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Turing Pattern in the Oregonator Revisited

Authors: Elragig Aiman, Dreiwi Hanan, Townley Stuart, Elmabrook Idriss


In this paper, we reconsider the analysis of the Oregonator model. We highlight an error in this analysis which leads to an incorrect depiction of the parameter region in which diffusion driven instability is possible. We believe that the cause of the oversight is the complexity of stability analyses based on eigenvalues and the dependence on parameters of matrix minors appearing in stability calculations. We regenerate the parameter space where Turing patterns can be seen, and we use the common Lyapunov function (CLF) approach, which is numerically reliable, to further confirm the dependence of the results on diffusion coefficients intensities.

Keywords: Diffusion driven instability, common Lyapunov function (CLF), turing pattern, positive-definite matrix.

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[1] Q. Hong, and J. D. Murray, A simple method of parameter space determination for diffusion driven instability with three species, Applied Math. Letters. 14 (2001) 405-411.
[2] A. Elragig and S. Townley, A New necessary condition for Turing instabilities Mathematical biosciences. 239(2012)131-138.
[3] J. D. Murray, Mathematical Biology : I , Springer, Berlin, 2008.
[4] J. Zhow, Applied Math. Letters Bifurcation analysis of the Oregonator model, 52 (2016) 192198.
[5] R. Peng and F. Sun, Turing pattern of the Oregonator model, Nonlinear Analysis: Theory, Methods & Applications, 72 (5) (2010) 23372345.
[6] R. Field and R. Noyea,Oscillations in chemical systems, Part IV. Limit cycle behaver in a model of a real chemical reaction, J. Chem. Phus. 60 (1974) 1877-1884.
[7] P. Beker and R. Field, Stationary concentration patterns in the Oregonator model of the Belousov-Zha- botinskii reaction, J. Phys. Chem. 89 (1985) 118-128.
[8] N. Kopell and L. Howard, Pattern formation in the Belousov reaction, Lectures on Math. in the Life Sciences, 7 ((1974) 201-216
[9] A. Turing, The chemical basis of morphogenesiss, Phil. Trans. R. Soc. Lond. B237 (1952)37-73.
[10] P. Maini, K. Painter, and H. Chau, Spatial pattern formation in chemical and biological systems, Faraday Trans., 93 (1997) 3601-3610.
[11] J. Murray, Mathematical Biology I: An introduction. Springer, Berlin,2008.
[12] H. Meinhardt, Models of Biological Pattern Formation, Academic Press, London, 1982.
[13] S. Kauffman, R. Shymko, and K. Trabert, Control of sequential compartment in drosophila, Science, 270 (1978) 199-259.
[14] K. Painter, P. Maini, and H. Othmer, Development and applications of a model for cellular response to multiple chemotactic cues, J. Mathematical Biology, 314 (2000)41-285.
[15] C. Varea, J. Aragon, and R. Barrio, Confined Turing patterns in growing systems, Phys. Rev., 56 (1997) 1250-1253.
[16] M. Chaplain, M. Ganesh, and I. Graham, Spatio-temporal pattern formation on spherical surfaces: Numerical simulation and application to solid tumour growth, Bull. Math.Biol., 42 (2001) 387-423.
[17] A. Gierer and H. Meinhardt, A theory of biological pattern formation,Kybernetik 12 (1972) 30-39.
[18] I. Epstein and K. Showalter, Nonlinear chemical dynamics: oscillations, patterns and chaos. J. Phys. Chem, 100 (1996) 13132-13147.
[19] M. Cross and P. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys, 65 (1993) 851-1112.
[20] K. A. J. White and C. A. Gilligan, Spatial heterogeneity in three-species, plant-parasite-hyperparasite systems, Phil. Trans. R. Soc. Lond. (B) (353) (1998) 543-557.
[21] W. Wilson, S. Harrison, A. Hastings, and K. McCann, Exploring stable pattern formation in models of tussock moth populations, J. Anim. Ecol, 68 (1999)94-107.
[22] M. Wang, Stability and hopf bifurcation for prey-predator model with prey-stage structure and diffusion, Mathematical Biosciences, 212 (2008) 149-160.
[23] L. Segel and J. Jackson, Dissipative structure: an explanation and an ecological example, J. Theo. Biol, 37 (1972)545-559.
[24] H. Malchow, S. Petrovskii, and V. Venturino, Spatio-temporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulation, Chapman and Hall/CRC, 2007.
[25] J. McNair, A reconciliation of simple and complex models of age-dependent predation, Theor. Popul. Biol., 32 (1987) 383-392.
[26] A. Edelstein-Keshet, Mathematical Models in Biology, McGraw-Hill Companies, 1988.
[27] B. C. Goodwin and L. E. H Trainor, Tip and whorl morphogenesis in acetabularia by calcium-regulated strain fields, Journal of theoretical biology, 117 (1985) 79-106.
[28] W. Dessaul, H. V. D. Mark, K. V. D Mark, and S. Fischer, Changes in the patterns of collagens and fibronectin during limb-bud chondrogenesis, J Embryol Exp Morphol, 57(1980) 51-60.
[29] K. J. Painter, Chemotaxis as a mechanism for morphogensis. PhD thesis, Brasenose college, University of Oxford, 1997.
[30] P. D. Kepper, V. Castets, E. Dulos, and J. Biossonade, Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction, Physica D, 49 (1991) 161-169.
[31] J. Horvath, I. Szalai, and P. D. Kepper, An experimental design method leading to chemical turing patterns, Science, 324 (2009) 772-775.
[32] J. Merkin, Travelling waves in the oregonator model for the bz reaction, IMA J. Appl. Math, 74 (2009) 622-643.
[33] R. Field and R. Noyes, Oscillations in chemical systems. iv. limit cycle behaviour in a model of a real chemical reaction, J.Chem.Phys., 60 (1974)1877-1884.
[34] I. Prigogine and R. Lefever, Symmetry-breaking instabilities in dissipative systems ii, J. Chem. Phys, 48 (1968)1695-1700.
[35] J. Field and F. W. Schneier, Oscillating chemical reactions and nonlinear dynamics, J. Chem. Educ., 66 (1989)195-204.
[36] J. D, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, Berlin, 2003.
[37] M. Zhu and J. D. Murray, Parameter domain for generating spatial patterns: a comparison of reaction-diffusion and cell chemotaxis models, Int. J. Bifurc. Chaos, 5 (1995) 1503-1524.
[38] J. D. Murray, parameter space for Turing instability in reaction diffusion mechanism: a comparison of models, J. Theo. Biol, 98(1982) 143-163.
[39] R. B. Hoyle, Pattern formation: An Introduction to Methods, Cambridge University Press, 2003.
[40] L. Wang, M. Y. Michael, Diffusion-driven Instability in reaction-diffusion systems, J. Math. Anal. Appl., 254 (2001) 138-153.
[41] G. Xiaoqing, A. Murat, A sufficient condition of d-stability and applications to reaction diffusion models, J. Contr., 77(2005)598-605.
[42] M. G. Neubert, H. Caswell, J. D. Murray, Transient dynamics and pattern formation: reactivity is necessary for Turing instabilities, Mathematical biosciences, 175(1) (200) 1-11.