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 derive 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.
Degond, P., & Navoret, L. (2015). A multi-layer model for self-propelled disks interacting through alignment and volume exclusion. Mathematical Models and Methods in Applied Sciences, 25(13), 2439–2475. https://doi.org/10.1142/S021820251540014X