zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Turing pattern formation for reaction-convection-diffusion systems in fixed domains submitted to toroidal velocity fields. (English) Zbl 1228.76154
Summary: We have studied the effect of advection on reaction-diffusion equations by using toroidal velocity fields. Turing patterns formation in diffusion-advection-reaction problems was studied specifically, considering the Schnackenberg and glycolysis reaction kinetics models. Four cases were analyzed and solved numerically using finite elements. For glycolysis models, the advective effect modified the form of Turing patterns obtained with diffusion-reaction; whereas for Schnackenberg problems, the original patterns distorted themselves slightly, making them rotate in direction of the velocity field. We have also determined that the advective effect surpassed the diffusive one for high values of velocity and instability driven by diffusion was eliminated. On the other hand the advective effect is not considerable for very low values in the velocity field, and there was no modification in the original Turing pattern.
76R99Diffusion and convection (fluid mechanics)
35K57Reaction-diffusion equations
37N10Dynamical systems in fluid mechanics, oceanography and meteorology
[1]D.A. Garzón Alvarado, Reaction Diffusion Process Simulation: An Approach to Bone Morphogenesis (in Spanish), University of Zaragoza, Ph.D. Thesis, 2007.
[2]White, D.: The planforms and onset of convection with a temperature dependent viscosity, J. fluid mech. 191, 247-286 (1988)
[3]Hirayama, O.; Takaki, R.: Thermal convection of a fluid with temperature-dependent viscosity, Fluid dynam. Res. 12, No. 1, 35-47 (1988)
[4]Ardes, M.; Busse, F.; Wicht, J.: Thermal convection in rotating spherical shells, Phys. Earth planet. Interiors 99, 55-67 (1997)
[5]Lir, J.; Lin, T.: Visualization of roll patterns in Rayleigh Bénard convection of air in rectangular shallow cavity, Int. J. Heat mass transfer 44, 2889-2902 (2001) · Zbl 1002.76504 · doi:10.1016/S0017-9310(00)00340-9
[6]Balkarei, Y.; Grigoryants, A.; Rhzanov, Y.; Elinson, M.: Regenerative oscillations, spatial – temporal single pulses and static inhomogeneous structures in optically bistable semiconductors, Opt. commun. 66, 161-166 (1988)
[7]V.I. Krinsky, Self-organisation: auto-waves and structures far from equilibrium, 1984.
[8]Zhang, L.; Liu, S.: Stability and pattern formation in a coupled arbitrary order of autocatalysis system, Appl. math. Model. 33, 884-896 (2009) · Zbl 1168.35385 · doi:10.1016/j.apm.2007.12.013
[9]Crauste, F.; Lhassan, M.; Kacha, A.: A delay reaction – diffusion model of the dynamics of botulinum in fish, Math. biosci. 216, 17-29 (2008) · Zbl 1151.92032 · doi:10.1016/j.mbs.2008.07.012
[10]Rossi, F.; Ristori, S.; Rustici, M.; Marchettini, N.; Tiezzi, E.: Dynamics of pattern formation in biomimetic systems, J. theor. Biol. 255, 404-412 (2008)
[11]Madzvamuse, A.; Wathen, A.; Maini, P.: A moving grid finite element method applied to a model biological pattern generator, J. comput. Phys. 190, 478-500 (2003) · Zbl 1029.65113 · doi:10.1016/S0021-9991(03)00294-8
[12]Frederik, H.; Maini, P.; Madzvamuse, A.; Wathen, A.; Sekimura, T.: Pigmentation pattern formation in butterflies: experiments and models, C. R. Biol. 326, 717-727 (2003)
[13]Yi, F.; Wei, J.; Shi, J.: Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator – prey system, J. differ. Equat. 246, No. 5, 1944-1977 (2009) · Zbl 1203.35030 · doi:10.1016/j.jde.2008.10.024
[14]Baurmanna, M.; Gross, T.; Feudel, U.: Instabilities in spatially extended predator – prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J. theor. Biol. 245, 220-229 (2007)
[15]Rothschild, B.; Ault, J.: Population-dynamic instability as a cause of patch structure, Ecol. model. 93, 237-239 (1996)
[16]Nozakura, T.; Ikeuchi, S.: Formation of dissipative structures in galaxies, Astrophysics 279, 40-52 (1984)
[17]Madzvamuse, A.: A numerical approach to the study of spatial pattern formation in the ligaments of arcoid bivalves, Bull. math. Biol. 64, 501-530 (2002)
[18]García-Aznar, J.; Kuiper, J.; Gómez-Benito, M.; Doblaré, M.; Richardson, J.: Computational simulation of fracture healing: influence of interfragmentary movement on the callus growth, J. biomech. 40, 1467-1476 (2007)
[19]Ferreira, S.; Martins, M.; Vilela, M.: Reaction – diffusion model for the growth of avascular tumor, Phys. rev. 65, No. 2, 1467-1476 (2002)
[20]Turing, A.: The chemical basis of morphogenesis, Philos. trans. Roy. soc. Lond. 237, 37-72 (1952)
[21]A. Madzvamuse, A Numerical Approach to the Study of Spatial Pattern Formation, Ph.D. Thesis, University of Oxford, 2000.
