Spatiotemporal patterns driven by autocatalytic internal reaction noise

Abstract

The influence that intrinsic local-density fluctuations can have on solutions of mean-field reaction-diffusion models is investigated numerically by means of the spatial patterns arising from two species that react and diffuse in the presence of strong internal reaction noise. The dynamics of the Gray–Scott (GS) model [P. Gray and S. K. Scott, Chem. Eng. Sci. 38, 29 (1983); and ibid.39, 1087 (1984); and J. Phys. Chem. 89, 22 (1985)] with a constant external source is first cast in terms of a continuum field theory representing the corresponding master equation. We then derive a Langevin description of the field theory and use these stochastic differential equations in our simulations. The nature of the multiplicative noise is specified exactly without recourse to assumptions and turns out to be of the same order as the reaction itself, and thus cannot be treated as a small perturbation. Many of the complex patterns obtained in the absence of noise for the GS model are completely obliterated by these strong internal fluctuations, but we find novel spatial patterns induced by this reaction noise in the regions of parameter space that otherwise correspond to homogeneous solutions when fluctuations are not included.