PapersNo Access

EXAMPLES OF THE EFFECT OF GROWTH AND STRAIN ON TURING PATTERN FORMATION DYNAMICS

DIEGO ALEXANDER GARZÓN-ALVARADO

Mechanical and Mechatronics Engineering Department, Universidad Nacional de Colombia, Bogotá, Colombia

https://doi.org/10.1142/S0218127412500393Cited by:1 (Source: Crossref)

Abstract

Reaction–diffusion (RD) equation behavior has been studied in various fields, including biology, bioengineering and chemistry. Their solution leads to the formation of patterns which are stable in time and unstable in space, especially when RD system parameters are in the Turing space. Such patterns can be changed due to the growth of the domain where the reaction takes place. This article presents RD equations concerning growing domains in 2D and 3D. Several numerical examples have been solved using different geometries to study the effect of growth on pattern formation. The finite element method was used in conjunction with the Newton–Raphson method for the numerical solution to approximate nonlinear partial differential equations. It was found that growth affected Turing pattern formation, thereby generating complex structures in the domain.

Keywords:

References

E. Allgower and K. Georg, Handbook of Numer. Anal. 5, 3 (1997), DOI: 10.1016/S1570-8659(97)80002-6. Crossref, Web of Science, Google Scholar
T. Belytschko , W. Liu and B. Moran , Nonlinear Finite Elements for Continua and Structures ( John Wiley and Sons , 2000 ) . Google Scholar
M. Chaplain, A. Ganesh and I. Graham, J. Math. Biol. 42, 387 (2001), DOI: 10.1007/s002850000067. Crossref, Web of Science, Google Scholar
E. J. Crampin, E. A. Gaffney and P. K. Maini, J. Math. Biol. 44, 107 (2002), DOI: 10.1007/s002850100112. Crossref, Web of Science, Google Scholar
A. De Wit, Adv. Chem. Phys. 109, 453 (1999). Web of Science, Google Scholar
H. Frederik, P. Maini and A. Madzvamuse, C. R. Biologies 326, 717 (2003). Web of Science, Google Scholar
Garzón, D. [2007] "Simulación de procesos de reacción–difusión: Aplicación a la morfogénesis del tejido óseo," PhD thesis, Universidad de Zaragoza . Google Scholar
Garzón-Alvarado, D., Galeano, C. & Mantilla, J., "Experimentos numéricos sobre ecuaciones de reacción convección difusión con divergencia nula del campo de velocidad," Revista Internacional de Métodos Numéricos en Ingeniería . Google Scholar
A. Geirer and H. Meinhardt, Kybernetik 12, 30 (1972). Crossref, Web of Science, Google Scholar
J. Hoffman , Numerical Methods for Engineers and Scientists ( Ed McGraw Hill , 1992 ) . Google Scholar
G. Holzapfel , Nonlinear Solid Mechanics ( John Wiley , 2000 ) . Google Scholar
T. Hughes , The Finite Element Method: Linear Static and Dynamic Finite Element Analysis ( Courier Dover Publications , NY , 2003 ) . Google Scholar
R. Kapral and K. Showalter , Chemical Waves and Patterns ( Kluwer , 1995 ) . Crossref, Google Scholar
Madzvamuse, A. [2000] "A numerical approach to the study of spatial pattern formation," D Phil thesis. University of Oxford . Google Scholar
A. Madzvamuseet al., Bull. Math. Biol. 64, 501 (2002), DOI: 10.1006/bulm.2002.0283. Crossref, Web of Science, Google Scholar
A. Madzvamuse and T. Sekimura, Morphogenesis and Pattern Formation in Biological Systems, Experiments and Models (2003) pp. 59–65. Crossref, Google Scholar
A. Madzvamuse, P. K. Maini and A. J. Wathen, J. Comp. Phys. 190, 478 (2003), DOI: 10.1016/S0021-9991(03)00294-8. Crossref, Web of Science, Google Scholar
A. Madzvamuse, J. Comp. Phys. 24, 239 (2005). Google Scholar
A. Madzvamuse and P. K. Maini, J. Comput. Phys. 225, 100 (2007), DOI: 10.1016/j.jcp.2006.11.022. Crossref, Web of Science, Google Scholar
A. Madzvamuse, Nonlin. Anal. Th. Meth. Appl. 12, 2250 (2009). Google Scholar
A. Madzvamuse, E. Gaffney and P. Maini, J. Math. Biol. 61, 133 (2010), DOI: 10.1007/s00285-009-0293-4. Crossref, Web of Science, Google Scholar
P. Maini, K. Painter and H. Chau, J. Chem. Soc. Faraday Trans. 93, 3601 (1997), DOI: 10.1039/a702602a. Crossref, Google Scholar
J. Marsden and T. Hughes , Mathematical Foundations of Elasticity ( Courier Dover Publications , NY , 1983 ) . Google Scholar
Z. Mei , Numerical Bifurcation Analysis for Reaction–Diffusion Equations ( Springer Verlag , Alemania , 2000 ) . Crossref, Google Scholar
H. Meinhardt , Models of Biological Pattern Formation ( Academic Press , NY , 1982 ) . Google Scholar
S. Nair , Introduction to Continuum Mechanics ( Cambridge University Press , 2009 ) . Crossref, Google Scholar
Oñate, E. [1995] "Calculo de estructuras por el método de elementos finitos. análisis estético lineal, Ed. Centro Internacional de Métodos Numéricos en Ingeniería . Google Scholar
S. Rüdiger, M. Ernesto and J. Nicola, Phys. Rep. 447, 79 (2007). Google Scholar
S. Ruuth, J. Math. Biol. 34, 148 (1995), DOI: 10.1007/BF00178771. Crossref, Web of Science, Google Scholar
F. Sagués, D. Míguez and E. Nicola, Physica D 199, 235 (2004). Crossref, Web of Science, Google Scholar
L. Taber , Nonlinear Theory of Elasticity: Applications in Biomechanics ( World Scientific , Singapore , 2004 ) . Link, Google Scholar