🌀 nonlinear sciences

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

This paper introduces a comprehensive mathematical framework for "Behavioural Swarms" that overcomes the limitations of classical models by incorporating dynamic internal activity variables to rigorously model the adaptive, heterogeneous, and agency-driven collective dynamics of living systems.

Original authors: Rene Fabregas, Jie Liao, Nisrine Outada

Published 2026-08-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Rene Fabregas, Jie Liao, Nisrine Outada

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Living things do not move like stones or planets. A rock follows the laws of physics with perfect predictability, but a bird, a bacterium, or a human being carries an internal world that changes how it acts. This internal world includes feelings like fear, confidence, or stress, which shift from moment to moment and alter how an individual interacts with others. For decades, scientists have tried to write mathematical rules that describe how groups of these living things move together. The most famous attempts treated every member of the group as a simple machine, following fixed rules to match the speed and direction of its neighbors. While these models explained how flocks of birds or schools of fish could stay together, they missed a crucial truth: living creatures are not machines. They make choices based on their own changing moods and strategies.

This gap in understanding is what a new study from researchers in Spain, China, and Morocco seeks to fill. They have developed a fresh mathematical theory called the "theory of behavioural swarms." Instead of treating every member of a group as a static object, this new framework gives each individual a hidden, shifting variable called "activity." Think of this activity as a measure of how alert, stressed, or motivated an individual is at any given second. This number is not fixed; it changes as the individual interacts with others, and in turn, it changes how the individual moves. The researchers built a set of rules that links this internal state directly to physical motion, creating a system where a person's fear can instantly change their walking speed, or a bird's confidence can alter its flight path.

The study begins by looking at older models that worked well for simple alignment but failed to capture the complexity of life. In those older systems, if a bird wanted to turn, it simply averaged the directions of the birds around it. The new theory adds a layer of decision-making. It proposes that before a bird or a person moves, it first updates its internal state based on what it sees and feels. If a pedestrian in a crowd feels a rising sense of panic, that feeling (the activity) immediately influences how fast they walk and which way they turn. The researchers showed that this internal state and the physical movement are locked in a loop: the environment changes the feeling, and the feeling changes the movement.

To test if this idea works, the team applied it to two very different real-world situations: the movement of people in a crowd and the buying and selling of goods in a market. In the crowd scenario, they simulated a group of pedestrians where one person suddenly became panicked. The model showed how that panic would spread. When the panicked person saw others, their fear influenced those neighbors, who then became fearful themselves. The researchers found that the shape of the panic wave depended heavily on how far people could see. If everyone had a wide field of view, the panic spread faster and covered a broader area. If their view was narrow, the panic moved more slowly in a tight line. This matched what we might expect in real life, but the model proved it mathematically by tracking the changing "activity" of every single person in the simulation.

In the second application, the researchers looked at a market where buyers and sellers interact. Here, the "activity" represented how much a buyer wanted a product or how much a seller was willing to lower their price. The model showed that when buyers acted as a group, hunting for the best deal, they could create sudden waves of demand that forced prices to change. Unlike older economic models that assumed a central authority sets prices, this system showed how prices could emerge naturally from the chaotic, individual decisions of many people. The simulation revealed that if sellers could adjust their prices based on how many buyers were looking at them, the market would eventually find a stable price on its own, without anyone telling them what to do.

The researchers also explored what happens when a group is not just moving together, but also thinking together. They simulated a scenario where individuals tried to agree on a direction to go, but they were also trying to agree on an internal opinion. They discovered something surprising: the group could reach a perfect agreement on where to walk, even if they remained deeply divided on what they believed. The physical movement of the group became unified, while their internal states stayed split into different camps. This suggests that in real life, a crowd can move as one unit even if the people inside it are arguing or holding different views.

While the theory is powerful, the authors are careful to note its limits. The current version of the model assumes the number of people in the group stays the same; it does not yet account for people being born or dying, or for new agents appearing out of nowhere. This makes it less suitable for studying things like the spread of a virus where the number of infected people changes constantly. However, for systems where the group size is stable, the theory offers a much clearer picture of how internal feelings drive external actions.

The paper concludes by looking toward the future, suggesting that this theory could be combined with modern computer learning tools. By feeding real data from animal flocks or human crowds into these models, scientists might one day be able to automatically discover the exact rules that govern how activity changes. This would move the field from guessing how people behave to actually predicting it. The work does not claim to have solved the mystery of life, but it provides a new, more honest language for describing it. By admitting that living things carry an internal world that shifts and changes, the theory of behavioural swarms offers a way to understand the complex, adaptive dance of life without reducing it to simple mechanics.

Technical Summary: The Mathematical Theory of Behavioural Swarms

Problem Statement
The paper addresses a fundamental limitation in classical swarm modeling, specifically within frameworks like Cucker–Smale (CS) and Vicsek. While these models successfully describe collective alignment and flocking based on mechanical rules (position and velocity), they treat agents as homogeneous automata governed by predetermined rules. They fail to capture the intrinsic adaptive strategies, heterogeneity, and internal decision-making processes characteristic of living systems. The authors argue that a "calculus of life" is required—one that moves beyond inert matter descriptions to formalize how internal states (e.g., stress, motivation, confidence) co-evolve with mechanical dynamics through nonlocal interactions.

