|Linear diffusion is an established model for the spatial spread in biological systems, including movement of cell populations. However, for interacting, closely packed cell populations, simple diffusion is inappropriate because a cell will stop moving when it encounters another cell. Introducing a nonlinear diffusion term will reflect this phenomenon, known as contact inhibition. I will look at traveling waves in two competing cell populations with contact inhibition, this is motivated by the very early stages of solid tumor growth. The traveling wave solutions will correspond to a moving interface between the two cell populations. Using a phase space analysis, the minimum wave speed arises via new behavior in the traveling wave equations. This is joint work with Jonathan Sherratt of the Center for Theoretical Modeling in Medicine in the Department of Mathematics at Heriot-Watt University.