  We present a generalized Keller-Segel model where an arbitrary number of
chemical compounds react, some of which are produced by a species, and one of
which is a chemoattractant for the species. To investigate the stability of
homogeneous stationary states of this generalized model, we consider the
eigenvalues of a linearized system. We are able to reduce this infinite
dimensional eigenproblem to a parametrized finite dimensional eigenproblem. By
matrix theoretic tools, we then provide easily verifiable sufficient conditions
for destabilizing the homogeneous stationary states. In particular, one of the
sufficient conditions is that the chemotactic feedback is sufficiently strong.
Although this mechanism was already known to exist in the original Keller-Segel
model, here we show that it is more generally applicable by significantly
enlarging the class of models exhibiting this instability phenomenon which may
lead to pattern formation.
