Turing Instabilities in Hyperbolic Reaction-Diffusion Equations

1 Turing Instabilities in Hyperbolic Reaction-Diffusion Equations 

We first use hyperbolic reaction-diffusion equations, see Sect. 2.2.1, to study the effect of inertia on Turing instabilities [206]. Specifically, we consider two-variable HRDEs, 

Let G0u(- ) Pv(- )) = (Pu Pv) be auniform steady state (USS) of (10.52), where (Pu Pv) is a stable steady state of the well-mixed system (10.21), i.e., the stability conditions (10.23) are satisfied. To determine the stability of the USS, we consider as usual the evolution of small, spatially nonuniform perturbations. 

The uniform steady state is stable if all roots have a negative real part for all k. The necessary and sufficient conditions for this to hold are the Routh-Hurwitz conditions, see Theorem 1.2. The Turing bifurcation corresponds to a real root k crossing the imaginary axis for some nonzero kj, i.e., k/ = 0. This occurs if condition (1.35) is violated, i.e., 4 = 0 for some 7 0. A bifurcation to oscillatory patterns, i.e., a spatial Hopf bifurcation, corresponds to a pair of complex conjugate roots crossing the imaginary axis for some nonzero %, i.e., = 0. 

According to (1.38), a Hopf bifurcation occurs if the Hurwitz determinant A3 vanishes, A3 = 0. 

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