Abstract
Starting from the Gierer–Meinhardt setting, we propose a stochastic model to characterize pattern formation on seashells under the influence of random space–time fluctuations. We prove the existence of a positive solution for the resulting system and perform numerical simulations in order to assess the behavior of the solution in comparison with the deterministic approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allen EJ, Novosel SJ, Zhang Z (1998) Finite element and difference approximation of some linear stochastic partial differential equations. Stoch Stoch Rep 64: 117–142
Assing S (1999) Comparison of systems of stochastic partial differential equations. Stoch Process Appl 82: 259–282
Bolley C, Crouzeix M (1978) Conservation de la positivité lors de la discrétization des problèmes d’évolution paraboliques. RAIRO Anal Numer 12(3): 237–245
Chow PL (2007) Stochastic partial differential equations. Chapman & Hall, London
Da Prato G, Zabczyk J (1992) Stochastic equations in infinite dimensions. Cambridge University Press, London
Eriksson K, Johnson C, Logg A (2002) Explicit time-stepping for stiff ODE’s. Chalmers Finite Element Center PREPRINT 2002-7, Göteborg
Evans LC (1999) Partial differential equations. AMS, Providence
Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetik 12: 30–39
Granero-Porati MI, Porati A (1984) Temporal organization in a morphogenetic field. J Math Biol 20: 153–157
Ishikawa M, Miyajima K (2005) Analyses of pattern formation processes in stochastic activator-inhibitor systems with saturation in growth domains. Int J Innov Comp Inf Control 1(3): 303–311
Jiang H (2006) Global existence of solutions of an activator-inhibitor system. Discrete Contin Dyn Syst 14: 737–751
Kelkel J, Surulescu C (2009) A weak solution approach to a reaction-diffusion system modeling pattern formation on seashells. Math Methods Appl Sci (to appear)
Kloeden PE, Platen E (1992) Numerical solution of stochastic differential equations. Springer, Berlin
Meinhardt H (2003) The algorithmic beauty of seashells. Springer, Berlin
Meinhardt H, Gierer A (1974) Applications of a theory of biological pattern formation based on lateral inhibition. J Cell Sci 15: 321–346
Meinhardt H, Klingler M (1987) A model for pattern formation on the shells of molluscs. J Theor Biol 126: 63–69
Moro E, Schurz H (2005) Non-negativity preserving numerical algorithms for stochastic differential equations. arXiv: arXiv:math/0509724v1 [math.NA]
Murray JD (1982) Parameter space for turing instability in reaction diffusion mechanisms: a comparison of models. J Theor Biol 98: 143–163
Murray JD (2003) Mathematical biology II. Springer, Berlin
Rothe F (1984) Global solutions of reaction-diffusion systems. Lecture notes in math, vol 1072. Springer, Berlin
Taniguchi T (1990) On sufficient conditions for non-explosion of solutions to stochastic differential equations. J Math Anal Appl 153: 549–561
Taniguchi T (1992) Successive approximation to solutions of stochastic differential equations. J Differ Equ 96: 152–169
Turing AM (1952) The chemical basis of morphogenesis. Philos Trans R Soc Lond B 237: 37
Walsh JB (1986) An introduction to stochastic partial differential equations. Lecture notes in mathematics, vol 1180. Springer, Berlin, pp 265–439
Wei J, Winter M (2001) Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case. J Nonlinear Sci 11(6): 415–458
Wei J, Winter M (2002) Spikes for the Gierer-Meinhardt system in two dimensions: the strong coupling case. J Differ Equ 178(2): 478–518
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kelkel, J., Surulescu, C. On a stochastic reaction–diffusion system modeling pattern formation on seashells. J. Math. Biol. 60, 765–796 (2010). https://doi.org/10.1007/s00285-009-0284-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-009-0284-5