Stability of flocking in the reciprocal two-species Vicsek model: Effects of relative population, motility, and noise

Natural flocks need to cope with various forms of heterogeneities, for instance, their composition, motility, interaction, or environmental factors. Here, we study the effects of such heterogeneities on the flocking dynamics of the reciprocal two-species Vicsek model [Phys. Rev. E 107, 024607 (2023)], which comprises two groups of self-propelled agents with anti-aligning inter-species interactions and exhibits either parallel or anti-parallel flocking states. The parallel and anti-parallel flocking states vanish upon reducing the size of one group, and the system transitions to a single-species flock of the majority species. At sufficiently low noise (or high density), the minority species can exhibit collective behavior, anti-aligning with the liquid state of the majority species. Unequal self-propulsion speeds of the two species strongly encourage anti-parallel flocking over parallel flocking. However, when activity landscapes with region-dependent motilities are introduced, parallel flocking is retained if the faster region is given more space, highlighting the role of environmental constraints. Under noise heterogeneity, the colder species (subjected to lower noise) attain higher band velocity compared to the hotter one, temporarily disrupting any parallel flocking, which is subsequently restored. These findings collectively reveal how different forms of heterogeneity, both intrinsic and environmental, can qualitatively reshape flocking behavior in this class of reciprocal two-species models.

💡 Research Summary

The paper investigates how various forms of heterogeneity affect collective motion in the reciprocal two‑species Vicsek model (TSVM), a minimal framework for studying flocking of self‑propelled particles with anti‑aligning inter‑species interactions. In the homogeneous case the model exhibits two ordered states: parallel flocking (PF), where bands of both species travel in the same direction, and anti‑parallel flocking (APF), where the bands move oppositely. The authors introduce four distinct heterogeneities—population imbalance, motility differences, spatially varying activity landscapes, and noise asymmetry—and explore their impact on the stability and prevalence of PF and APF.

Population heterogeneity is quantified by the imbalance parameter m₀ = (N_A − N_B)/N. Simulations show that as m₀ increases, the minority species (B) forms fewer high‑density bands; beyond a critical imbalance (≈ 0.6) B can no longer sustain bands and remains in a gaseous state while the majority species (A) forms a single‑species flock reminiscent of the classic Vicsek model. The probability distribution of the order parameters v_s (global alignment) and v_a (anti‑alignment) evolves from a bimodal shape (reflecting stochastic switching between PF and APF) to a single peak, indicating collapse of the two ordered states into a single‑species flock.

Motility heterogeneity is introduced by assigning different constant speeds v_A ≠ v_B while keeping populations equal. The faster species dominates the dynamics: when the speed ratio exceeds ≈ 1.5, the anti‑parallel order parameter ⟨v_a⟩ surpasses the parallel one ⟨v_s⟩, and APF becomes the only stable ordered phase. This asymmetry arises because the anti‑ferromagnetic coupling between species is amplified when one species moves more quickly, effectively biasing the system toward opposite‑direction bands.

Spatial heterogeneity is implemented by dividing the simulation box into two regions. In one region species A moves faster than B, while in the opposite region B is faster. Although the global average speed is unchanged, the relative size of the “fast” region controls the ordered state. If the region where a given species is faster occupies more than roughly 60 % of the total area, the PF state persists; otherwise APF dominates. Thus, environmental constraints that allocate more space to the faster species can rescue parallel flocking even when intrinsic motility differences would otherwise favor anti‑parallel motion.

Noise heterogeneity is modeled by assigning different angular noise amplitudes η_A ≠ η_B. The low‑noise (“colder”) species exhibits higher alignment and a larger band propagation speed, temporarily disrupting PF and producing transient APF‑like dynamics. However, over long simulation times the system self‑organizes back into a PF configuration, demonstrating that noise asymmetry induces only a metastable disturbance while the underlying anti‑aligning interaction ultimately restores parallel order.

Technical analysis relies on two order parameters: v_s = |∑_i σ_i|/N (global polar order) and v_a = |∑_i s_i σ_i|/N (species‑weighted polar order). PF corresponds to ⟨v_s⟩ > 0, ⟨v_a⟩ ≈ 0; APF to ⟨v_a⟩ > 0, ⟨v_s⟩ ≈ 0. By measuring these quantities across ensembles and time, the authors map phase boundaries in the (density, noise, speed) parameter space for each heterogeneity scenario.

Overall, the study reveals that intrinsic heterogeneities (population ratio, speed differences) can fundamentally reshape the collective state, often suppressing one of the ordered phases or reducing the system to a single‑species flock. In contrast, extrinsic heterogeneities (spatial activity landscapes, differential noise) can either preserve or temporarily destabilize parallel flocking depending on the spatial allocation or magnitude of the asymmetry. These insights provide a roadmap for controlling multi‑species active matter—whether biological swarms, synthetic colloidal swimmers, or robotic agents—by tuning species composition, propulsion speeds, or environmental cues to achieve desired flocking configurations.