  Chemotaxis phenomena govern the directed movement of micro-organisms in
response to chemical stimuli. In this paper, we investigate two Keller--Segel
systems of reaction-advection-diffusion equations modeling chemotaxis on thin
networks. The distinction between two systems is driven by the rate of
diffusion of the chemo-attractant. The intermediate rate of diffusion is
modeled by a coupled pair of parabolic equations, while the rapid rate is
described by a parabolic equation coupled with an elliptic one. Assuming the
polynomial rate of growth of the chemotaxis sensitivity coefficient, we prove
local well-posedness of both systems on compact metric graphs, and, in
particular, prove existence of unique classical solutions. This is achieved by
constructing sufficiently regular mild solutions via analytic semigroup methods
and combinatorial description of the heat kernel on metric graphs. The
regularity of mild solutions is shown by applying abstract semigroup results to
semi-linear parabolic equations on compact graphs. In addition, for logistic
type Keller--Segel systems we prove global well-posedness and, in some special
cases, global uniform boundedness of solutions.
