# The Role of Sensory Cues in Collective Dynamics: A Study of Three-Dimensional Vicsek Models
## 3 Results
For example, when \(\eta=0.1\) and \(\phi=2\pi/15\), polarization in the composite mode (Figure 4c) reaches a near-perfect value of one, indicating a perfect group alignment. In contrast, at these parameter values, polarization in auditory (Figure 4a) and visual modalities (Figure 4b) is lower (around 0.6 and 0.8, respectively). Alternatively, at this noise intensity, the visual mode requires a larger sensing angle of \(\phi=6\pi/15\) to achieve perfect polarization, whereas the auditory mode can achieve this level of alignment with a smaller sensing region of \(\phi=4\pi/15\), compared to visual sensing. In comparison, at this noise intensity, the composite mode attains perfect polarization with an even smaller sensing angle of \(\phi=3\pi/15\). Based on these observations, we conclude that the auditory mode enables particles to achieve higher group alignment within smaller sensing regions compared to the visual mode under low noise conditions. However, as noise levels increase, both auditory and visual modes require larger sensing regions to maintain high polarization. Despite this, auditory sensing continues to facilitate better alignment than visual sensing using an equivalent sensing region. Furthermore, combining auditory and visual sensing in composite sensing allows for better alignment at smaller sensing angles (\(\phi\leq 5\pi/15\)) and higher noise intensities, compared to auditory or visual sensing alone. For example, comparing the three sensing modes at \(\eta=0.4\), the composite mode can achieve moderate polarization at \(\phi=2\pi/15\), whereas the auditory and visual modes achieve moderate polarization at \(\phi=4\pi/15\) and \(\phi=5\pi/15\), respectively. This is attributed to the composite mode's ability to combine information from both auditory and visual sensing, enhancing the particles' ability to align effectively. An intriguing result emerges that is not observed in earlier studies using a 2D Vicsek model with similar modifications [31; 46]. In the visual mode, at a fixed noise level, the variation in polarization exhibits a non-monotonic trend as the sensing region increases. For instance, at \(\eta=0.5\), polarization initially increases, reaches a maximum at \(\phi=8\pi/15\), and subsequently decreases as the sensing region approaches a full sphere. This effect becomes more pronounced at higher noise levels, indicating the presence of an optimal sensing region that maximizes group alignment when using visual sensing. Another interesting phenomenon different from previous 2D results is that, as the sensing region grows (\(0<\phi\leq 10\pi/15\)) at noise intensities \(\eta\leq 0.6\), polarization gradually increases in the auditory mode (Figure 4a), contrasting the sharp increases from low to high polarization seen in the visual mode (Figure 4b). This trend is also seen at higher noise intensities; however, at \(\eta>0.6\), the auditory mode only achieves low-to-moderate polarization for the entire range of sensing angles. The sharp transitions from low to high polarization in the visual mode persists at slightly higher noise intensities with a noticeable reduction in effect for \(\eta>0.8\). The transition from a disordered to ordered state, in terms of group alignment, is more discontinuous when using visual sensing. When increasing sensing angle, visual sensing results in a first-order-like phase transition (for \(\eta\leq 0.8\)), whereas auditory sensing results in behavior akin to a second-order phase transition (low-to-moderate or low-to-moderate-to-high polarization) [60]. The trends seen in the composite mode are similar to that of the second-order phase transition seen in the auditory sensing, indicating that the auditory mechanics play a dominant role when combined with the visual mechanics to produce this composite mode. Next, we compare the behavior of the group in terms of the mean cohesion (Figure 4d-f) and the mean largest cluster size (Figure 4g-i) between the modalities.