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Pattern formation in random walks with inertia (Invited Paper)

We study the Turing instability to stationary spatial patterns in reaction-transport systems with inertia. After a brief discussion of hyperbolic reaction-diffusion equations and reaction-Cattaneo equations we focus on reaction random walks, which are the most natural generalization of reaction-diffusion equations. We analyze the effect of inertia in the transport on spatial instabilities of the homogeneous steady state. Direction-dependent reaction walks, where the interaction between particles depends on the direction in which the particles are moving, allow us to take account of an energy requirement for reactive interactions, i.e., an activation energy, in the reaction-transport equation. We compare bifurcation conditions for direction-independent reaction walks and for direction-dependent reaction walks to assess the effect of inertia in the transport and the effect of activation energies in the kinetics on the Turing instability.