Directional switches in time-delay swarming systems induced by speed-direction trade-off

Collective motion in biological systems often exhibits emergent orderly directional switching behavior. The classical Vicsek model can partially simulate this behavior but fails to fully account for the complex effects arising from the interplay of individual trade-off between speed and movement direction, social interactions, and time delays. In this paper, we propose a mathematical framework to compute the average directional switching time in a time-delay system that incorporates speed-direction trade-offs. Using this framework, we analyze the effects of time delay and social interactions on collective motion. Theoretical and numerical results indicate the existence of an optimal speed-direction trade-off and social interaction strength that maximize the orderliness of directional switching in the system.