Turing Patterns in the BZ-AOT Reaction Theory

Vanag and Epstein have formulated a four-variable model to understand pattern formation in the BZ-AOT system [449,453]. Their model builds on the Oregonator, see Sect. 1.4.8. It assumes that the chemistry within the water core of the droplets is well described by the two-variable Oregonator rate equations (1.131). It further assumes that the species in the oil phase are inert, since they lack reaction partners, the key reactants all being confined to the aqueous core of the droplets. Consequently, only transfer reactions occur for the activator B1O2 and inhibitor Br2 in the oil phase. The rate terms for the two transfer reactions are added to the rate terms of the two-variable Oregonator model. The reaction-diffusion equations of the four-variable model of the BZ-AOT system are given in nondimensionalized form by 

Here U = HBr02, V is the oxidized form of the catalyst, W = Br02 in the oil phase, and Z = Bf2 in the oil phase. The parameters e, 2, xj, a, p, y, x, and q depend on the rate constants, the bromate concentration, and the droplet volume fraction (p, h is an adjustable stoichiometric parameter and e, 63 1. The diffusion coefficients 

The four-variable reaction-diffusion system (12.35) possesses a nontrivial uniform steady state given by 

To assess the stability of the uniform steady state (12.36), we carry out again a hnear stability analysis  

The growth rates of the A th spatial mode are given by the eigenvalues of the Jacobian K = k ) 

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