A multi-layer model for self-propelled disks interacting through alignment and volume exclusion

TL;DR

This paper develops a multi-layer model for self-propelled disks that interact through alignment and volume exclusion, deriving macroscopic equations from individual behaviors, with potential applications in biological collective dynamics.

Contribution

The paper introduces a coupled multi-layer macroscopic model derived from individual-based interactions, incorporating both alignment and volume exclusion effects.

Findings

Numerical validation shows agreement between individual-based and macroscopic models.

The model captures the influence of inter-layer volume exclusion on disk orientations.

Potential application to sperm-cell collective behavior.

Abstract

We present an individual-based model describing disk-like self-propelled particles moving inside parallel planes. The disk directions of motion follow alignment rules inside each layer. Additionally, the disks are subject to interactions with those of the neighboring layers arising from volume exclusion constraints. These interactions affect the disk inclinations with respect to the plane of motion. We formally de-rive a macroscopic model composed of planar Self-Organized Hydrodynamic (SOH) models describing the transport of mass and evolution of mean direction of motion of the disks in each plane, supplemented with transport equations for the mean disk inclination. These planar models are coupled due to the interactions with the neighboring planes. Numerical comparisons between the individual-based and macroscopic models are carried out. These models could be applicable, for instance, to describe sperm-cell collective dynamics.