Selection and Orientation of Reaction-Diffusion Patterns on Anisotropically Growing Domains

Keywords: Reaction-diffusion

Abstract: Reaction-diffusion (RD) systems are a cornerstone of mathematical biology, yet the role of domain dynamics, particularly growth geometry, in pattern selection remains an active area of research. This work presents a systematic computational study demonstrating how anisotropic domain growth can act as a powerful mechanism for selecting and orienting Turing patterns. We first map the parameter space of the Gray-Scott model to identify distinct pattern regimes. We then focus on a stripe-forming regime to systematically vary the anisotropy of the domain growth, from purely isotropic to fully anisotropic. To quantify the global alignment of the resulting patterns, we introduce an Orientation Order Parameter (OOP) derived from the 2D power spectrum. Our results show a clear transition: as growth becomes more anisotropic, the system evolves from a disordered state of randomly oriented stripe domains to a globally aligned state. This transition is quantified by the OOP, which shows a systematic increase with the anisotropy of the growth. Furthermore, by analyzing the angular distribution of power in Fourier space, we provide a deeper insight into how specific orientations are selected. We provide a theoretical discussion, explaining this phenomenon through the lens of Fourier space, where anisotropic growth selectively disfavors wave vectors aligned with the growth direction. This work provides strong quantitative evidence that the geometry of growth is a critical factor in morphogenesis, capable of imposing large-scale order on local pattern-forming instabilities.

Submission Number: 311