Spatial Hopf Instabilities in Direction-Independent Reaction Walks

2 Spatial Hopf Instabilities in Direction-Independent Reaction Walks 

Spatial Hopf bifurcations or wave bifurcations can never occur in two-variable reaction-diffusion systems, see Sect. 10.1.2. This is no longer the case for reaction-transport systems with inertia. As shown in Sects. 10.2.1 and 10.2.2, spatial Hopf bifurcations are in principle possible in two-variable hyperbolic reaction-diffusions 

Since T is negative according to the stability conditions (10.23), Ci is always positive. The positivity of the Hurwitz determinant A2 and C4 at the Hopf bifurcation imply that C3 0 there and that uh,dirw is well defined [205]. Gathering terms of equal powers in k, we rewrite the Hopf condition as 

The uniform steady state of a DIRW can undergo a spatial Hopf or wave bifurcation only if the right-hand side of (10.115) is positive. In other words, the rate of activation must exceed the loss rate of the activator in the steady state. If a spatial Hopf instability occurs, then all spatial modes with wavenumbers bigger than jj, the positive root of (10.107), are unstable. 

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. 

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