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Front Location for Go or Grow Models of Aerotaxis

Mete Demircigil, Christopher Henderson

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.

Front Location for Go or Grow Models of Aerotaxis

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 precision in all cases, including the delicate boundary ``pushmi-pullyu'' case.

Paper Structure

This paper contains 37 sections, 27 theorems, 371 equations, 1 table.

Table of Contents

  1. Introduction
  2. The model and its context
  3. The model
  4. The biological motivation
  5. A broader reaction–diffusion–advection class
  6. Pushed, pulled, and pushmi-pullyu fronts
  7. The shape defect function
  8. Main Results
  9. Connection Between the Local Model and a Model of Berestycki, Brunet, and Derrida
  10. Organization of the paper
  11. Notation
  12. The Shape Defect Function $\omega$
  13. Equation of the shape defect function $\omega$.
  14. Properties of the Traveling Wave Profile Functions $\eta^P$ and $\eta^u$.
  15. Well-Posedness and Comparison Principle
  16. ...and 22 more sections

Key Result

Theorem 1.1

Fix $\chi \geq 0$, $\gamma \in (0,\chi^{-1})$, and let $c=c_*(\chi)$ be as in e.c_star_chi. Consider $u_{\rm in} \in L^2(\mathbb R, \zeta_\gamma dx)$ taking values in $[0,1]$ such that $u_{\rm in} \not\equiv 0,1$. Suppose that, in the distributional sense, where $\eta^u$ is the wave profile function associated with the wave speed $c$. Then, there exists a unique weak solution $u\in C([0,+\infty);

Theorems & Definitions (49)

  • Theorem 1.1: Well-posedness for the Local Model
  • Theorem 1.2: Front asymptotics
  • Proposition 2.1
  • proof : Proof of [S]l.w_eqn
  • Lemma 2.2: Properties of $\eta^u$ and $Q^u$.
  • Lemma 2.3: Properties of $\eta^P$ and $Q^P$.
  • proof : Proof of [S]l.l.eta_loc
  • proof : Proof of [S]l.l.eta_nonloc.
  • Proposition 3.1: Well-posedness for a Lipschitz continuous flux $A$
  • Corollary 3.2: Comparison Principle for a Lipschitz continuous flux $A$
  • ...and 39 more