[22]Sekimura, T.; Madzvamuse, A.; Wathen, A.; Maini, P.: A model for colour pattern formation in the butterfly wing of papilio dardanus, Proc. roy. Soc. lond. Ser. B 26, 851-859 (2000)
[23]Madzvamuse, A.; Wathen, A.; Maini, P.: A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains, J. sci. Comput. 24, 247-262 (2005) · Zbl 1080.65091 · doi:10.1007/s10915-004-4617-7
[24]Madzvamuse, A.; Thomas, R.; Maini, P.; Wathen, A.: A numerical approach to the study of spatial pattern formation in the ligaments of arcoid bivalves, Bull. math. Biol 64, 501-530 (2002)
[25]Madzvamuse, A.; Sekimura, T.; Wathen, A.; Maini, P.: A predictive model for colour pattern formation in the butterfly wing of papilio dardanus, Hiroshima math. J. 32, 325-336 (2002) · Zbl 1007.92007
[26]Madzvamuse, A.; Maini, P.: Velocity-induced numerical solution of reaction – diffusion systems on continuously growing domains, Journal of computational physics 225, 100-119 (2007) · Zbl 1122.65076 · doi:10.1016/j.jcp.2006.11.022
[27]E. Crampin, Reaction and Diffusion on Growing Domains, Ph.D. Thesis, University of Oxford, 2000.
[28]H. Meinhardt, Models of Biological Pattern Formation, 1982.
[29]Murray, J.: A prepattern formation mechanism for animal coat markings, J. theor. Biol. 88, 161-199 (1981)
[30]Kondo, S.; Asai, R.: A reaction – diffusion wave on the skin of the marine anglefish, pomacanthus, Nature 376, 765-768 (1995)
[31]Sekimura, T.; Madzvamuse, A.; Wathen, A.; Maini, P.: A model for colour pattern formation in the butterfly wing of papilio dardanus, Proc. roy. Soc. lond. Ser. B 26, 851-859 (2000)
[32]Chaplain, M.; Ganesh, A.; Graham, I.: Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumor growth, J. math. Biol. 42, 387-423 (2001) · Zbl 0988.92003 · doi:10.1007/s002850000067
[33]Barrio, R.; Varea, C.; Aragón, J.; Maini, P.: A two-dimensional numerical study of spatial pattern formation in interacting systems, Bull. math. Biol. 61, 483-505 (1999)
[34]Crampin, E.; Gaffney, E.; Maini, P.: Reaction and diffusion on growing domains: scenarios for robust pattern formation, Bull. math. Biol. 61, 1093-1120 (1999)
[35]A. Kassam, L. Trefethen, Solving reaction – diffusion equations 10 times faster, Oxford University: Numerical Analysis Group Research, Report 16, 2003.
[36]Madzvamuse, A.: Time-stepping schemes for moving grid finite element applied to reaction – diffusion systems on fixed and growing domains, J. comput. Phys. 214, 239-263 (2006) · Zbl 1089.65098 · doi:10.1016/j.jcp.2005.09.012
[37]O. Zienkiewicz, R. Taylor, Finite Element Method: Fluid Dynamics, vol. 3, 2003.
[38]J. Hoffman, Numerical Methods for Engineerings and Scientist, 1992. · Zbl 0823.65006
[39]Schnakenberg, J.: Simple chemical reaction systems with limit cycle behaviour, J. theor. Biol. 81, No. 3, 389-400 (1979)
[40]M. Chaturvedi, R. Huang, C. Izaguirre, J. Newman, S. Glazier, J. Alber, A Hybrid Discrete-Continuum Model for 3-D Skeletogenesis of the Vertebrate Limb, Springer-Verlag, ACRI 2004, pp. 543 – 552. · Zbl 1116.92304 · doi:10.1007/b102055
[41]Chaturvedi, R.; Huang, C.; Kazmierczak, B.; Schneider, T.; Izaguirre, J.; Glimm, T.; Hentschel, H.; Glazier, J.; Newman, S.; Alber, M.: On multiscale approaches to three-dimensional modelling of morphogenesis, J. roy. Soc. interf. 2, 237-253 (2005)
[42]Doblaré, M.; Garzón-Alvarado, D.; García-Aznar, J.: A reaction – diffusion model for long bones growth, Biomech. model. Mechanobiol. 8, No. 5, 381-395 (2009)
[43]Revelli, R.; Ridolfi, L.: Generalized collocation method for two-dimensional reaction – diffusion problems with homogeneous Neumann boundary conditions, Comput. math. Appl. 56, 2360-2370 (2008) · Zbl 1165.65394 · doi:10.1016/j.camwa.2008.05.041
[44]Vanegas, J.; Landínez, N.; Garzón-Alvarado, D. A.: Modelo matemático de la coagulación en la interfase hueso implante dental, Revista cubana de investigaciones biomédicas 28, No. 2 (2009)
[45]K.J. Painter, Chemotaxis as a Mechanism for Morphogenesis, University of Oxford, Ph.D. Thesis, 1997.