Stability analysis of non-autonomous reaction-diffusion systems: The effects of growing domains

Journal of Mathematical Biology

Volumn 61, Issue 1, 2010, Pages 133-164

  Madzvamuse, Anotida ^a   Gaffney, Eamonn A ^b   Maini, Philip K ^b  

^a UNIVERSITY OF SUSSEX   (United Kingdom)

^b UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Convection reaction diffusion systems; Domain induced diffusively driven instability; Growing domains asymptotic theory; Pattern formation; Turing diffusively driven instability

Indexed keywords

ARTICLE; BIOLOGICAL MODEL; CHEMICAL MODEL; DIFFUSION; GROWTH; KINETICS; STATISTICAL MODEL;

DIFFUSION; GROWTH; KINETICS; MODELS, BIOLOGICAL; MODELS, CHEMICAL; MODELS, STATISTICAL;

EID: 77952289005     PISSN: 03036812     EISSN: 14321416     Source Type: Journal    
DOI: 10.1007/s00285-009-0293-4     Document Type: Article

References (30)

1. Castets V, Dulos E, Boissonade J, De Kepper P (1990) Experimental evidence of a sustained Turing-type equilibrium chemical pattern. Phys Rev Lett 64(3): 2953-2956.
2. Chaplain M, Ganesh AJ, Graham IG (2001) Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumor growth. J Math Biol 42: 387-423.
3. Connell McCluskey C (2005) A strategy for constructing Lyapunov functions for non-autonomous linear differential equations. Linear Algebra Appl 409: 100-110.
4. Crampin EJ, Gaffney EA, Maini PK (1999) Reaction and diffusion on growing domains: scenarios for robust pattern formation. Bull Math Biol 61: 1093-1120.
5. Crampin EJ, Gaffney EA, Maini PK (2002) Mode doubling and tripling in reaction-diffusion patterns on growing domains: a piecewise linear model. J Math Biol 44: 107-128.
6. Crampin EJ, Hackborn WW, Maini PK (2002) Pattern formation in reaction-diffusion models with nonuniform domain growth. Bull Math Biol 64: 746-769.
7. Edelstein-Keshet L (1988) Mathematical models in biology. Random House, New York.
8. Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetik 12: 30-39.
9. Gjorgjieva J, Jacobsen J (2007) Turing patterns on growing spheres: the exponential case. Dynamical systems and differential equations. In: Proceedings of the 6th AIMS international conference. Discrete continuous dynamical systems supplement, USA, pp 436-445.
10. Golub GH, van Loan CF (1996) Matrix computations. JHU Press ISBN 0801854148.
11. Kondo S, Asai R (1995) A reaction-diffusion wave on the skin of the marine anglefish, Pomacanthus. Nature 376: 765-768.
12. Liaw SS, Yang CC, Liu RT, Hong JT (2001) Turing model patterns of lady beetles. Phys Rev E 64: 041909.
13. Madzvamuse A (2005) Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains. J Comp Phys 24(1): 239-263.
14. Madzvamuse A (2008) Stability analysis of reaction-diffusion systems with constant coefficients on growing domains. Int J Dyn Diff Eq 1(4): 250-262.
15. Madzvamuse A, Maini PK (2007) Velocity-induced numerical solutions of reaction-diffusion systems on fixed and growing domains. J Comp Phys 225: 100-119.
16. Madzvamuse A, Maini PK, Wathen AJ (2003) A moving grid finite element method applied to a model biological pattern generator. J Comp Phys 190: 478-500.
17. Madzvamuse A, Wathen AJ, Maini PK (2005) A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains. J Sci Comp 24(2): 247-262.
18. Maini PK, Crampin EJ, Madzvamuse A, Wathen AJ, Thomas RDK (2002) Implications of domain growth in morphogenesis. In: Capaso V (ed) Mathematical modelling and computing in biology and medicine. Proceedings of the 5th European conference for mathematics and theoretical biology: conference, Milan, vol 153, pp 67-73.
19. Maini PK, Baker RE, Chong CM (2006) The Turing model comes of molecular age, (Invited Perspective). Science 314: 1397-1398.
20. Murray JD (2002) Mathematical biology I and II, 3rd edn. Springer, Berlin.
21. Oster AM, Bressloff PC (2006) A developmental model of ocular dominance column formation on a growing cortex. Bull Math Biol 68: 73-98.
22. Ouyang Q, Swinney HL (1991) Transition from a uniform state to hexagonal and striped Turing patterns. Nature 352: 610-612.
23. Painter KJ, Othmer HG, Maini PK (1999) Stripe formation in juvenile Pomacanthus explained by a generalized Turing mechanism with chemotaxis. Proc Natl Acad Sci 96: 5549.
24. Plaza RG, Sánchez-Garduño F, Padilla P, Barrio RA, Maini PK (2004) The effect of growth and curvature on pattern formation. J Dynam Diff Equ 16(4): 1093-11214.
25. Prigogine I, Lefever R (1968) Symmetry breaking instabilities in dissipative systems. II. J Chem Phys 48: 1695-1700.
26. Schnakenberg J (1979) Simple chemical reaction systems with limit cycle behaviour. J Theor Biol 81: 389-400.
27. Sick S, Reinker S, Timmer J, Schlake T (2006) WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism. Science 314: 1447-1450.
28. Solnica-Krezel L (2003) Vertebrate development: taming the nodal waves. Curr Biol 13: R7-9.
29. Turing A (1952) The chemical basis of morphogenesis. Phil Trans R Soc Lond B 237: 37-72.
30. Varea C, Aragón JL, Barrio RA (1999) Confined Turing patterns in growing systems. Phys Rev E 60: 4588-4592.