  In this work, we focus on an autocatalytic reaction-diffusion model and carry
out multiple scale weakly nonlinear analysis. A cubic and a quadratic
autocatalytic reaction system is analysed. We develop a framework to identify
the critical surfaces in parameter space across which the nature of the Turing
bifurcation changes from supercritical to subcritical. These are verified by
direct numerical simulations of the system. Using weakly nonlinear analysis, we
derive equations up to the fifth order that governs the amplitude of the
spatial patterns. The limit point of the bifurcating solution is captured
accurately by extending the analysis to the fifth order for the case of
subcritical bifurcation. The numerical solutions are in good agreement with the
predictions of the weakly nonlinear analysis for supercritical bifurcations. We
show that when multiple steady states arise Turing patterns can coexist with
another spatially uniform steady states. Furthermore, we show that our
framework can be extended to get different patterns like squares and hexagons
in a two-dimensional domain. We show that the shape of Turing patterns is
influenced by the domain size. This shows that the geometry can influence the
kind of patterns formed in natural systems. This study will aid the
experimentalist identify operating conditions where Turing patterns can be
obtained.
