Stable pattern and standing wave formation in a simple isothermal cubic autocatalytic reaction scheme (Q1905554)

The authors study the formation of stable patterns in a reaction-diffusion system based on the cubic autocatalator, \(A+ 2B\to 3B\), \(B\to C\) with the reaction taking place within a closed regime, the reactant \(A\) being replenished by the slow decay of precursor \(P\) via the reaction \(P\to A\). Let \(D_a\) and \(D_b\) be the diffusion coefficients of chemical species \(A\) and \(B\) respectively and \(D= D_b/D_a\). The main result of this paper asserts that a necessary condition for the bifurcation of the steady state solutions to be stable, spatially non-uniform patterns is that \(D< 3- 2\sqrt 2\). It is also shown that both supercritical and subcritical bifurcations may occur. Several numerical evaluations are given to illustrate these results, as well as for showing that stable patterns can lose stability through supercritical Hopf bifurcations.