Wavefront propagation in a competition equation with a new motility term modelling contact inhibition between cell populations

Linear di¬usion is an established model for spatial spread in biological systems, including movement of cell populations. However, for interacting, closely packed cell populations, simple di¬usion is inappropriate, because di¬erent cell populations will not move through one another: rather, a cell will stop moving when it encounters another cell. In this paper, I introduce a nonlinear di¬usion term that re®ects this phenomenon, known as contact inhibition of migration. I study this term in the context of two competing cell populations, one of which has a proliferative advantage over the other; this is motivated by the very early stages of solid tumour growth. I focus in particular on travelling-wave solutions, corresponding to moving interfaces between the two cell populations. Numerical simulations indicate that there are wavefront solutions for wave speeds above a critical minimum value, and I present linear analysis that explains the selection of wave speeds by initial conditions. I obtain an approximation to the shape of these waves for high speeds, and show that the minimum speed arises via quite new behaviour in the travelling-wave equations, with the proportion of cells of each type approaching a step function as the wave speed decreases towards the minimum. Exploiting this structure, I use singular perturbation theory to investigate the wave shape for speeds close to the minimum.