  In this paper we consider kinetic and associated macroscopic models for
chemotaxis on a network. Coupling conditions at the nodes of the network for
the kinetic problem are presented and used to derive coupling conditions for
the macroscopic approximations. The results of the different models are
compared and relations to a Keller-Segel model on networks are discussed. For a
numerical approximation of the governing equations asymptotic preserving
relaxation schemes are extended to directed graphs. Kinetic and macroscopic
equations are investigated numerically and their solutions are compared for
tripod and more general networks.
