title: Diffusive instabilities in hyperbolic reaction-diffusion equations.
abstract: We investigate two-variable reaction-diffusion systems of the hyperbolic type. A linear stability analysis is performed, and the conditions for diffusion-driven instabilities are derived. Two basic types of eigenvalues, real and complex, are described. Dispersion curves for both types of eigenvalues are plotted and their behavior is analyzed. The real case is related to the Turing instability, and the complex one corresponds to the wave instability. We emphasize the interesting feature that the wave instability in the hyperbolic equations occurs in two-variable systems, whereas in the parabolic case one needs three reaction-diffusion equations.
authors.affiliation: Department of Continuum Mechanics, Dorodnicyn Computing Centre, Russian Academy of Sciences, Vavilova 40, 119333 Moscow, Russia.
authors.forename: Evgeny P
authors.initials: EP
authors.lastname: Zemskov
authors.affiliation: Department of Chemistry, Southern Methodist University, Dallas, Texas 75275-0314, USA.
authors.forename: Werner
authors.initials: W
authors.lastname: Horsthemke