Turing pattern formation for reaction-convection-diffusion in fixed domains submitted to toroidal velocity fields
Keywords:
diffusion instability, Schnackenberg, glycolysis, Turing patternsAbstract
This article studies the effect of the inclusion of the transport term in the reaction-diffusion equations, through toroidal velocity fields. The formation of Turing patterns in diffusion-advection-reaction problems is studied specifically, considering the Schnackenberg reaction kinetics and glycolysis models. Three cases are analyzed and solved numerically using finite elements. It is found that, for the glycolysis models, the advective effect totally modifies the form of the obtained Turing patterns with diffusion-reaction; whereas for the problems of Schnackenberg, the original patterns distort themselves slightly, making them to rotate in the direction of the velocity field. Also, this work was able to determine that for high values of velocity the advective effect surpasses the diffusive one and the instability by diffusion is eliminated. On the other hand, for very low values in the velocity field, the advective effect is not considerable and there is no modification of the original Turing pattern.
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D. Garzón. Simulación de procesos de reacción-difusión. Aplicación a la morfogénesis del tejido óseo, Ph.D. Thesis. Universidad de Zaragoza. 2007. pp. 10-50.
D. White. “The planforms and onset of convection with a temperature dependent viscosity”. J. Fluid Mech. Vol. 191. 1988. pp. 247-286.
O. Hirayama, R. Takaki. “Thermal convection of a fluid with temperature-dependent viscosity”. Fluid Dynamics Research. Vol. 12-1. 1988. pp. 35-47.
M. Ardes, F. Busse, J. Wicht. “Thermal convection in rotating spherical shells”. Physics of the Earth and Planetary Interiors. Vol. 99. 1997. pp. 55-67.
J. Lir, T. Lin. “Visualization of roll patterns in Rayleigh–Bénard convection of air in rectangular shallow cavity”. International Journal of Heat and Mass Transfer. Vol. 44. 2001. pp. 2889-2902.
Y. Balkarei, A. G. Yants, Y. Rhzanov, M. Elinson. “Regenerative oscillations, spatial-temporal single pulses and static inhomogeneous structures in optically bistable semiconductors”. Opt. Commun. Vol. 66. 1988. pp. 161-166.
V. I. Krinsky. “Self-organisation”. Auto-Waves and structures far from equilibrium. Ed. Springer. Berlin. 1984. pp. 111-118.
L. Zhang, S. Liu. “Stability and pattern formation in a coupled arbitrary order of autocatalysis system”. Applied Mathematical Modelling. Vol. 33. 2009. pp. 884-896.
F. Crauste, M. Lhassan, A. Kacha. “A delay reaction-diffusion model of the dynamics of botulinum in fish”. Mathematical Biosciences. Vol. 216. 2008. pp. 17-29.
Rossi, S. Ristori, M. Rustici, N. Marchettini, E. Tiezzi. “Dynamics of pattern formation in biomimetic systems”. Journal of Theoretical Biology. Vol. 255. 2008. pp. 404-412.
A. Madzvamuse, A. Wathen, P. Maini. “A moving grid finite element method applied to a model biological pattern generator”. Journal of Computational Physics. Vol. 190. 2003. pp. 478-500.
H. Frederik, P. Maini, A. Madzvamuse, A. Wathen, T. Sekimura. “Pigmentation pattern formation in butterflies: experiments and models”. C. R. Biologies. Vol. 326. 2003. pp. 717-727.
F. Yi, J. Wei, J. Shi. “Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator–prey system”. Journal of Differential Equations. Vol. 246. 2009. pp. 1944-1977.
M. Baurmanna, T. Gross, U. Feudel. “Instabilities in spatially extended predator–prey systems: Spatio-temporal patterns in the neighborhood of Turing–Hopf bifurcations”. Journal of Theoretical Biology. Vol. 245. 2007. pp. 220-229.
B. Rothschild, J. Ault. “Population-dynamic instability as a cause of patch structure”. Ecological Modelling. Vol. 93. 1996. pp. 237-239.
T. Nozakura, S. Ikeuchi. “Formation of dissipative structures in galaxies”. Astrophys. J. Vol. 279. 1984. pp. 40-52.
A. Madzvamuse. “A Numerical Approach to the Study of Spatial Pattern Formation in the Ligaments of Arcoid Bivalves”. Bulletin of Mathematical Biology. Vol. 64. 2002. pp. 501-530.
A. Turing. “The chemical basis of morphogenesis”. Phil. Trans. Roy. Soc. Lond. Vol. 237. 1952. pp. 37-72.
E. Crampin, E. Gaffney, P. Maini. “Reaction and diffusion on growing domains: Scenarios for robust pattern formation”. Bulletin of Mathematical Biology. Vol. 61(6). 1999. pp. 1093-1120.
