Numerical framework and stability analysis of the Gray-Scott model

Turing models of pattern formation provide insight into an intriguing question in developmental biology, like how nature exhibits various structures, shapes, and organized patterns. These natural patterns include the pattern and texture on a desert dune, spots and stripes on the skin of various animals, the growth of a body from a single cell, and the formation of fingerprints. The current work emphasized stability analysis and parameter settings to obtain diverse patterns in nature. As growth is an inevitable continuous process, it is responsible for producing different structures and patterns in living beings. The proposed numerical framework and simulation exhibit realistic natural patterns using reaction-diffusion (RD) models driven by Turing-type instability in the Gray-Scott model. The study proposes a parameter space for Turing patterns using stability analysis. The implemented mathematical model discretizes the continuous partial differential equations into their discrete counterpart by employing a finite-difference scheme. The presented framework combines state-of-the-art spatial and temporal discretization techniques together with stability analysis to mirror stable Turing-type patterns. The proposed numerical scheme is robust, efficient, accurate, and capable of exhibiting diverse biological patterns for the set of parameters in the Turing space, which are validated through stability analysis.