Travelling waves in hybrid chemotaxis models

TL;DR

This paper investigates travelling wave solutions in hybrid chemotaxis models combining agent-based bacteria with PDE-based chemical signals, revealing conditions for wave existence, speed agreement, and oscillatory behaviors.

Contribution

It introduces a comprehensive analysis of travelling waves in hybrid chemotaxis models, including derivation of equations, existence proofs, and numerical comparisons with continuum models.

Findings

Cell proliferation is essential for stationary travelling waves.

Good agreement in wave speeds between hybrid and continuum models under weak chemotaxis.

Oscillations occur in slow adaptation scenarios, not explained by mean-field models.

Abstract

Hybrid models of chemotaxis combine agent-based models of cells with partial differential equation models of extracellular chemical signals. In this paper, travelling wave properties of hybrid models of bacterial chemotaxis are investigated. Bacteria are modelled using an agent-based (individual-based) approach with internal dynamics describing signal transduction. In addition to the chemotactic behaviour of the bacteria, the individual-based model also includes cell proliferation and death. Cells consume the extracellular nutrient field (chemoattractant) which is modelled using a partial differential equation. Mesoscopic and macroscopic equations representing the behaviour of the hybrid model are derived and the existence of travelling wave solutions for these models is established. It is shown that cell proliferation is necessary for the existence of non-transient (stationary) travelling waves in hybrid models. Additionally, a numerical comparison between the wave speeds of the continuum models and the hybrid models shows good agreement in the case of weak chemotaxis and qualitative agreement for the strong chemotaxis case. In the case of slow cell adaptation, we detect oscillating behaviour of the wave, which cannot be explained by mean-field approximations.