Published Oct 17, 2012



PLUMX
Almetrics
 
Dimensions
 

Google Scholar
 
Search GoogleScholar


Diego Garzón-Alvarado, MSc

Angélica Ramírez-Martínez, PhD

Carlos Duque-Daza, PhD

##plugins.themes.bootstrap3.article.details##

Abstract

In this paper we present several numerical tests on reaction-diffusion equations in the space of Turing, under the Schnakenberg reaction mechanism. The tests were performed on 2D unit square, to which is imposed on random initial conditions and Neumann zero on the boundary. The parameters that define the behavior of the equations are modeled as stochastic fields, specifically, are used: the diffusion and reactive parameters as values of random type. Thus, combines the standard method of finite element Newton-Raphson with the finite element method spectral stochastic. The parameters of each equation described by Karhunen-Loève expansion, while the unknown is represented by the expansion of the polynomials of chaos. The objective of this article is to obtain the patterns of each coefficient of the polynomial chaos expansion. The results show the versatility of the method to solve different physical problems. Furthermore, it achieves statistical description of the solution. The results (for the unknown coefficients stochastic) show bands complex patterns and mixing points, which can not be predicted from the dynamics of the system.

Keywords

Stochastic finite elements, reactiondiffusion Turing patterns, Schnakenberg reaction mechanismElementos finitos estocásticos, reacción-difusión, patrones de Turing, mecanismo de reacción de Schnakenberg

References
ARDES, M.; BUSSE, F. y WICHT, J. Thermal convection in rotating spherical shells. Physics of the Earth and Planetary Interiors. 1997, vol. 99, pp. 55-67.
BABUSKA, I.; IHLENBURG, F.; PAIK, E. et al. A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution. Computer Methods in Applied Mechanics and Engineering. 1995, vol. 128, pp. 325-359.
BALKAREI, Y.; GRIGORYANTS, A. y RHZANOV, Y. et al. Regenerative oscillations, spatial-temporal single pulses and static inhomogeneous structures in optically bitable semiconductors. Optics Communications. 1988, vol. 66, pp. 161-166.
BAURMANNA, M.; GROSS, T. y FEUDEL, U. Instabilities in spatially extended predator–prey systems: Spatio-temporal patterns in the neighborhood of Turing–Hopf bifurcations. Journal of Theoretical Biology. 2007, vol. 245, pp. 220-229.
CHAPLAIN, M.; GANESH, M. y GRAHAM, I. Spatio-temporal pattern formation on spherical surfaces: Numerical simulation and application to solid tumor growth. Journal of Mathematical Biology. 2001, vol. 42, núm. 5, pp. 387-423.
CRAUSTE, F.; LHASSAN, M. y KACHA, A. A delay reaction-diffusion model of the dynamics of botulinum in fish. Mathematical Biosciences. 2008, vol. 216, pp. 17-29.
FERRAGUT, L.; ASENSIO, M. y MONEDERO, S. A numerical method for solving convection–reaction–diffusion multivalued equations in fire spread modelling. Advances in Engineering Software. 2007, vol. 38, pp. 366-371.
FERREIRA, S.; MARTINS, M. y VILELA, M. Reaction-diffusion model for the growth of avascular tumor. Physical Review. 2002, vol. 65, núm. 2, pp. 1-8.
FREDERIK, H.; MAINI, P.; MADZVAMUSE, A. et al. Pigmentation pattern formation in butterflies: experiments and models. C. R. Biologies. 2003, vol. 326, pp. 717-727.
GARCÍA-AZNAR, J.; KUIPER, J. y GÓMEZ-BENITO, M. et al. Computational simulation of fracture healing: Influence of interfragmentary movement on the callus growth. Journal of Biomechanics. 2007, vol. 40, núm. 7, pp. 1467-1476.
GARZÓN, D. “Simulación de procesos de reacción-difusión: Aplicación a la morfogénesis del tejido óseo”. Tesis doctoral. Universidad de Zaragoza. 2007.
GARZÓN-ALVARADO, D.; GARCÍA-AZNAR, J. y DOBLARÉ M. Appearance and location of secondary ossification centres may be explained by a reaction-diffusion mechanism. Computers in Biology and Medicine. 2009, vol. 39, pp. 554-561.
HIRAYAMA, O. y TAKAKI, R. Thermal convection of a fluid with temperature-dependent viscosity. Fluid Dynamics Research. 1988, vol. 12, núm. 1, pp. 35-47.
KONDO, S. y ASAI, R. A reaction-diffusion wave on the skin of the marine anglefish, Pomacanthus. Nature. 1995, vol. 376, pp. 765-768.
KRINSKY, V. I. Self-organization: Auto-waves and structures far from equilibrium. Berlín: Editorial Springer, 1984.
LIR, J. y LIN, T. Visualization of roll patterns in Rayleigh-Bénard convection of air in rectangular shallow cavity. International Journal of Heat and Mass Transfer. 2001, vol. 44, pp. 2889-2902.
MADZVAMUSE, A. “A numerical approach to the study of spatial pattern formation”. Tesis doctoral. Oxford University. UK. 2000.
MADZVAMUSE, A. A Numerical approach to the study of spatial pattern formation in the ligaments of arcoid bivalves. Bulletin of Mathematical Biology. 2002, vol. 64, pp. 501-530.
MADZVAMUSE, A.; WATHEN, A. y MAINI, P. A moving grid finite element method applied to a model biological pattern generator. Journal of Computational Physics. 2003, vol. 190, pp. 478-500.
MEI, Z. Numerical bifurcation analysis for reaction-diffusion equations. Berlín: Springer Verlag, 2000.
NOZAKURA, T. y IKEUCHI, S. Formation of dissipative structures in galaxies. Astrophys Journal. 1984, vol. 279, pp. 40-52.
RICHTER, O. Modelling dispersal of populations and genetic information by finite element methods. Environmental Modelling & Software. 2008, vol. 23, núm. 2, pp. 206-214.
ROSSI, F.; RISTORI, S. y RUSTICI, M. et al. Dynamics of pattern formation in biomimetic systems. Journal of Theoretical Biology. 2008, vol. 255, pp. 404-412.
ROTHSCHILD, B. y AULT, J. Population-dynamic instability as a cause of patch structure. Ecological Modelling. 1996, vol. 93, pp. 237-239.
SMITH, R. Optimal and near-optimal advection-diffusion finite-difference schemes iii. Black-Scholes equation. Mathematical, Physical and Engineering Sciences. 2000, vol. 456, pp. 1019-1028.
YI, F.; WEI, J. y SHI, J. Bifurcation and spatio-temporal patterns in a homogeneous diffusive predator–prey system. Journal of Differential Equations. 2009, vol. 246, núm. 5, pp. 1944-1977.
ZHANG, L. y LIU, S. Stability and pattern formation in a coupled arbitrary order of autocatalysis system. Applied Mathematical Modelling. 2009, vol. 33, pp. 884-896.
How to Cite
Garzón-Alvarado, D., Ramírez-Martínez, A., & Duque-Daza, C. (2012). On Turing pattern formation under stochastic considerations. Ingenieria Y Universidad, 16(2), 471. https://doi.org/10.11144/Javeriana.iyu16-2.otpf
Section
Articles

Most read articles by the same author(s)