A moving grid finite element method applied to a model biological pattern generator

Journal of Computational Physics

Volumn 190, Issue 2, 2003, Pages 478-500

  Madzvamuse, Anotida ^a   Wathen, Andrew J ^b   Maini, Philip K ^b  

^a AUBURN UNIVERSITY   (United States)

^b UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Butterfly wing; Colour pattern formation; Moving grid finite elements; Papilio dardanus; Reaction diffusion systems; Turing instability

Indexed keywords

BIOINFORMATICS; DIFFUSION IN LIQUIDS; FINITE ELEMENT METHOD;

BUTTERFLY WINGS; COLOR PATTERN; COLOR PATTERN FORMATION; MOVING GRID; MOVING GRID FINITE ELEMENT; MOVING GRID FINITE ELEMENT METHOD; PAPILIO DARDANI; PATTERN FORMATION; REACTION DIFFUSION SYSTEMS; TURING INSTABILITY;

DYNAMIC MODELS;

EID: 0141515669     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0021-9991(03)00294-8     Document Type: Article

References (50)

1. R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
2. J.L. Aragón, C. Varea, R.A. Barrio, P.K. Maini, Spatial patterning in modified turing systems: application to pigmentation patterns on Marine fish, Forma 13 (1998) 213-221.
3. J.L. Aragón, M. Torres, D. Gil, R.A. Barrio, P.K. Maini, Turing patterns with pentagonal symmetry, Phys. Rev. E 65 (2002) 1-9.
4. M.J. Baines, Moving finite elements, Monographs on Numerical Analysis, Clarendon Press, Oxford, 1994.
5. M.J. Baines, A.J. Wathen, Moving finite element methods for evolutionary problems. I. Theory, J. Comp. Phys. 79 (1988) 245-269.
6. Y.I. Balkarei, A.V. Grigor̈Yants, Y.A. Rhzanov, M.I. Elinson, Regenerative oscillations, spatial-temporal single pulses and static inhomogeneous structures in optically bistable semiconductors, Opt. Commun. 66 (1988) 161-166.
7. R.A. Barrio, C. Varea, J.L. Aragón, P.K. Maini, A two-dimensional numerical study of spatial pattern formation in interacting systems, Bull. Math. Biol. 61 (1999) 483-505.
8. R.A. Barrio, P.K. Maini, J.L. Aragón, M. Torres, Size-dependent symmetry breaking in models for morphogenisis, Physica D 2920 (2002) 1-12.
9. J.T. Batima, Unsteady Euler airfoil solutions using unstructured dynamic meshes, AIAA J. 28 (8) (1990) 1381-1388.
10. J.F. Blom, Considerations of the spring analogy, Int. J. Numer. Meth. Fluids 12 (2000) 647-668.
11. S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer, Berlin, 1994.
12. N.N. Carlson, K. Miller, Design and application of a gradient-weighted moving finite element code I: in one dimension, SIAM J. Sci. Comput. 19 (3) (1998) 728-765.
13. N.N. Carlson, K. Miller, Design and application of a gradient-weighted moving finite element code II: in two dimensions, SIAM J. Sci. Comput. 19 (3) (1998) 766-798.
14. E.J. Crampin, Reaction-diffusion patterns on growing domains, D. Phil. Thesis, University of Oxford, 2000.
15. E.J. Crampin, E.A. Gaffney, P.K. Maini, Reaction and diffusion on growing domains: scenarios for robust pattern formation, Bull. Math. Biol. 61 (2002) 1093-1120.
16. L. Edelstein-Keshet, Mathematical Models in Biology, Random House, New York, 1988.
17. P. Gray, S.K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability, Chem. Eng. Sci. 38 (1) (1983) 29-43.
18. P. Gray, S.K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and the instabilities in the system A + 2B → 3B, B → C, Chem. Eng. Sci. 39 (6) (1984) 1087-1097.
19. A. Gierer, H. Meinhardt, A theory of biological pattern formation, Kybernetik 12 (1972) 30-39.
20. P.K. Jimack, A.J. Wathen, Temporal derivatives in the finite element method on continuously deforming grids, SIAM J. Numer. Anal. 28 (4) (1991) 990-1003.
21. C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, MA, 1987.
22. S. Kondo, R. Asai, A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus, Nature 376 (1995) 765-768.
23. V.I. Krinsky, Self-organisation, Auto-Waves and Structures Far From Equilibrium, Springer, Berlin, 1984.
