Boundary effects on Turing pattern formation in a spiral growing domain

Abstract

In living organisms, growth is an ubiquitous process that influences the dynamics of various chemical processes, particularly pattern formation. Although significant insights into the effects of growth on pattern formation have been obtained through studies using inorganic chemical reactions, few works have explored realistic biological scenarios involving physical boundaries, chemical sources, and accelerated growth modes. In this study, we investigate the effects of physical boundaries and different boundary conditions on Turing pattern formation in both linearly and nonlinearly growing spiral domains using the Lengyel–Epstein model of the chlorine dioxide-iodine-malonic acid (CDIMA) reaction. We identify trends in pattern morphology under various boundary conditions. Specifically, Dirichlet boundary conditions for the activator and inhibitor favor the formation of parallel and spiral-striped patterns, respectively. We find that the chemical activity of the inhibitor plays an important role in inducing spiral patterns of different multiplicities, which depend on the growth speed. We identify that the stability of these spiral patterns is significantly influenced by the presence of internal boundaries. Finally, under non-linear growth conditions, we observe the emergence of complex spiral patterns exhibiting different local multiplicities. Our work provides evidence that boundary conditions are key in inducing pattern variability in reacting growing domains.