  We study the existence of weak solutions for a model of cell invasion into
the extracellular matrix (ECM), which consists of a non-linear partial
differential equation for the density of cells, coupled with an ordinary
differential equation (ODE) describing the ECM density. The model contains
cross-species density-dependent diffusion and proliferation terms that capture
the role of the ECM in providing structural support for the cells during
invasion while also preventing growth via volume-filling effects. Furthermore,
the model includes ECM degradation by the cells. We present an existence result
for weak solutions which is based on carefully exploiting the partial gradient
flow structure of the problem which allows us to overcome the non-regularising
nature of the ODE involved.
  In addition, we present simulations based on a finite difference scheme that
illustrate that the system exhibits travelling wave solutions, and we
investigate numerically the asymptotic behaviour as the ECM degradation rate
tends to infinity.
