The generation mechanisms of Turing patterns have proven to be instrumental in representing biological systems, particularly those involving morphogenesis and pigmentation. While reactiondiffusion models of the molecules responsible for these biological processes have been extensively studied, the impact of interconnected domains on Turing pattern formation remains unexplored. This article delves into the intricate dynamics of Turing pattern formation within interconnected reactiondiffusion systems using the Schnakenberg model. Two coupling mechanisms are examined: (a) diffusive partition walls with variable width (5-100%) and diffusivity (d = 0.1, 10), and (b) direct contact boundaries where domains share a common interface. We examine how partition walls and direct contact boundaries between domains influence the morphology and stability of the resulting patterns. Finite element simulations of modal combinations (1, 1), (2, 2), (3, 3), and (4, 4) demonstrate that interconnected domains exhibit pattern generation mechanisms fundamentally different from the classical ‘local activation and long-range inhibition’ paradigm. Specifically, we observe modal competition, where lower-frequency modes suppress higher-frequency ones, and critical thresholds beyond which pattern formation is completely inhibited, particularly when the partition width exceeds 50% at low diffusivity (d = 0.1). Direct contact boundaries generate greater morphological diversity, giving rise to lobe-shaped and triangular structures not present in isolated domains, indicating that the immediacy of interdomain interaction strongly influences pattern morphology. These findings provide insights into the self-organization mechanisms governing biological pattern formation and offer potential applications in understanding pigmentation patterns and morphogenesis. Furthermore, these results open new perspectives for exploring alternative geometrical configurations, heterogeneous diffusion processes, and the extension of reaction-diffusion models to both two- and three-dimensional interconnected domains.
Villamizar et al. (Wed,) studied this question.