  We investigate the point spectrum associated with travelling wave solutions
in a Keller-Segel model for bacterial chemotaxis with small diffusivity of the
chemoattractant, a logarithmic chemosensitivity function and a constant,
sublinear or linear consumption rate. We show that, for constant or sublinear
consumption, there is an eigenvalue at the origin of order two. This is
associated with the translation invariance of the model and the existence of a
continuous family of solutions with varying wave speed. These point spectrum
results, in conjunction with previous results in the literature, imply that in
these cases the travelling wave solutions are absolute unstable if the
chemotactic coefficient is above a certain critical value, while they are
transiently unstable otherwise.
