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

Abstract

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.

Figures (20)

• Figure 1: (color online) Schematic of the different heterogeneities applied on the TSVM. Particles of species A (B) are represented by red (blue) balls. (a) Population heterogeneity: $N_{\rm A} \neq N_{\rm B}$ with $v_{\rm A} = v_{\rm B}$ and $\eta_{\rm A} = \eta_{\rm B}$; (b) Motility heterogeneity: $v_{\rm A} \neq v_{\rm B}$ with $N_{\rm A} = N_{\rm B}$ and $\eta_{\rm A} = \eta_{\rm B}$; (c) Spatial heterogeneity: $v_{\rm A} > v_{\rm B}$ in left region and $v_{\rm A} < v_{\rm B}$ in right region, where the dotted line represents the inter-region interface, $N_{\rm A} = N_{\rm B}$ and $\eta_{\rm A} = \eta_{\rm B}$; (d) Noise heterogeneity: $\eta_{\rm A} \neq \eta_{\rm B}$ with $N_{\rm A} = N_{\rm B}$ and $v_{\rm A} = v_{\rm B}$, i.e. the red particles experience a higher noise amplitude (a "hotter" environment) compared to the blue particles (a "colder" environment). In all subsequent snapshots, $L_x$ and $L_y$ represent horizontal and vertical system sizes, respectively.
• Figure 2: (color online) Steady-state snapshots for varying population heterogeneity. Particles of species A (B) are represented with red (blue) dots, and a local particle density is color-coded according to the color bar. (a) The homogeneous TSVM features an equal number of bands for A and B species. (b--c) The band number of species B decreases with increasing $m_{0}$. (d--e) B-particles can not form bands due to scarcity in numbers. Parameters: $\rho = 1$, $\eta = 0.3$, $v_{0}=0.5$, $L_{x}=800$, and $L_{y}=100$. A movie (movie1) of the same can be found at Ref. zenodo.
• Figure 3: (color online) Probability distribution $P(v_{s},v_{a})$ for varying population heterogeneity. (a) Representation of the homogeneous TSVM ($N_{\rm A} = N_{\rm B}$) exhibiting stochastic switching between the PF and APF states. (b--i) The two peaks progressively converge as $m_0$ increases, signifying a collapse into a single state. Parameters: $\rho=0.5$, $\eta=0.24$, $v_{0}=0.5$, $L_{x} = 256$, and $L_{y} = 32$. A movie (Movie S1) of the same can be found at Ref. SM.
• Figure 4: (color online) Order parameters for population heterogeneity.$\langle v_{a} \rangle$ and $\langle v_{s} \rangle$ are shown in the restricted APF (blue squares), PF (red circles), and SSF (black stars) ensembles for varying $m_0$. (a) $\langle v_{a} \rangle$ remains relatively constant in the APF ensemble but increases monotonically in the PF ensemble. (b) $\langle v_{s} \rangle$ increases monotonically in the APF ensemble while remaining relatively constant in the PF ensemble. Parameters: $\rho = 0.5$, $\eta = 0.24$, $v_{0}=0.5$, $L_{x} = 256$, and $L_{y} = 32$.
• Figure 5: (color online) Single species and both species flocking at large population heterogeneity. Snapshots of (a) SSF $(\eta=0.45)$ and (b) flocking of both species in an APF state $(\eta=0.2)$ are shown for a $20 \times 10$ section of a $800 \times 100$ simulation box. Red and blue arrows represent the orientation of A and B particles, respectively. (c) The time- and ensemble-averaged order parameters $\langle v_\pm \rangle$ as a function of $\eta$; $L_x=256$, $L_y=32$. Parameters: $\rho=2$, $v_0=0.5$, and $m_0=0.9$.