Effect of domain growth rates on Turing patterns during embryo development

Abstract

This work examines reaction–diffusion dynamics for Turing pattern formation on expanding domains, with the aim of explaining how growth drives pattern selection. Under exponential growth, new analytical insights that predict mode jumping (both to neighboring modes and to higher, non-adjacent modes) consistent with prior theoretical accounts have been obtained. Under linear growth, analogous phenomena are shown using FEM where simple closed-form analysis is not possible. For exponential growth, the computational findings align with the analysis, revealing regimes where growth promotes, inhibits, or redirects Turing modes, while for linear growth, they enable a comparison with prior results that establishes analogies between the two regimes. By integrating analytical and numerical perspectives, this study delineates the mechanisms by which domain growth reorganizes the pattern spectrum and shapes the pathways through which form emerges.