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

Abstract 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, as well as the 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.