Turing Instability in Small Inhomogeneous Arrays

We consider inhomogeneous networks with Lengyel-Epstein kinetics where the substrate concentration, i.e., the value of a, can differ from reactor to reactor [208]. All other parameters are independent of the reactor number i. The evolution equations are 

Generally, inhomogeneities in parameters of an array of reactors lead to nonuniform steady states. This is not the case for Lengyel-Epstein networks with inhomogeneities in the parameter a, as is clear from the structure of (13.139). The network still has a unique uniform steady state given by (13.52). We use the Routh-Hurwitz criterion to determine the stability boundaries of this USS. Note that the Routh-Hurwitz analysis is general and can deal with the case where inhomogeneities in parameters lead to nonuniform steady states. Let 

Therefore, the Turing bifurcation and the Hopf bifurcation occur together, if 

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