| For foraging or migratory animals,such as bison,flocks,fish shoals,colonies,etc.,the direction of collective motion always depends on a small number of individuals,which are called as leaders.These leaders are almost the same with ordinary individuals,except having pertinent information,such as the location of a food source or migration route.The leadership effect is caused only by information gap.For fish shoals,only relatively few informed individuals could lead the collective foraging behavior,while the bee colony needs only a few leaders（about 5%）to navigate towards a destination.To investigate the phenomenon,we discuss the Vicsek Model with vectorial noise and leaders.The Vicsek Model is one of the most widely used models of studying the animal collective motion,and about the vectorial noise case,the model is a first-order phase transition.When we input leaders,we find that the phase transition of the system will changes as the proportion of leader increases,and it will be a second-order phase transition beyond a certain proportion threshold.Second,we show that with periodic boundary conditions,only a single leader is needed to lead the direction of the entire group motion.While open boundary conditions,the proportion of leaders has to reach a certain threshold to control the movement of the group,and the larger the system size,the smaller the proportion of leaders required.Finally,we respectively study the relationship between the efficiency of leader leading group movement and leader proportion,system density and noise intensity.Our simulation found that the larger the proportion of leaders,the greater the system density,and the higher the efficiency of the leader.For the noise case,when the noise intensity exceeds the critical noise or nearly achieve the critical noise,the system appears as a chaotic movement,and the individuals couldn’t achieve a consensus.With small noise,we find that the leader’s efficiency will increases with higher noise level. |
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