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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:76493", author = "Elragig Aiman and Dreiwi Hanan and Townley Stuart and Elmabrook Idriss", title = "Turing Pattern in the Oregonator Revisited", abstract = "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.", volume = "11", number = "7", pages = "315-5", }
