Mass migration of cells (via wave motion) plays an important role in many
biological processes, particularly chemotaxis. We study the existence of travelling
wave solutions for a chemotaxis model on a microscopic scale. The interaction between
nutrients and chemoattractants are considered. Unlike previous approaches,
we allow for diffusion of substrates, degradation of chemoattractants and cell growth
(constant and linear growth rate). We apply asymptotic methods to investigate the
behaviour of the solutions when cells are highly sensitive to extracellular signalling.
Explicit solutions are demonstrated, and their biological implications are presented.
The results presented here extend and generalize known results.