Oregonator generalization as a minimal model of quorum sensing in Belousov–Zhabotinsky reaction with catalyst confinement in large populations of particles

We consider the case of the Belousov-Zhabotinsky (BZ) oscillating reaction where the metal catalyst is confined in a discrete phase of cation-exchange resin particles. We show that modelling of the sudden transitions from the steady state to fully synchronized oscillations, which were previously described only in terms of quorum sensing, does not necessarily require fine-graining of the models into individual particles. As a minimal representation of the system, we propose a generalized version of the Oregonator with only one additional parameter phi that corresponds to the total volume fraction of the catalytic phase in the system. We show that the kinetic consequences of catalyst confinement can be interpreted through phi acting as the second variable stoichiometric factor in the Oregonator mechanism that decreases the order of autocatalysis and simultaneously weakens the negative feedback. Based on the proposed model, an analysis of the so-called quorum sensing is provided in terms of the standard concepts of Hopf bifurcation. The model reproduces correctly that (1) oscillations are only exhibited above a certain critical threshold of phi; (2) as phi decreases, so do the amplitudes of oscillations; whereas (3) the periods of oscillations are almost independent of phi. The model thus appears to be a reasonable minimal representation of uniform states in the BZ reaction with catalyst confinement, and therefore might be a convenient starting point for further refinements representing more complicated transitions in this system.