Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray--Scott equations

Hausenblas E, Randrianasolo TA, Thalhammer M (2019)
Journal of Computational and Applied Mathematics 364: 112335.

Zeitschriftenaufsatz | E-Veröff. vor dem Druck | Englisch
 
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Autor*in
Hausenblas, Erika; Randrianasolo, Tsiry AvisoaUniBi ; Thalhammer, Mechthild
Abstract / Bemerkung
Mathematical models based on systems of reaction-diffusion equations provide fundamental tools for the description and investigation of various processes in biology, biochemistry, and chemistry; in specific situations, an appealing characteristic of the arising nonlinear partial differential equations is the formation of patterns, reminiscent of those found in nature. The deterministic Gray–Scott equations constitute an elementary two-component system that describes autocatalytic reaction processes; depending on the choice of the specific parameters, complex patterns of spirals, waves, stripes, or spots appear. In the derivation of a macroscopic model such as the deterministic Gray–Scott equations from basic physical principles, certain aspects of microscopic dynamics, e.g. fluctuations of molecules, are disregarded; an expedient mathematical approach that accounts for significant microscopic effects relies on the incorporation of stochastic processes and the consideration of stochastic partial differential equations. The present work is concerned with a theoretical and numerical study of the stochastic Gray–Scott equations driven by independent spatially time-homogeneous Wiener processes. Under suitable regularity assumptions on the prescribed initial states, existence and uniqueness of the solution processes is proven. Numerical simulations based on the application of a time-adaptive first-order operator splitting method and the fast Fourier transform illustrate the formation of patterns in the deterministic case and their variation under the influence of stochastic noise
Stichworte
Mathematical and theoretical biology; Mathematical biochemistry; Mathematical chemistry; Reaction-diffusion systems; Gray–Scott equations; Turing patterns; Stochastic partial differential equations; Wiener processes; Numerical approximation; Operator splitting methods; Fast Fourier transform.
Erscheinungsjahr
2019
Zeitschriftentitel
Journal of Computational and Applied Mathematics
Band
364
Art.-Nr.
112335
ISSN
0377-0427
Page URI
https://pub.uni-bielefeld.de/record/2934005

Zitieren

Hausenblas E, Randrianasolo TA, Thalhammer M. Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray--Scott equations. Journal of Computational and Applied Mathematics. 2019;364: 112335.
Hausenblas, E., Randrianasolo, T. A., & Thalhammer, M. (2019). Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray--Scott equations. Journal of Computational and Applied Mathematics, 364, 112335. doi:10.1016/j.cam.2019.06.051
Hausenblas, Erika, Randrianasolo, Tsiry Avisoa, and Thalhammer, Mechthild. 2019. “Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray--Scott equations”. Journal of Computational and Applied Mathematics 364: 112335.
Hausenblas, E., Randrianasolo, T. A., and Thalhammer, M. (2019). Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray--Scott equations. Journal of Computational and Applied Mathematics 364:112335.
Hausenblas, E., Randrianasolo, T.A., & Thalhammer, M., 2019. Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray--Scott equations. Journal of Computational and Applied Mathematics, 364: 112335.
E. Hausenblas, T.A. Randrianasolo, and M. Thalhammer, “Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray--Scott equations”, Journal of Computational and Applied Mathematics, vol. 364, 2019, : 112335.
Hausenblas, E., Randrianasolo, T.A., Thalhammer, M.: Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray--Scott equations. Journal of Computational and Applied Mathematics. 364, : 112335 (2019).
Hausenblas, Erika, Randrianasolo, Tsiry Avisoa, and Thalhammer, Mechthild. “Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray--Scott equations”. Journal of Computational and Applied Mathematics 364 (2019): 112335.
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2019-07-25T07:16:11Z
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