24. A. Madzvamuse, A numerical approach to the study of spatial pattern formation, D. Phil. Thesis, University of Oxford, 2000.
25. A. Madzvamuse, P.K. Maini, A.J. Wathen, T. Sekimura, A predictive model for color pattern formation in the butterfly wing of Papilio dardanus, Hiroshima Math. J. 32 (2002) 325-336.
26. A. Madzvamuse, R.D.K. Thomas, P.K. Maini, A.J. Wathen, A numerical approach to the study of spatial pattern formation in the Ligaments of Arcoid Bivalves, Bull. Math. Biol. 64 (2002) 501-530.
27. P.K. Maini, M. Solursh, Cellular mechanisms of pattern formation in the development of limb, Int. Rev. Cytol. 129 (1991) 91-133.
28. H. Meinhardt, The Algorithmic Beauty of Sea Shells, Springer, Heidelberg, New York, 1995.
29. K. Miller, R.N. Miller, Moving finite elements. Part I, SIAM J. Numer. Anal. 18 (1981) 1019-1032.
30. K. Miller, Moving finite elements. Part II, SIAM J. Numer. Anal. 18 (1981) 1033-1057.
31. J.D. Müller, P.L. Roe, H. Deconinck, A frontal approach for internal node generation for Delaunay triangulation, Int. J. Numer. Methods Fluids 17 (3) (1993) 241-256.
32. J.D. Murray, On pattern formation mechanisms for lepidopteran wing patterns and mammalian coat markings, Philos. Trans. Roy. Soc. London B 295 (1981) 473-496.
33. J.D. Murray, Mathematical Biology, Springer, Heidelberg, New York, 1993.
34. H.F. Nijhout, A comprehensive model for colour pattern formation in butterflies, Proc. Roy. Soc. London B 239 (1990) 81-113.
35. T. Nozakura, S. Ikeuchi, Formation of dissipative structures in galaxies, Astrophys. J. 279 (1984) 40-52.
36. K.J. Painter, Chemotaxis as a mechanism for morhogenesis, D. Phil. Thesis, University of Oxford, 1997.
37. K.J. Painter, P.K. Maini, H.G. Othmer, Stripe formation in juvenile Pomacanthus explained by a generalised Turing mechanism with chemotaxis, Proc. Natl. Acad. Sci. USA 96 (1999) 5549-5554.
38. T.N. Reddy, An Introduction to the Finite Element Method, McGraw-Hill, New York, 1984.
39. Y. Saad, SPARSEKIT, a basic tool kit for sparse matrix computations, 1994, http://www-users.cs.umn.edu/ saad/.
40. Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publishing, 1996.
41. J. Schnakenberg, Simple chemical reaction systems with limit cycle behavior, J. Theoret. Biol. 81 (1979) 389-400.
42. L.A. Segel, J.L. Jackson, Dissipative structure: an explanation and an ecological example, J. Theoret. Biol. 37 (1972) 545-559.
43. T. Sekimura, P.K. Maini, J.B. Nardi, M. Zhu, J.D. Murray, Pattern formation in lepidopteran wings, Comm. Theoret. Biol. 5 (2-4) (1998) 69-87.
44. T. Sekimura, M. Zhu, J. Cook, P.K. Maini, J.D. Murray, Pattern formation of scale cells in lepidoptera by differential origin-dependent cell adhesion, Bull. Math. Biol. 61 (1999) 807-827.
45. T. Sekimura, A. Madzvamuse, A.J. Wathen, P.K. Maini, A model for colour pattern formation in the butterfly wing of Papilio dardanus, Proc. Roy. Soc. London B 267 (2000) 851-859.
46. W. Sun, T. Tang, J.M. Ward, J. Wei, Numerical challenges for resolving spike dynamics for two reaction-diffusion systems, Stud. Appl. Math. III (2003) 41-84.
47. D. Thomas, Artificial enzyme membrane, transport, memory and oscillatory phenomena, in: D. Thomas, J.-P. Kervenez (Eds.), Analysis and Control of Immobilised Enzyme Systems, Springer, Berlin, Heidelberg, New York, 1975, pp. 115-150.
48. R.D.K. Thomas, A. Madzvamuse, P.K. Maini, A.J. Wathen, Growth patterns of noetiid ligaments: implications of developmental models for the origin of an evolutionary novelty among arcoid bivalves, in: E.M. Harper, J.D. Taylor, J.A. Crame (Eds.), The Evolutionary Biology of the Bivalvia, vol. 177, Geological Society Special Publication, 2000, pp. 279-289.
49. A.M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London B 237 (1952) 37-72.
50. D.B. White, The planforms and onset of convection with a temperature dependent viscosity, J. Fluid Mech. 191 (1988) 247-286.