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

TL;DR

This paper introduces a new mathematical framework for behavioural swarms that models adaptive, heterogeneous behaviors in living systems, enabling better prediction of collective dynamics through internal activity variables.

Contribution

It formalizes the concept of Behavioural Swarms, integrating adaptive decision-making and heterogeneity into differential models, advancing beyond classical swarm theories.

Findings

The framework captures emergent macroscopic patterns from individual behavioural states.

Application to crowd dynamics demonstrates realistic collective movement predictions.

Comparison shows advantages over kinetic and agent-based models.

Abstract

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.