  We analyze travelling wave (TW) solutions for nonlinear systems consisting of
an ODE coupled to a degenerate PDE with a diffusion coefficient that vanishes
as the solution tends to zero and blows up as it approaches its maximum value.
Stable TW solutions for such systems have previously been observed numerically
as well as in biological experiments on the growth of cellulolytic biofilms. In
this work, we provide an analytical justification for these observations and
prove existence and stability results for TW solutions of such models.
  Using the TW ansatz and a first integral, the system is reduced to an
autonomous dynamical system with two unknowns. Analysing the system in the
corresponding phase-plane, the existence of a unique TW is shown, which
possesses a sharp front and a diffusive tail, and is moving with a constant
speed. The linear stability of the TW in two space dimensions is proven under
suitable assumptions on the initial data. Finally, numerical simulations are
presented that affirm the theoretical predictions on the existence, stability,
and parametric dependence of the TWs.
