Signal propagation and linear response in the delay Vicsek model

Persistent and responsive collective motion with adaptive time delay

“…The absolute value of Ο R ranges from 0 to 1, indicating the group's degree of rotation. By performing parametric sweeping of the fixed τ, our experimental observations reproduce two distinctive features of timedelay collective motion, which are consistent with the predictions by several previous simulation studies (12,30,35). First, the time-averaged |Ο R | (〈|Ο R |〉) initially increases and then decreases as τ increases (figs.…”

“…To better quantify the persistence of collective structures in experiments, we introduce a method called "active perturbation. " Accordingly, by manipulating particle-particle interactions based on the time-delay Vicsek model (12,30,31,35), our experimental observations quantitatively confirm that a fixed time delay can enhance the persistence of collective structures against perturbations while diminishing their responsivity. However, when the time delay of each particle adapts to its surrounding environment, the collective structure can maintain its persistence while experiencing an approximate 1.6-fold increase in responsivity, implying a time saving of ~38% for the structure to reach the steady state.…”

“…S6). In addition, as the responsivity of a collective structure reflects the rate at which it responds to the perturbation and approaches the presumed steady state ( 35 , 36 ), we adopt the straightforward definition of d ∣ Ο R ∣/ dt to quantify the collective structures’ global response speed, i.e., their responsivity, in this study. An R -fold increase in responsivity implies that the collective structure can reach the steady state R−1R*100% faster.…”

“…To enable time-delay interaction for each particle, real-time images are captured and stored in a sequential manner within the control program. This allows us to have a time-sequence history of each particle’s positions, which is essential for implementing the nonphysical interaction between particles based on the time-delay Vicsek model ( 12 , 30 , 31 , 35 ). Specifically, the original Vicsek model posits that each particle determines its new direction by averaging the current velocities of itself and its neighbors, incorporating some random perturbation.…”

“…Notably, by incorporating proper time delay, the collective structures can have high persistence to maintain their movement in noisy environments ( 30 , 34 ). However, since most of the agents cannot efficiently distinguish nontrivial information (e.g., food sources and predators) from noise, the fixed time delay can also reduce the responsivity of collective structures, which slows down their global response to nontrivial external stimuli ( 35 ). Consequently, the fixed time delay in collective motion inherently leads to tradeoff between persistence and responsivity ( 36 ).…”