Turing Threshold

As discussed in Sect. 1.4.9, the CDIMA reaction in a single CSTR is well described by the two-variable Lengyel-Epstein model. The evolution of a homogeneous network of n reactors with Lengyel-Epstein kinetics is governed by the set of ordinary differential equations 

The system (13.46) has a unique uniform steady state given by 

To obtain the threshold condition for the Turing bifurcation of this USS, we need to determine all conditions for which the eigenvalues of Jg can vanish. According to Theorem 13.8, we need to determine the conditions such that 

For b sufficiently large, the uniform steady state of Q, given by pi = p, is stable. As mentioned above, with a large enough, the Turing bifurcation occurs before the Hopf bifurcation as b decreases. Define the set F as 

This implies that a Turing instability can occur only if a is larger than a°, see (1.159), i.e., the iodide ion must be an aetivator. Since r = —fi and fij P2 j 2, the maximum value of k for which a Turing instability can occur is 

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In order for the Turing bifureation to occur first, the Turing threshold must lie below the Hopf threshold of the well-mixed system, b - < bn-... 

Remark 10.3 The analysis of all three approaches to two-variable reaction-transport systems with inertia establishes that the Turing instability of reaction-diffusion systems is structurally stable. The threshold conditions are either the same, HRDEs and reaction-Cattaneo systems, or approach the reaction-diffusion Turing threshold smoothly as the inertia becomes smaller and smaller, t 0. Further, inertia effects induce no new spatial instabilities of the uniform steady state in the diffusive regime, T small. A spatial Hopf bifurcation to standing wave patterns can only occur in the opposite regime, the ballistic regime. 

The coefficients of this equation differ from those of the corresponding equation for the Brusselator DIRW (10.97) by terms that are first order in x, namely the terms with 1 hx and 1/ /Iv, and a term that is second order in r, namely the term with l/(/u,v). In the Brusselator DDRW with activated UV complex, inertia and activation energy both affect the threshold of the Turing bifurcation at first order, in contrast to the DDRW with activated U dimer. We find for the Turing threshold T,uv = o,uv + i,uv + ... 

If the loss rate of the activator is nonlinear, then we need to solve the Turing threshold condition C2 — 4c4 = 0, which provides two solutions for the ratio 9 ... 

A plot of this curve, 6 vs y, is shown in Fig. 10.1. The ratio of the diffusion constant of the inhibitor and the effective diffusion constant of the activator at the Turing threshold increases as the motion becomes more subdiffusive, y ... 

This curve has a single minimum, (A x, bj), which corresponds to the Turing instability of the uniform steady state. The Turing threshold bj and the critical wavenumber kj depend on q and read in parametric form... 

Note that A (AT) does not depend on and that G factors out the complex-ation reaction has no effect on the Turing condition. The (2+1)-variable activator-inhibitor-substrate system has the same Turing threshold as the two-variable activator-inhibitor system without substrate. Equation (10.32),... 

We systematically survey the Turing threshold conditions and the structural mode that becomes unstable, for arrays of two, three, and four coupled reactors with Lengyel-Epstein kinetics. The results illustrate the importance of the value of the coupling strength on the occurrence of a Turing instability. They also provide the... 

There is only one nonuniform structural mode, and we obtain the Turing threshold... 

The nonzero eigenvalue is doubly degenerate, and so is the eigenvalue of Jg that passes through zero. The Turing threshold is given by... 

The expressions for the coefficients c of the characteristic polynomial and the Hurwitz determinants A in terms of b, a, d, ay, and 02 are obtained using computational algebra software, such as Mathematica (Wolfram Research, Inc., Champaign, IL, 2(X)2) and Maple (Waterloo Maple Inc., Waterloo, Ontario, 2002). They are very lengthy already for only two coupled reactors and will not be displayed here. The condition C4 = 0 yields the Turing threshold... 

Determine the Turing threshold for a linear three-reactor array, a circular three-reactor array, and a five-node star graph of Brusselators. 

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