  We study the long-range, long-time behavior of the reactive-telegraph
equation and a related reactive-kinetic model. The two problems are equivalent
in one spatial dimension. We point out that the reactive-telegraph equation,
meant to model a population density, does not preserve positivity in higher
dimensions. In view of this, in dimensions larger than one, we consider a
reactive-kinetic model and investigate the long-range, long-time limit of the
solutions. We provide a general characterization of the speed of propagation
and we compute it explicitly in one and two dimensions. We show that a phase
transition between parabolic and hyperbolic behavior takes place only in one
dimension. Finally, we investigate the hydrodynamic limit of the limiting
problem.
