Abstract: "The amoeba Dictyostelium discoideum exhibits aerotaxis: in experiments, when a colony is covered, cells quickly consume the available oxygen and then move outwards at a constant speed to find new oxygen. In this talk, we discuss a reaction-diffusion equation, arising from a large population limit of an stochastic differential equation, for this behavior that was proposed by Demircigil and Tomašević. This model is quite degenerate, making even the well-posedness difficult to show. The main goal of the talk is to discuss the existence, uniqueness, and long-time behavior of solutions. We show a transition in the character of the behavior as the strength of the aerotactic response varies. Our results use crucially the shape defect function, an object that measures the distance from the solution to the traveling wave in a novel way related to Stein's method in probability. This is a joint work with Mete Demircigil."