A. Madzvamuse. A numerical approach to the study of spatial pattern formation. Ph.D. Thesis. University of Oxford. Oxford. 2000. pp. 10-147.
H. Meinhardt. Models of Biological Pattern Formation. Ed. Academic Press. London. 1982. pp. 53-58.
J. Murray. “A prepattern formation mechanism for animal coat markings”. J. Theor. Biol. Vol. 88. 1981. pp. 161-199.
S. Kondo, R. Asai. “A reaction-diffusion wave on the skin of the marine anglefish, Pomacanthus”. Nature. Vol. 376. 1995. pp. 765-768.
T. Sekimura, A. Madzvamuse, A. Wathen, P. Maini. “A model for colour pattern formation in the butterfly wing of Papilio dardanus”. Proc. Roy. Soc. London. Vol. 26. 2000. pp. 851-859.
J. García-Aznar, J. Kuiper, M. Gómez-Benito, M. Doblaré, J. Richardson. “Computational simulation of fracture healing: Influence of interfragmentary movement on the callus growth”. Journal of Biomechanics. Vol. 40. 2007. pp. 1467-1476.
S. Ferreira, M. Martins, M. Vilela. “Reaction-diffusion model for the growth of avascular tumor”. Physical Review. Vol. 65. 2002. pp. 1-8.
M. Chaplain, A. Ganesh, I. Graham, “Spatio-temporal pattern formation on spherical surfaces: Numerical simulation and application to solid tumor growth”. J. Math. Biol. Vol. 42. 2001. pp. 387-423.
R. Barrio, C. Varea, J. Aragón, P. Maini. “A two-dimensional numerical study of spatial pattern formation in interacting systems”. Bull. Math. Biol. Vol. 61. 1999. pp. 483-505.
E. Crampin, E. Gaffney, P. Maini. “Reaction and diffusion on growing domains: Scenarios for robust pattern formation”. Bull. Math. Biol. Vol.61. 1999. pp. 1093-1120.
T. Sekimura, A. Madzvamuse, A. Wathen, P. Maini. “A model for colour pattern formation in the butterfly wing of Papilio dardanus”. Proc. Roy. Soc. London. Vol. 26. 2000. pp. 851-859.
A. Madzvamuse, A. Wathen, P. Maini. “A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains”. J. Sci. Comp. Vol. 24 . 2005. pp. 247-262.
A. Madzvamuse, R. Thomas, P. Maini, A. Wathen. “A numerical approach to the study of spatial pattern formation in the ligaments of arcoid bivalves”. Bull. Math. Biol. Vol. 64. 2002. pp. 501-530.
A. Madzvamuse, T. Sekimura, A. Wathen, P. Maini. “A predictive model for colour pattern formation in the butterfly wing of Papilio dardanus”. Hiroshima Math. J. Vol. 32. 2002. pp. 325-336.
A. Kassam, L. Trefethen. “Solving reaction-diffusion equations 10 times faster”. Numerical Analysis Group Research Report No. 16. Oxford University. Oxford. 2003. pp. 1-13.
A. Madzvamuse, P. Maini. “Velocity-induced numerical solution of reaction–diffusion systems on continuosly growing domains”. Journal of computational physics. Vol. 225. 2007. pp. 100-119.
A. Madzvamuse. “Time-stepping schemes for moving grid finite element applied to reaction-diffusion systems on fixed and growing domains”. Journal of computational physics. Vol. 214. 2006. pp. 239-263.
O. Zienkiewicz, R. Taylor. Problemas de convección dominante: Aproximaciones de elementos finitos a la ecuación de difusión-advección. Finite Element Method. Ed. Butterworth-Heinemann College. Oxford. Vol. 3. 2000. pp. 5-150.
J. Schnakenberg. “Simple chemical reaction systems with limit cycle behaviour”. J. Theor Biol. Vol. 81. 1979. pp.389-400.
R. Revelli, L. Ridolfi. “Generalized collocation method for two-dimensional reaction-diffusion problems with homogeneous Neumann boundary conditions”. Computers and Mathematics with Applications. Vol. 56. 2008. pp. 2360-2370.
D. Garzón Alvarado, J. García-Aznar, M. Doblare. “A reaction-diffusion model for long bones growth”, Biomechanics and modeling in mechanobiology. Vol. 8. 2009. pp. 381 - 395
D. Garzón Alvarado, J. García-Aznar, M. Doblare. “Appearance and location of secondary ossification centres may be explained by a reaction-diffusion mechanism”. Computers in biology and medicine. Vol. 39. 2009. pp. 554-561.
J. Vanegas, N. Landinez, D. Garzón-Alvarado. “Análisis de Inestabilidad de Turing en Modelos Biológicos”. Revista DYNA. Vol. 76. 2009. pp. 123-134.
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