Chemotactic Cellular Migration: Smooth and Discontinuous Travelling Wave Solutions

A simple model of chemotactic cell migration gives rise to travelling wave solutions. By varying the cellular growth rate and chemoattractant production rate, travelling waves with both smooth and discontinuous fronts are found using phase plane analysis. The phase plane exhibits a curve of singularities whose position relative to the equilibrium points in the phase plane determines the nature of the heteroclinic orbits, where they exist. Smooth solutions have trajectories connecting the steady states lying to one side of the singular curve. Travelling shock waves arise by connecting trajectories passing through a special point in the singular curve and recrossing the singular curve, by way of a discontinuity. Hyperbolic partial differential equation theory gives the necessary shock condition. Conditions on the parameter values determine when the solutions are smooth travelling waves versus discontinuous travelling wave solutions. These conditions provide bounds on the travelling wave speeds, corresponding to bounds on the chemotactic velocity or bounds on cellular growth rate. This analysis gives rise to the possibility of representing sharp fronts to waves of invading cells through a simple chemotactic term, without introducing a nonlinear diffusion term. This is more appropriate when cell populations are sufficiently dense.