp983210v. Google Scholar [42] J. A. Vastano, J. E. Pearson, W. Horsthemke and H.L. Swinney, Turing patterns in an open reactor,, J. Chem. Phys., 88 (1988). doi: 10.1063/1.454456. Google Scholar

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##### References:
 [1] A. M. Turing, The chemical basis of morphogenesis,, Phil. Trans. Roy. Soc. Ser B, 237 (1952), 37. doi: 10.1098/rstb.1952.0012. Google Scholar [2] J. D. Murray, "Mathematical Biology I-II,'', 3rd edition, (2002). doi: 10.1007/b98868. Google Scholar [3] H. Ikeda, M. Mimura and Y. Nishiura, Global bifurcation phenomena of travelling wave solutions for some bistable reaction-diffusion systems,, Nonl. Anal. TMA, 13 (1989). doi: 10.1016/0362-546X(89)90061-8. Google Scholar [4] A. Hagberg and E. Meron, Complex patterns in reaction-diffusion systems: a tale of two front instabilities,, Chaos, 4 (1994). doi: 10.1063/1.166047. Google Scholar [5] G. Nicolis and I. Prigogine, "Self Organization in Nonequilibrium Systems,'', Wiley, (1977). Google Scholar [6] H. Meinhardt, "Models of Biological Pattern Formation,'', Academic Press, (1982). Google Scholar [7] E. Ammelt, Y. A. Astrov and H. G. Purwins, Stripe Turing structures in a two-dimensional gas discharge system,, Physical Review E, 55 (2001), 6731. doi: 10.1103/PhysRevE.55.6731. Google Scholar [8] L. A. Lugiato, C. Oldano and L. M. Narducci, Cooperative frequency locking and stationary spatial structures in lasers,, Journal of the Optical Society of America B -Optical Physics, 5 (1988), 879. doi: 10.1364/JOSAB.5.000879. Google Scholar [9] S. A. Levin, The problem of pattern and scale in ecology,, Ecology, 73 (1992), 1943. doi: 10.2307/1941447. Google Scholar [10] F. Borgogno, P. D'Odorico and F. Laio, et al, Mathematical models of vegetation pattern formation in ecohydrology,, Reviews of Geophysics, 47 (2009). doi: 10.1029/2007RG000256. Google Scholar [11] P. De Kepper, J. Boissonade and I. Szalai, From sustained oscillations to stationnary reaction-diffusion patterns,, in, (2009), 978. doi: 10.1007/978-90-481-2993-5\_1. Google Scholar [12] V. Castets, E. Dulos, J. Boissonade and P. De Kepper, Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern,, Phys. Rev. Lett., 64 (1990), 2953. doi: 10.1103/PhysRevLett.64.2953. Google Scholar [13] K. J. Lee, W. D. McCormick, Q. Ouyang and H. L. Swinney, Pattern formation by interacting chemical fronts,, Science, 261 (1993), 192. doi: 10.1126/science.261.5118.192. Google Scholar [14] E. Dulos, P. W. Davies, B. Rudovics and P. De Kepper, From quasi-2D to 3D Turing patterns in ramped systems,, Physica D, 98 (1996), 53. doi: 10.1016/0167-2789(96)00072-3. Google Scholar [15] P. De Kepper, J.-J. Perraud, B. Rudovics and E. Dulos, Experimental study of stationary Turing patterns and their interaction with traveling waves in a chemical system,, Int. J. Bifurcation and Chaos, 6 (1994), 1077. doi: 10.1142/S0218127494000915. Google Scholar [16] K. J. Lee, W. D. McCormick, H. L. Swinney and J. E. Pearson, Experimental observation of self-replicating spots in a reaction-diffusion system,, Nature, 369 (1994), 215. doi: 10.1038/369215a0. Google Scholar [17] J. Horváth, I. Szalai and P. De Kepper, An experimental design method leading to chemical turing patterns,, Science, 324 (2009), 772. doi: 10.1126/science.1169973. Google Scholar [18] V. K. Vanag and I. R. Epstein, Pattern formation in a tunable medium: The Belousov-Zhabotinsky reaction in an aerosol ot microemulsion,, Phys. Rev. Lett., 87 (2001). doi: 10.1103/PhysRevLett.87.228301. Google Scholar [19] P. Érdi and J. Tóth, "Mathematical Models of Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models,'', Princeton University Press, (1989). Google Scholar [20] J. Boissonade and P. De Kepper, Transitions from bistability to limit cycle oscillations. Theoretical analysis and experimental evidence in an open chemical system,, J. Phys. Chem., 84 (1980), 501. doi: 10.1021/j100442a009. Google Scholar [21] G. Dangelmayr and J. Guckenheimer, On a four parameter family of planar vector fields,, Archive for Rational Mechanics and Analysis, 97 (1987), 321. doi: 10.1007/BF00280410. Google