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All issues Volume 11 / No 5 (2016) Math. Model. Nat. Phenom., 11 5 (2016) 4-32 References
Free Access
Issue
Math. Model. Nat. Phenom.
Volume 11, Number 5, 2016
Bifurcations and Pattern Formation in Biological Applications
Page(s) 4 - 32
DOI https://doi.org/10.1051/mmnp/201611502
Published online 07 December 2016
  1. W. Bangerth, T. Heister, L. Heltai, G. Kanschat, M. Kronbichler, M. Maier, B. Turcksin, T.D. Young (2013). The deal.II Library, Version 8.1, arXiv preprint. [Google Scholar]
  2. R. Barreira, C.M. Elliott, A. Madzvamuse, The surface finite element method for pattern formation on evolving biological surfaces, Journal of Math. Bio., 63 (2011), 1095–1119. [Google Scholar]
  3. S. Bianco, F. Tewes, L. Tajber, V. Caron, O.I. Corrigan, A.M. Healy. Bulk, Surface properties and water uptake mechanisms of salt/acid amorphous composite systems. Int. J. Pharm., (2013), 456143–152. [Google Scholar]
  4. H.R. Chang, R.L. Grossman. Evaluation of bulk surface flux algorithms for light wind conditions using data from the Coupled Ocean-Atmosphere Response Experiment (COARE). Q. J. R. Meteorol. Soc. 125 (1999), 1551–1588. [Google Scholar]
  5. A.V. Chechkin, I.M. Zaid, M.A. Lomholt, I.M. Sokolov, R. Metzler, Bulk-mediated diffusion on a planar surface: Full solution. Phys. Rev. E. 86 (2012), 041101. [Google Scholar]
  6. G. Dziuk, C.M. Elliott. Surface finite elements for parabolic equations. J. Comp. Math., 25 (2007), 385–407. [Google Scholar]
  7. C.M. Elliott, T. Ranner, Finite element analysis for a coupled bulk-surface partial differential equation. IMA J. Num. Anal., 33, (2) (2012), 377–402. [Google Scholar]
  8. C.M. Elliott, B. Stinner, C. Venkataraman, Modelling cell motility and chemotaxis with evolving surface finite elements. J. Roy. Soc. Inter., 9 (76) (2013), 3027–3044. [Google Scholar]
  9. I. Garate, L. Glazman, Weak localization and antilocalization in topological insulator thin films with coherent bulk-surface coupling. Phys. Rev. B, 86 (2012), 035422. [Google Scholar]
  10. A. Gierer, H. Meinhardt, A theory of biological pattern formation. Kybernetik. 12 (1972), 30–39. [CrossRef] [PubMed] [Google Scholar]
  11. A. Hahn, K. Held, L. Tobiska, Modelling of surfactant concentration in a coupled bulk surface problem. PAMM Proc. Appl. Math. Mech., 14 (2014), 525–526. [CrossRef] [Google Scholar]
  12. G. Hetzer, A. Madzvamuse, W. Shen, Characterization of Turing diffusion-driven instability on evolving domains. Discrete and Continuous Dynamical Systems,- Series A, 32 (11) (2012), 3975–4000. [CrossRef] [MathSciNet] [Google Scholar]
  13. G.C. Holmes. The use of hyperbolic cosines in solving cubic polynomials. Mathematical Gazette, 86 (2002), 473–477. [CrossRef] [Google Scholar]
  14. D. Köster, O. Kriessl, K.G. Siebert. Design of finite element tools for coupled surface and volume meshes. Mathematik. Technical Report, 2008-01, (2008). [Google Scholar]
  15. Y. Kwon, J.J. Derby. Modeling the coupled effects of interfacial and bulk phenomena during solution crystal growth. J. Cry. Growth, 230 (2001), 328–335. [CrossRef] [Google Scholar]
  16. O. Lakkis, A. Madzvamuse, C. Venkataraman, Implicit–Explicit Timestepping with Finite Element Approximation of Reaction–Diffusion Systems on Evolving Domains. SINUM, 51 (2013), 2309–2330. [CrossRef] [Google Scholar]
  17. H. Levine, W.J. Rappel. Membrane-bound Turing patterns. Phys. Rev. E., 72(6) (2005). [Google Scholar]
  18. C.B. Macdonald, B. Merrimanb, S.J. Ruuth. Simple computation of reaction-diffusion processes on point clouds. Proc. Nat. Acad. Sci., 110 (2013), 9209–9214. [CrossRef] [MathSciNet] [Google Scholar]
  19. G. Macdonald, J.A. Mackenzie, M. Nolan, R.H. Insall. A computational method for the coupled solution of reaction-diffusion equations on evolving domains and manifolds: Application to a model of cell migration and chemotaxis. Journal of Computational Physics, 309 (2016), 207–226. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  20. A. Madzvamuse. A numerical approach to the study of spatial pattern formation. DPhil Thesis. University of Oxford, (2000). [Google Scholar]
  21. A. Madzvamuse, E.A. Gaffney, P.K. Maini. Stability analysis of non-autonomous reaction-diffusion systems: The effects of growing domains. Journal of Mathematical Biology, 61 (1) (2010), 133–164. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  22. A. Madzvamuse, A.H.W. Chung. Fully implicit time-stepping schemes and non-linear solvers for systems of reaction-diffusion equations. Appl. Math. Comp., 244 (2014), 361–374. [CrossRef] [Google Scholar]
