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