Travelling wave solutions to a haptotaxis-dominated model of malignant invasion

A haptotaxis-dominated model of cell invasion is considered for small cell diffusion and fast protease adjustment to the cell-collagen matrix interaction. A simplified limit model has travelling wave cell invasion profiles that are blunt, that is end with a 'shock-like' step, and that evolve stably from initial data that lie to one side of some initial plane. In common with diffusion-dominated systems, the travelling wave which evolves from such initial data has the minimum wavespeed permissible in the model. This minimum wavespeed is not, however, determined by the local stability of the steady states in the travelling wave phase plane, but by a novel combination of singular behaviour within the phase plane and hyperbolic shock conditions. It is shown that more accurate models including the detailed fast dynamics of the protease require small amounts of diffusion (of the same order as the fast dynamics timescale) in order to remain stable. However, small diffusion and fast protease adjustment then give physically relevant and interesting solutions that evolve from semi-compact initial data and stably invade at speeds well predicted by the simple model.