Scholar [22] I. R. Epstein and J. A. Pojman, "An Introduction to Nonlinear Chemical Dynamics Oscillations, Waves, Patterns, and Chaos,'', Oxford University Press, (1998). Google Scholar [23] P. Gray and S. K. Scott, "Chemical Oscillations and Instabilities,'', Clarendon Press, (1990). Google Scholar [24] P. Blanchedeau and J. Boissonade, Resolving an experimental paradox in open spatial reactors: The role of spatial bistability,, Phys. Rev. Lett., 81 (1998), 5007. doi: 10.1103/PhysRevLett.81.5007. Google Scholar [25] K. Benyaich, T. Erneux T, S. Metens, S, S. Villain and P. Borckmans, Spatio-temporal behaviors of a clock reaction in an open gel reactor,, Chaos, 16 (2006). doi: 10.1063/1.2219703 . Google Scholar [26] J. Boissonade, E. Dulos, F. Gauffre, M. N. Kuperman and P. De Kepper, Spatial bistability and waves in a reaction with acid autocatalysis,, Faraday Discuss., 120 (2001), 353. doi: 10.1039/b103240m. Google Scholar [27] P. Blanchedeau, J. Boissonade and P. De Kepper, Theoretical and experimental studies of spatial bistability in the chlorine-dioxide-iodide reaction,, Physica D, 147 (2000), 283. doi: 10.1016/S0167-2789(00)00169-X. Google Scholar [28] Z. Virányi, I. Szalai, J. Boissonade and P. De Kepper, Sustained Spatiotemporal Patterns in the Bromate-Sulfite Reaction,, J. Phys. Chem. A, 111 (2007), 8090. doi: 10.1021/jp0723721. Google Scholar [29] I. Szalai and P. De Kepper, Spatial bistability, oscillations and excitability in the Landolt reaction,, Phys. Chem. Chem. Phys., 8 (2006), 1105. doi: 10.1039/b515620c. Google Scholar [30] I. Szalai and P. De Kepper, Pattern formation in the ferrocyanide-iodate-sulfite reaction: The control of space scale separation,, Chaos, 18 (2008). doi: 10.1063/1.2912719. Google Scholar [31] J. Horváth, I. Szalai and P. De Kepper, Pattern formation in the Thiourea-Iodate-Sulfite system: spatial bistability, waves, and stationary patterns,, Physica D, 239 (2010), 776. doi: 10.1016/j.physd.2009.07.005. Google Scholar [32] I. Szalai and P. De Kepper, Patterns of the Ferrocyanide-Iodate-Sulfite reaction revisited: the role of immobilized carboxylic functions,, J. Phys. Chem. A, 112 (2008), 783. doi: 10.1021/jp711849m. Google Scholar [33] S. Ponce Dawson, M. V. D'Angelo and J. E. Pearson, Towards a global classification of excitable reaction-diffusion systems,, Phys. Lett. A, 265 (2000), 346. doi: 10.1016/S0375-9601(00)00008-6. Google Scholar [34] I. Lengyel and I. R. Epstein, A chemical approach to design Turing patterns in reaction-diffusion systems,, Proc. Natl. Acad. Sci. USA, 89 (1992), 3977. doi: 10.1073/pnas.89.9.3977. Google Scholar [35] J. E. Pearson and W. Bruno, Pattern formation in an N+Q component reaction-diffusion system,, Chaos, 2 (1992), 513. doi: 10.1063/1.165893. Google Scholar [36] D. E. Strier and S. P. Dawson, Turing patterns inside cells,, PLoS ONE, 2 (2007). doi: 10.1371/journal.pone.0001053. Google Scholar [37] D. Horváth and Á. Tóth, Diffusion-driven front instabilities in the chlorite-tetrathionate reaction,, J. Chem. Phys., 108 (1998). doi: 10.1063/1.475355. Google Scholar [38] D. Horváth and Á. Tóth, Turing patterns in a single-step autocatalytic reaction,, J. Chem. Soc. Farad. Trans., 93 (1997). doi: 10.1039/a705895k. Google Scholar [39] I. Szalai and P. De Kepper, Turing patterns, spatial bistability, and front instabilities in a reaction-diffusion system,, J. Phys. Chem. A, 108 (2004), 5315. doi: 10.1021/jp049168n. Google Scholar [40] I. Szalai N. Takács, J. Horváth and P. De Kepper, Sustained self-organizing pH patterns in hydrogen peroxide driven aqueous redox systems,, preprint, (2011). Google Scholar [41] B. Rudovics, E. Barillot, P. W. Davies, E. Dulos, J. Boissonade and P. De Kepper, Experimental Studies and Quantitative Modeling of Turing Patterns in the (Chlorine Dioxide, Iodine, Malonic Acid) Reaction,, J. Phys. Chem. A, 103 (1999), 1790. doi: 0.1021/jp983210v. Google Scholar [42] J. A. Vastano, J. E. Pearson, W. Horsthemke and H.L. Swinney, Turing patterns in an open reactor,, J. Chem. Phys., 88 (1988). doi: 10.1063/1.454456. Google Scholar
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