Diffusive and Chemotactic Cellular Migration: Smooth and Discontinuous Traveling Wave Solutions

A mathematical model describing cell migration by diffusion and chemotaxis is considered. The model is examined using phase plane, numerical, and perturbation techniques. For a proliferative cell population, travelingwave solutions are observed regardless of whether the migration is driven by diffusion, chemotaxis, or a combination of the two mechanisms. For pure chemotactic migration, both smooth and discontinuous solutions with shocks are shown to exist using phase plane analysis involving a curve of singularities, and identical results are obtained numerically. Alternatively, pure diffusive migration and combinations of diffusive and chemotactic migration yield smooth solutions only. For all cases the wave speed depends on the exponential decay rate of the initial cell density, and it is bounded by a minimum value which is numerically observed whenever the initial cell distribution has compact support. The minimum wave speed $c_{min}$ is proportional to $\sqrt{\chi}$ or $\sqrt {D}$ for pure chemotaxis ...