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

This paper considers a chemical reaction-diffusion model for studying pattern formation with the Sel’kov–Schnakenberg model. Firstly, the stability conditions of the positive equilibrium and the existing conditions of the Hopf bifurcation are established for the local system. Then, Turing instability (diffusion-driven), which causes the spatial pattern is investigated and the existing condition of the Turing bifurcation is obtained. In addition, the dynamic behaviors near the Turing bifurcation are also studied by employing the method of weakly nonlinear analysis. The theoretical analysis shows that spatio-temporal patterns change from the spot, mixed (spot-stripe) to stripe with the variation of parameters, which can be verified by a series of numerical simulations. These numerical simulations give a visual representation of the evolution of spatial patterns. Our results not only explain the evolution process of reactant concentration, but also reveal the mechanism of spatio-temporal patterns formation.