A macroscopic model for a system of swarming agents using curvature control

TL;DR

This paper introduces a new macroscopic model called PTWA that combines Vicsek alignment and curvature control, providing insights into collective swarming behavior with a focus on fish trajectories.

Contribution

The paper derives a novel macroscopic limit for the PTWA model using generalized collisional invariants, linking microscopic curvature control to macroscopic swarming dynamics.

Findings

The macroscopic limit involves density and mean velocity with hyperbolic, non-conservative equations.

Numerical results show model coefficients closely match those of Vicsek hydrodynamics.

The PTWA model generalizes the Vicsek model, capturing swarming behavior with curvature-based motion.

Abstract

In this paper, we study the macroscopic limit of a new model of collective displacement. The model, called PTWA, is a combination of the Vicsek alignment model and the Persistent Turning Walker (PTW) model of motion by curvature control. The PTW model was designed to fit measured trajectories of individual fish. The PTWA model (Persistent Turning Walker with Alignment) describes the displacements of agents which modify their curvature in order to align with their neighbors. The derivation of its macroscopic limit uses the non-classical notion of generalized collisional invariant. The macroscopic limit of the PTWA model involves two physical quantities, the density and the mean velocity of individuals. It is a system of hyperbolic type but is non-conservative due to a geometric constraint on the velocity. This system has the same form as the macroscopic limit of the Vicsek model (the 'Vicsek hydrodynamics') but for the expression of the model coefficients. The numerical computations show that the numerical values of the coefficients are very close. The 'Vicsek Hydrodynamic model' appears in this way as a more generic macroscopic model of swarming behavior as originally anticipated.