  A hyperbolic model for diffusion, nonlinear transport (or advection) and
production of a scalar quantity, is considered. The model is based on a
constitutive law of Cattaneo-Maxwell type expressing non-Fickian diffusion by
means of a relaxation time relation. The production or source term is assumed
to be of logistic type. This paper studies the existence and spectral stability
properties of spatially periodic traveling wave solutions to this system. It is
shown that a family of subcharacteristic periodic waves emerges from a local
Hopf bifurcation around a critical value of the wave speed. These waves have
bounded fundamental period and small-amplitude. In addition, it is shown that
these waves are spectrally unstable as solutions to the hyperbolic system. For
that purpose, it is proved that the Floquet spectrum of the linearized operator
around a wave can be approximated by a linear operator whose point spectrum
intersects the unstable half plane of complex numbers with positive real part.
