# The Role of Sensory Cues in Collective Dynamics: A Study of Three-Dimensional Vicsek Models
## 1 Introduction  
In many animal groups, coordinated group-level patterns are formed by local interactions between individuals. Each individual acts based on information received from local neighbors, leading to organized motion without any central leader. This collective behavior is prevalent in various biological systems, such as ant colonies [1], bird flocks [2], fish schools [3, 4, 5], and mosquito swarms [6]. A significant advantage of group living is enhanced information access, aiding social animals in locating food [7], evading predators [8], and finding mates [9]. To understand the mechanisms behind the formation of group-level patterns from local interactions, various models have been proposed to simulate group coordination [10, 11, 12, 13, 14, 15, 16, 17]. The mathematical modeling of collective behavior employs different approaches, such as continuous medium modeling [10], continuous time modeling [12], and agent-based modeling in discrete time [18]. A widely used agent-based model is the original Vicsek model [19], which operates on behavioral rules at the individual level. According to the model, each individual is represented as a particle. The sensing region of a particle is a circular region centered around it, with neighbors being particles located within it. In theoriginal Vicsek model, each particle is moving at a constant speed within a two-dimensional space, aligning its heading direction with the mean orientation of its neighbors, while accounting for intrinsic noise to represent individual stochastic behavior. When the number of individuals or the intensity of the noise changes, the simulations show a phase transition from disordered to ordered states [19]. In the literature, a modification of the original Vicsek model is also considered that includes extrinsic noise, which means errors in environmental assumptions or information [20]. Studies indicate that intrinsic noise leads to a continuous phase transition, whereas extrinsic noise results in a discontinuous transition [20]. The simplicity of the Vicsek model has inspired numerous variants, including models with attractive and repulsive interactions [21; 22] and adaptations that extend the original model into more realistic three-dimensional scenarios [23]. Real-world biological swarms may not have omnidirectional vision, which requires modifications of the sensing region from a circular disk to a sector [24; 25; 26]. Most existing models of collective behavior assume visual cues for communication [24; 25; 27]. However, some social animals, such as bats and dolphins, rely on auditory cues [28]. Few models incorporate auditory interactions, such as those inspired by acoustic sensing in midges [29] and echolocation in bats [30]. A study in [31] explores auditory sensing in a modified two-dimensional Vicsek model, comparing it to visual sensing and demonstrating differences in group-level behavior. Auditory sensing models ultrasonic beams as sectors of a circle, mimicking their directivity patterns observed in biological systems [32]. The results in [31] reveal that auditory sensing results in higher alignment and lower aggregation compared to visual sensing. Despite advancements in studying group behavior through individual sensing cues, few models integrate multiple sensory modalities. However, empirical evidence, such as with bats, shows that they employ multimodal sensing, including audition, vision, somatosensory input, vestibular perception, and chemoreception, for navigation and communication [32; 33; 34]. Bats gather complementary information from vision and audition; vision helps to detect long-range objects, while audition helps to accurately detect small objects [34; 35]. Multisensory integration offers several advantages, including reduced reaction times [36]. Studies show that bats are more dependent on multimodal sensing in low light conditions and adjust their behavior based on visual and auditory inputs [37; 38; 39; 40; 41].