  In this paper, it is proved that the KdV-Burgers equation with a monostable
source term of Fisher-KPP type has small-amplitude periodic traveling wave
solutions with finite fundamental period. These solutions emerge from a
subcritical local Hopf bifurcation around a critical value of the wave speed.
Moreover, it is shown that these periodic waves are spectrally unstable as
solutions to the PDE, that is, the Floquet (continuous) spectrum of the
linearization around each periodic wave intersects the unstable half plane of
complex values with positive real part. To that end, classical perturbation
theory for linear operators is applied in order to prove that the spectrum of
the linearized operator around the wave can be approximated by that of a
constant coefficient operator around the zero solution, which intersects the
unstable complex half plane.
