Existence of Traveling Wave Solutions for a Model of Tumor Invasion

The existence of traveling wave solutions to a haptotaxis dominated model is analyzed. A version of this model has been derived in Perumpanani et al. [Phys. D, 126 (1999), pp. 145--159] to describe tumor invasion, where diffusion is neglected as it is assumed to play only a small role in the cell migration. By instead allowing diffusion to be small, we reformulate the model as a singular perturbation problem, which can then be analyzed using geometric singular perturbation theory. We prove the existence of three types of physically realistic traveling wave solutions in the case of small diffusion. These solutions reduce to the no-diffusion solutions in the singular limit as diffusion as is taken to zero. A fourth traveling wave solution is also shown to exist, but that is physically unrealistic as it has a component with negative cell population. The numerical stability, in particular the wavespeed of the traveling wave solutions, is also discussed.