TY  - JOUR
AB  - 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
DA  - 2019
DO  - 10.1016/j.cam.2019.06.051
KW  - Mathematical and theoretical biology
KW  - Mathematical biochemistry
KW  - Mathematical chemistry
KW  - Reaction-diffusion systems
KW  - Gray–Scott equations
KW  - Turing patterns
KW  - Stochastic partial differential equations
KW  - Wiener processes
KW  - Numerical approximation
KW  - Operator splitting methods
KW  - Fast Fourier transform.
LA  - eng
PY  - 2019
SN  - 0377-0427
T2  - Journal of Computational and Applied Mathematics
TI  - Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray--Scott equations
UR  - https://nbn-resolving.org/urn:nbn:de:0070-pub-29340058
Y2  - 2024-11-22T01:08:43
ER  -