Simulation and analysis of collective self-propelled movement in the presence of a leader

In recent years, the study of collective behavior in living organisms has become a prominent research focus in both physics and biology. This paper aims to investigate the mechanisms behind the collective motion observed in certain living systems, focusing on the role of physical interactions at both microscopic and macroscopic scales. Specifically, we model a scenario in which individuals within a group follow a designated leader, with the goal of understanding how collective behavior emerges under this dynamic. Our numerical study employs a model that incorporates both position and velocity updates, where individuals adjust their speeds and directions to align with the leader's movement. The used model is based on elastic interaction. Importantly, our results demonstrate that collective motion is achieved not purely through aligning interactions, but rather through a combination of position-based regrouping and velocity matching with the leader. Hence, before individuals can match the speed of the leader, they undergo a phase of regrouping where they adjust their positions and velocities to form a cohesive cluster. Once this regrouping phase is complete, they accelerate to match the leader's speed. However, the time required for regrouping depends on the size of the system, with smaller systems achieving collective motion more rapidly. Morever, noise, denoted by η, plays a critical role in determining the success of directed movement. Low noise levels facilitate the emergence of coordinated motion, whereas higher noise intensifies random fluctuations (thermal agitation), preventing the group from achieving stable, directed movement. This study underscores the importance of system size and noise control in the emergence of collective behavior, offering insights into the conditions necessary for achieving cohesive and directed motion in group dynamics.