Turing instability and pattern formation in a diffusive Sel’kov–Schnakenberg system

An Efficient Linearized Difference Algorithm for a Diffusive Sel′kov–Schnakenberg System

“…And when a < a T , the positive equilibrium E * is unstable. When a = a T with k = k c , the system undergoes Turing bifurcation, where a T , k c are the parameters in literature [7].…”

“…Turing's theory reveals the principles of the relationship between the patterns produced by convection-diffusion systems and these phenomena [3]. As a famous example related to cellular processes in biochemical systems, the Sel ′ kov-Schnakenberg system has attracted the attention of many scholars on the stability and the existence of steady-state solutions [4][5][6][7]. The Sel ′ kov-Schnakenberg model, as an extension of the Sel'kov model [8] and Schnakenberg model [9], can describes the limit cycle behavior:…”

“…Numerous works in the literature are devoted to the study of ( 2)-( 5), including the Sel ′ kov model and the Schnakenberg model; See [1,2,7,[10][11][12][13][14], and the references therein. In 2015, Zhou and Shi [14] investigate stability, instability, time-periodic orbits and spatiotemporal patterns through bifurcation methods and Leray-Schauder degree theory.…”

“…In 2021, Al Noufaey [2] used the Galerkin method to study the singularity behavior and stability of the Sel ′ kov-Schnakenberg system. In 2023, Wang and Zhou et al [7] studied the Turing instability (diffusion drive) causing spatial patterns and obtained the conditions for the existence of Turing bifurcations, then the changes in the spatiotemporal pattern that depend on the parameters were theoretically analyzed, and a series of numerical experimental simulations were conducted to verify the analysis results through the finite difference method. To the best of our knowledge, there is relatively limited research on numerical algorithms for the Sel ′ kov-Schnakenberg system at present.…”