Front Location for Go or Grow Models of Aerotaxis

TL;DR

This paper analyzes models of aerotaxis-driven invasion fronts, establishing well-posedness and characterizing front locations, including complex boundary cases, for both nonlocal and local reaction-advection-diffusion models.

Contribution

It introduces a local model as a counterpart to a nonlocal one and develops new methods to analyze front locations and well-posedness.

Findings

Established well-posedness for both models.

Characterized front location up to O(1) precision.

Analyzed the boundary pushmi-pullyu case.

Abstract

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.