Traveling Shock Waves Arising in a Model of Malignant Invasion

Abstract

Members of a family of traveling wave solutions for a simple two-variable model of malignant invasion driven by haptotaxis are shown to have shocks (which satisfy an entropy condition) and to be computationally stable. By seeking traveling wave solutions, a PDE model, equivalent to three first order PDEs defined on the real space and time axes, may be reduced to an ordinary differential equation system, which can be studied in a two-dimensional phase plane. This phase plane is shown to contain a singular barrier and a "hole" in this barrier which admits two singular phase trajectories. Thus orbits are able to smoothly cross this barrier along either of the two singular trajectories. Once this has occurred, the barrier seems to prevent any heteroclinic steady-state connections. However, we will show that shocks admitted by the PDEs allow the orbit to jump back over the singular barrier. This process leads to a previously unnoticed family of solutions for malignant invasion which have slower wavespeeds than the family of smooth traveling wave solutions (extending over the whole real axis) admitted by the model. The existence of a minimum-speed wave with semi-infinite support is demonstrated. The practical stability of these new shock solutions is confirmed via numerical solutions of the model. The minimum speed solution is found to evolve from specific nonsmooth initial data, whereas smooth initial data extending over the whole real axis evolves to faster waves.