Motivated by recent experimental and computational results that show a motility-induced clustering transition in self-propelled particle systems, we study an individual model and its corresponding Self-Organized Hydrodynamic model for collective behaviour that incorporates a density-dependent velocity, as well as inter-particle alignment. The modal analysis of the hydrodynamic model elucidates the relationship between the stability of the equilibria and the changing velocity, and the formation of clusters. We find, in agreement with earlier results for non-aligning particles, that the key criterion for stability is (ρv(ρ))' ≥ 0, i.e. a nondecreasing mass flux ρv(ρ) with respect to the density. Numerical simulation for both the individual and hydrodynamic models with a velocity function inspired by experiment demonstrates the validity of the theoretical results.
This work has been supported by the Agence Nationale pour la Recherche (ANR) under grant ‘MOTIMO’ (ANR-11-MONU-009-01), by the Engineering and Physical Sciences Research Council (EPSRC) under grant ref. EP/M006883/1, and by the National Science Foundation (NSF) under grant RNMS
11-07444 (KI-Net). P. D. is on leave from CNRS, Institut de Math ́ematiques, Toulouse, France. He acknowledges support from the Royal Society and the Wolfson foundation through a Royal Society Wolfson Research Merit Award. H. Y. wishes to acknowledge the hospitality of the Department of Mathematics, Imperial College London, where this research was conducted. P. D. and H. Y. wish to thank F. Plourabou ́e (IMFT, Toulouse, France) for enlighting discussions.
- Active matter
- Alignment interaction
- Collective dynamics
- Density-dependent velocity
- Hydrodynamic limit
- Motility induced phase separation
- Relaxation model