Methodology
The paper formalizes the Mathematical Theory of Behavioural Swarms (BST), a framework where the state of each agent (an "active particle" or a-particle) is augmented by a dynamic internal variable termed activity ( $u_i$ ). Unlike classical models where parameters are fixed, activity is a co-evolving state variable that modulates and is modulated by the agent's mechanical motion.

The core methodology involves a system of coupled second-order ordinary differential equations (ODEs) governing the evolution of four state variables for each particle $i$ :

Position ( $x_i$ ) and Velocity ( $v_i$ ): Mechanical dynamics.
Activity ( $u_i$ ): A dimensionless internal variable ( $u_i \in [0,1]$ ) representing behavioral states (e.g., stress, conviction, learning level).
Activity Momentum ( $z_i = \dot{u}_i$ ): The rate of change of activity.

The system is structured hierarchically to reflect causal flow in living systems:

Step 1 (Behavioral Update): The internal state $u_i$ evolves based on interactions within a dynamic interaction domain $\Omega_i$ (defined by the intersection of visibility and sensitivity). This evolution is driven by a "behavioral force" ( $\chi_{ij}$ ) representing social influences like imitation or learning.
Step 2 (Mechanical Update): The updated behavioral state informs the mechanical dynamics. The velocity $v_i$ evolves based on a "mechanical force" ( $\psi_{ij}$ ), which includes alignment, attraction, or repulsion, but is now weighted and modulated by the current activity levels of the interacting agents.

The interaction rate $\eta_{ij}$ is non-constant and depends on the full state of the interacting pair, allowing for heterogeneous and state-dependent interactions. The framework is general enough to be specialized into spatially homogeneous systems (focusing solely on opinion/sentiment dynamics) or network-based systems (replacing physical space with graph topology).

Key Contributions

Formalization of Activity as a Dynamic Variable: The primary contribution is the rigorous mathematical integration of an internal "activity" variable that is not a fixed parameter but a dynamic state. This creates a feedback loop where behavior drives motion, and motion (via environmental perception) drives behavior.
Unification of Models: The paper demonstrates that the BST framework is a natural generalization of existing models. By constraining the activity variable to be constant and simplifying interaction rules, the BST reduces to the classical Cucker–Smale model. Conversely, it can recover the Thermodynamic Cucker–Smale model by interpreting activity as temperature.
Dual Application Framework: The authors provide a unified structure applicable to two distinct classes of problems:
- Physical Swarms: Where dynamics occur in physical space (e.g., pedestrian crowds, animal flocks), and activity modulates speed and direction.
- Abstract Swarms: Where dynamics occur entirely in the state space of internal variables (e.g., opinion formation, market price dynamics), with no physical movement.
Critical Comparison: The paper offers a critical analysis contrasting BST with the Kinetic Theory of Active Particles (KTAP) and Fokker–Planck–Boltzmann (FPB) theories. It highlights that BST does not require the assumption of a large number of particles for continuity (unlike KTAP) but notes its computational limitations with large $N$ and its current inability to model proliferative/destructive events (birth/death).

Results and Numerical Demonstrations
The paper validates the theory through several numerical case studies and applications:

Stress and Swarm Behavior: Simulations show that agents with higher activity levels align more rapidly to external signals. When activity is dynamic and based on consensus, the swarm achieves better cohesion. Asymmetric influence (where only high-activity agents influence others) leads to faster convergence and the emergence of mobile leaders.
Market Dynamics (Cherry-Picking): In a decentralized market model, the framework captures "cherry-picking" behavior where buyers seek the lowest prices. The model demonstrates how local interactions lead to macroscopic phenomena like "winner-take-all" market structures or price consensus, depending on whether sellers adapt their prices.
Panic Propagation: Numerical experiments on pedestrian crowds reveal that the geometry of the interaction domain (visibility angle) critically affects panic spread. A wider visibility angle leads to faster, broader, and accelerating panic waves due to non-linear amplification of the emotional state.
Multi-Consensus Dynamics: The paper illustrates that agents can achieve kinematic consensus (moving in the same direction) while maintaining behavioral heterogeneity (holding different internal opinions). By introducing a "bounded confidence" mechanism (agents only interact if their internal states are close), the system fragments into distinct opinion clusters while still achieving global alignment in movement.

Significance and Claims
The paper positions BST as a necessary evolution in the mathematical modeling of living systems, bridging the gap between mechanical motion and internal agency. Its significance lies in:

Providing a rigorous ODE-based framework that explicitly models the reciprocal feedback between internal states and physical dynamics.
Offering a flexible structure capable of describing diverse systems, from biological aggregations to economic markets, under a single formalism.
Identifying the need for future development in data-driven discovery. The authors propose using Physics-Informed Neural Networks (PINNs) to solve inverse problems with sparse data and Sparse Identification of Nonlinear Dynamics (SINDy) to discover the analytical forms of interaction kernels from observational data. This aims to move the theory from descriptive to predictive by deriving interaction laws from first principles or data rather than empirical assumptions.

The authors conclude that while BST currently faces challenges in computational scalability and modeling non-conservative events (birth/death), it provides a foundational blueprint for a predictive science of collective behavior in adaptive living systems.

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