Front Location for Go or Grow Models of Aerotaxis

We investigate the pushed-to-pulled transition for a minimal model for invasive fronts influence by ``aerotaxis,'' that is, when organisms follow oxygen gradients. We consider two singular reaction-advection-diffusion models for this. The version of primary interest arises as a hydrodynamic limit of a system of branching, rank-based interacting Brownian particles and features a nonlinear, nonlocal advection. The second version is introduced here as a local counterpart. We establish well-posedness for both models, with the local case requiring a novel use of the ``shape defect function.'' We further characterize the front location up to $O(1)$ precision in all cases, including the delicate boundary ``pushmi-pullyu'' case.