标题： 行为集群的数学理论：迈向对活体系统集体动力学的建模 显示英文标题

标题： The Mathematical Theory of Behavioural Swarms: Towards Modelling the Collective Dynamics of Living Systems

摘要： 经典群体模型，以Cucker--Smale框架为例，提供了关于集体对齐的基础见解，但在捕捉生物系统固有的适应性和异质性行为方面存在根本性局限。 本文形式化了 \textit{行为集群}，一个综合框架，其中每个粒子的状态包含一个动态内部变量，即\textit{活动}，该变量通过非局部相互作用与位置和速度共同演化。 我们展示了这种方法如何通过将自适应决策机制和异质行为状态整合到严格的微分系统中，超越了先前的模型。 通过在行为经济学和人群动力学中的应用，我们确立了该理论从个体行为状态预测涌现宏观模式的能力。 我们的分析将该框架与活性粒子的动能理论和基于代理的方法进行了对比，揭示了在个体自主性驱动集体结果的系统建模中的显著优势。 显示英文摘要

摘要： Classical swarm models, exemplified by the Cucker--Smale framework, provide foundational insights into collective alignment but exhibit fundamental limitations in capturing the adaptive, heterogeneous behaviours intrinsic to living systems. This paper formalises the mathematical theory of \textit{Behavioural Swarms}, a comprehensive framework where each particle's state incorporates a dynamic internal variable, the \textit{activity} that co-evolves with position and velocity through nonlocal interactions. We demonstrate how this approach transcends prior models by integrating adaptive decision-making mechanisms and heterogeneous behavioural states into rigorous differential systems. Through applications in behavioural economics and crowd dynamics, we establish the theory's capacity to predict emergent macroscopic patterns from individual behavioural states. Our critical analysis positions this framework against kinetic theories of active particles and agent-based approaches, revealing distinct advantages for modelling systems where individual agency drives collective outcomes.