Semi-Analytical Solutions of the Schnakenberg Model in a Circularly Symmetric Reaction–Diffusion Annulus

This paper investigates the semi-analytical solutions of the Schnakenberg reaction system in a circularly symmetric reaction–diffusion annulus. The Galerkin method is used to approximate spatial concentration profiles of both the autocatalyst and reactant. This method transforms governing partial differential equations into a system of ordinary differential equations, allowing a semi-analytical investigation of this special geometric domain. Singularity theory is then used to determine the regions of the parameter space where different steady states are obtained, and a degenerate Hopf bifurcation analysis is used to determine the region where the Hopf bifurcation points are located. A novel result is that the width of the annulus significantly influences both static and dynamic multiplicity; specifically, a thicker annulus expands the region of parameter space where multiple steady-state solutions and Hopf bifurcations exist. Finally, the reliability and precision of these semi-analytical results are confirmed through direct comparison with numerical solutions of the original partial differential equations.