**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**31896

##### Turing Pattern in the Oregonator Revisited

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

**Abstract:**

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

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1132497

**References:**

[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.