Spatial instabilities in reaction random walks with direction-independent kinetics

We study spatial instabilities in reacting and diffusing systems, where dif fusion is modeled by a persistent random walk instead of the usual Brownian motion. Perturbations in these reaction walk systems propagate with finite speed, whereas in reaction-diffusion systems localized disturbances affect every part instantly. albeit with heavy damping. We present evolution equa tions for reaction random walks whose kinetics do not depend on the particl es' direction of motion. The homogeneous steady state of such systems can u ndergo two types of transport-driven instabilities. One type of bifurcation gives rise to stationary spatial patterns and corresponds to the Turing in stability in reaction-diffusion systems. The other type occurs in the balli stic regime and leads to oscillatory spatial patterns; it has no analog in reaction-diffusion systems. The conditions for these bifurcations are deriv ed and applied to two model systems. We also analyze the stability properti es of one-variable systems and find that small wavelength perturbations dec ay in an oscillatory manner. [S1063-651X(99)07409-7].