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In this paper, we present a model describing the collective motion of birds.We explore the dynamic relationship between followers and leaders, wherein aselect few agents, known as leaders, can initiate spontaneous changes indirection without being influenced by external factors like predators. Startingat the microscopic level, we develop a kinetic model that characterizes thebehaviour of large crowds with transient leadership. One significant challengelies in managing topological interactions, as identifying nearest neighbors inextensive systems can be computationally expensive. To address this, we proposea novel stochastic particle method to simulate the mesoscopic dynamics andreduce the computational cost of identifying closer agents from quadratic tologarithmic complexity using a $k$-nearest neighbours search algorithm with abinary tree. Lastly, we conduct various numerical experiments for differentscenarios to validate the algorithm's effectiveness and investigate collectivedynamics in both two and three dimensions.

Kinetic description of swarming dynamics with topological interaction and emergent leaders