Spatiotemporal Dynamics of Reaction–Diffusion System and Its Application to Turing Pattern Formation in a Gray–Scott Model

This paper deals with the effects of partial differential equations on the development of spatiotemporal patterns in reaction–diffusion systems. These systems describe how the concentration of a certain substance is distributed in space or time under the effect of two phenomena: the chemical reactions of different substances into some other product and the diffusion which results in the dispersion of a certain substance over a surface in space. Mathematical representation of these types of models are named the Gray–Scott model, which exhibits the formation of patterns and morphogenesis in living organisms, e.g., the initial formation of patterns that occur in cell development, etc. To explore the nonhomogeneous steady-state solutions of the model, we use a novel high-order numerical approach based on the Chebyshev spectral method. It is shown that the system is either in uniform stabilized steady states in the case of spatiotemporal chaos or lead to bistability between a trivial steady state and a propagating traveling wave. When the diffusion constant of each morphogen is different in any two species of the reaction–diffusion equation, diffusion-driven instability will occur. For the confirmation of theoretical results, some numerical simulations of pattern formation in the Gray–Scott model are performed using the proposed numerical scheme.