In this paper, we present a model describing the collective motion of birds. We explore the dynamic relationship between followers and leaders, wherein a select few agents, known as leaders, can initiate spontaneous changes in direction without being influenced by external factors like predators. Starting at the microscopic level, we develop a kinetic model that characterizes the behaviour of large crowds with transient leadership. One significant challenge lies in managing topological interactions, as identifying nearest neighbors in extensive systems can be computationally expensive. To address this, we propose a novel stochastic particle method to simulate the mesoscopic dynamics and reduce the computational cost of identifying closer agents from quadratic to logarithmic complexity using a $k$-nearest neighbours search algorithm with a binary tree. Lastly, we conduct various numerical experiments for different scenarios to validate the algorithm's effectiveness and investigate collective dynamics in both two and three dimensions.

翻译：本文提出一个描述鸟类集体运动的模型。我们探究了跟随者与领导者之间的动态关系，其中少数个体（称为领导者）能在不受捕食者等外部因素影响的情况下自发改变方向。从微观层面出发，我们建立了一个描述具有瞬态领导特征的大规模群体行为的动力学模型。关键挑战在于处理拓扑相互作用——在庞大系统中识别最近邻域的计算成本极高。为此，我们提出一种新型随机粒子方法来模拟介观动力学，通过采用基于二叉树的k-最近邻搜索算法，将识别邻近个体的计算复杂度从二次型降低至对数型。最后，我们针对不同场景开展多种数值实验，验证了算法的有效性，并在二维和三维空间中研究了集体动力学行为。