  23. A. Madzvamuse, R. Barreira, Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces. Physical Review E., 90 (2014), 043307-1-043307-14 [Google Scholar]
  24. A. Madzvamuse, A.H.W. Chung, C. Venkataraman. Stability analysis and simulations of coupled bulk-surface reaction-diffusion systems. Proc. Roy. Soc. A., 471 (2175) (2015), 20140546. [CrossRef] [Google Scholar]
  25. A. Madzvamuse, H.S. Ndakwo, R. Barreira, Cross-diffusion-driven instability for reaction-diffusion systems: analysis and simulations. Journal of Mathematical Biology, 70(4) (2015), 709–743. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  26. A. Madzvamuse, A.H.W. Chung. The bulk-surface finite element method for reaction-diffusion systems on stationary volumes. Finite Elements in Analysis and Design, 108 (2016), 9–21. [CrossRef] [Google Scholar]
  27. A. Madzvamuse, H.S. Ndakwo, R. Barreira, Stability analysis of reaction-diffusion models on evolving domains: the effects of cross-diffusion. Discrete and Continuous Dynamical Systems - Series A, 36 (4) (2016), 2133–217. [CrossRef] [MathSciNet] [Google Scholar]
  28. E.S. Medvedev, A.A. Stuchebrukhov. Proton diffusion along biological membranes. J. Phys. Condens. Matter., 23 (2011), 234103. [CrossRef] [PubMed] [Google Scholar]
  29. E.S. Medvedev, A.A. Stuchebrukhov. Mechanism of long-range proton translocation along biological membranes. FEBS Lettes, 587 (2013), 345–349. [CrossRef] [PubMed] [Google Scholar]
  30. J.D. Murray. Mathematical Biology II: Spatial models and biomedical applications. Third Edition. Springer, (2003). [Google Scholar]
  31. W. Nagata, H.R.Z. Zangeneh, D.M. Holloway. Reaction-diffusion patterns in plant tip morphogenesis: bifurcations on spherical caps. Bull. Math. Biol., 75 (2013), 2346–2371. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  32. D.R. Nisbet, A.E. Rodda, D.I. Finkelstein, M.K. Horne, J.S. Forsythe, W Shen. Surface and bulk characterisation of electrospun membranes: Problems and improvements. Colloids and Surfaces B: Biointerfaces., 71 (2009), 1–12. [CrossRef] [Google Scholar]
  33. I.L. Novak, F. Gao, Y-S. Choi, D. Resasco, J.C. Schaff, B.M. Slepchenko. Diffusion on a curved surface coupled to diffusion in the volume: Application to cell biology. J. Comp. Phys., 226 (2007), 1271–1290. [CrossRef] [Google Scholar]
  34. I. Prigogine, R. Lefever, Symmetry breaking instabilities in dissipative systems. II. J. Chem. Phys., 48 (1968), 1695–1700. [Google Scholar]
  35. A. Rätz, M. Röger. Turing instabilities in a mathematical model for signaling networks. J. Math. Biol., 65 (2012), 1215–1244. [Google Scholar]
  36. A. Rätz, M. Röger. Symmetry breaking in a bulk-surface reaction-diffusion model for signaling networks. arXiv:1305.6172v1, (2013). [Google Scholar]
  37. I. Rozada, S. Ruuth, M.J. Ward. The stability of localized spot patterns for the Brusselator on the sphere. SIADS, 13 (1) (2014), 564–627. [CrossRef] [Google Scholar]
  38. Y. Saad. Iterative Methods for Sparse Linear Systems (2nd ed.). SIAM., (2003), 231–234. ISBN 978-0-89871-534-7. [Google Scholar]
  39. J. Schnakenberg. Simple chemical reaction systems with limit cycle behaviour. J. Theor. Biol., 81 (1979), 389–400. [PubMed] [Google Scholar]
  40. N. Tuncer, A. Madzvamuse, Projected finite elements for systems of reaction-diffusion equations on closed evolving spheroidal surfaces. Communications in Computational Physics. Accepted (2016). [Google Scholar]
  41. A. Turing. On the chemical basis of morphogenesis. Phil. Trans. Royal Soc. B., 237 (1952), 37–72. [Google Scholar]
  42. H.A. Van der Vorst. A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems. SIAM J. Sci. and Stat. Comput., 13 (2) (1992), 631–644. doi:10.1137/0913035. [Google Scholar]
  43. C. Venkataraman, O. Lakkis, A. Madzvamuse, Global existence for semilinear reaction-diffusion systems on evolving domains. J. Math. Biol., 64 (2012), 41–67. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  44. C. Venkataraman, T. Sekimura, E.A. Gaffney, P.K. Maini, A. Madzvamuse, Modeling parr-mark pattern formation during the early development of Amago trout. Phys. Rev. E, 84 (4) (2011), 041923. [Google Scholar]

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