The Bifurcation Growth Rate for the Robust Pattern Formation in the Reaction–Diffusion System on the Growing Domain

Among living organisms, there are species that change their patterns on their body surface during their growth process and those that maintain their patterns. Theoretically, it has been shown that large-scale species do not form distinct patterns. However, exceptionally, even large-scale species like giraffes form and maintain patterns, and previous studies have shown that the growth plays a crucial role in pattern formation and transition. Here we show how the growth of the domain contributes to Turing bifurcation based on the reaction-diffusion system by applying the Gray-Scott model to the reaction terms, both analytically and numerically, focusing on the phenomenon of pattern formation and maintenance in large species like giraffes, where melanocytes are widely distributed. After analytically identifying the Turing bifurcation related to the growth rate, we numerically verify the pattern formation and maintenance in response to the finite-amplitude perturbations of the blue state specific to the Gray-Scott model near the bifurcation. Furthermore, among pairs of the parameters that form Turing patterns in a reaction-diffusion system on a fixed domain, we determine a pair of the parameters that maximizes the growth rate for the Turing bifurcation in a reaction-diffusion system on a time-dependently growing domain. Specifically, we conduct a numerical analysis to pursue the pair of the parameters in the Turing space that can be the most robust in maintaining the patterns formed on the fixed domain, even as the domain grows. This study may contribute to specifically reaffirming the importance of growth rate in pattern formation and understanding patterns that are easy to maintain even during growth.