We discuss travelling-wave solutions of a system of coupled reaction-diffusion equations used by the authors to describe the macroscopic behaviour of fungal mycelia. Such systems have been used in a multitude of applications; and, in particular, Merkin and Needham (1990, Proc. R. Soc. Lond. A 430, 315-45) have studied a certain formulation as a model for generic isothermal chemical reactions. We show that an alternative analysis provides more complete results in ascertaining the conditions under which travelling-wave solutions exist and that it allows a wider range of parameter values to be considered; this is essential to the application considered in the present case. Numerical investigations of travelling-wave solutions and the related initial-value problem are included to motivate and extend the analysis.