Behavioural Swarms in Adaptive Systems

• Behavioural swarms are multi-agent systems where each agent’s motion and internal state dynamically interact to yield complex collective behavior.
• They integrate an internal activity variable with spatial variables, enabling context-sensitive decision-making and adaptive responses.
• Applications include crowd dynamics, decentralized markets, and active matter, capturing phenomena like panic waves and spontaneous leadership.

Behavioural swarms are multi-agent systems—biological or artificial—where the collective dynamics result from the interplay between mechanical variables (position, velocity) and evolving internal behavioural states that drive agent-level adaptation, heterogeneity, and decision-making. Unlike classical swarm models that treat agents as undifferentiated automatons and focus on local mechanical alignment, behavioural swarm theory formalizes the feedback between microscopic internal variables (such as stress, commitment, or cognitive state) and group-scale outcomes, capturing the adaptive, context-responsive, and heterogeneous nature of living collectives (Fabregas et al., 16 Aug 2025).

1. Mathematical Formulation and Core Principles

The foundational distinguishing feature of behavioural swarms is the introduction of a dynamical internal activity variable, $u$, for each agent, in addition to the conventional spatial attributes (position $\mathbf{x}$, velocity $\mathbf{v}$). The generic microscopic state is therefore represented by the triplet $(\mathbf{x}_i, \mathbf{v}_i, u_i)$. The time evolution of the system is governed by a coupled system of ordinary differential equations (ODEs):

$$\begin{cases} \frac{du_i}{dt} = z_i \ \frac{dz_i}{dt} = \sum_{j \in \Omega_i} \eta_{ij} \, \chi_{ij} \ \frac{d\mathbf{x}_i}{dt} = \mathbf{v}_i \ \frac{d\mathbf{v}_i}{dt} = \sum_{j \in \Omega_i} \eta_{ij} \, \psi_{ij} \end{cases}$$

where $\Omega_i$ is the interaction domain (often a vision cone or metric neighborhood), and the interaction kernels $\eta_{ij}$, $\chi_{ij}$, and $\psi_{ij}$ depend nonlinearly on the full agent states—explicitly incorporating behavioral states into both mechanical and internal dynamics. Typical interpretations:

• $\eta_{ij}$: interaction rate modulated by distance and possibly behavioral states
• $\chi_{ij}$: forces driving the change in $u_i$, representing social influence, imitation, or emotional contagion
• $\psi_{ij}$: mechanical alignment, attraction, repulsion—modulated by internal state

This architecture encodes a feedback loop: internal state $u$ modifies agent-level kinetic behavior, while local mechanical stimuli and social signals dynamically reshape $u$ through repeated interactions. The activity variable is typically normalized ($u \in [0,1]$), where extremal values represent distinct behavioral modes (e.g., $u = 0$ for quiescence, $u = 1$ for full engagement/agitation).

2. Adaptive Decision-Making and Behavioral Feedback

Behavioural swarm theory (BST) formalizes adaptive individual decision-making by embedding internal state evolution hierarchically within the agent’s dynamical update. Specifically:

1. Internal Update: The agent’s $u$ is updated via $\chi_{ij}$—a function that captures social influence, learning from neighbors, or stress propagation. For example, in a crowd, $u$ may represent local panic driven by neighboring panic levels.
2. Mechanical Update: The newly updated $u$ then determines the agent’s mechanical response (speed, heading). E.g.,

$$s_i = \frac{1 + u_i}{2}, \qquad \boldsymbol{\omega}_i = (1 - u_i)\mathbf{v}^T_i + u_i \mathbf{v}_i^S$$

where $s_i$ is the preferred speed interpolating between baseline and maximal, $\mathbf{v}^T_i$ is a target direction, and $\mathbf{v}_i^S$ is a consensus direction, e.g., mean heading of neighbors.

This hierarchy gives rise to rich context-sensitive responses: agents dynamically adapt their movement and behavioral stance in response to both their own state and the local field induced by others. The adaptive feedback is critical for modeling context-driven transitions, e.g., panic waves, leadership emergence, segregation, or commitment in consensus tasks.

3. Heterogeneity and Macroscopic Pattern Formation

A central strength of the BST framework is the systematic incorporation of heterogeneity. Because $u_i$ evolves independently for every agent and modulates the agent's coupling to its neighbors, the collective can spontaneously segregate into subgroups or display nontrivial spatial-behavioral patterning. This may result in phenomena such as:

• Spontaneous leadership, when subpopulations of higher activity bias collective direction
• Behavioral phase separation, with quiescent and active bands as internal state segregates
• Transition points and group-level bifurcations, determined by the distribution/function of $u$ in the population, not merely mechanical thresholds

Crucially, in BST, the nontrivial functional dependence of interaction kernels on $u$ and the explicit inclusion of higher-order moments (e.g., quadratic or cubic in $u$) capture nonlinear amplification and suppression of social influences, which are essential for reproducing living system phenomenology.

4. Applications: Economics, Crowd Dynamics, Active Matter

The framework’s versatility is demonstrated via several paradigmatic applications:

Behavioral Economics & Decentralized Markets:

Agents (buyers/sellers) possess a dynamically evolving internal valuation or activity. Local price adjustments are governed by both the market average and agent-level states (e.g., cherry-picking behavior for buyers). The BST framework predicts the spontaneous emergence of price consensus—even under strictly local interaction and heterogeneous internal states—an effect that cannot be captured by standard Cucker–Smale or kinetic models.

Crowd Dynamics and Panic Propagation:

In group egress or escape scenarios, $u$ encodes panic, which is propagated through $\chi_{ij}$ based on visual perception domains (cones with prescribed angles and ranges). Macroscopically, the spread of high-$u$ patches models the traveling front of crowd agitation; the geometry of $\Omega_i$ dictates the speed and topology of panic waves, critically linking individual perception structures to collective escape dynamics.

Active Matter and Beyond:

Because internal state variables may encode any adaptive property (temperature, metabolic activity, commitment), BST generalizes beyond alignment/flocking and supplies a template for studying the coupling of mechanical and behavioral phenomena in cell sorting, tissue morphogenesis, or distributed robotic systems.

5. Comparison with Mean-Field, Kinetic, and Agent-Based Models

BST explicitly improves over previous modeling approaches:

Approach	Internal State	Adaptation	Heterogeneity	Macroscopic Predictability	Limitations
Kinetic theory (KTAP)	Absent	No	Weak	Good (in large $N$)	Assumes large $N$, no explicit adaptation
Agent-based models	Ad hoc	Ad hoc	Possible	Weak	Rule-based; lacks mathematical consistency
BST (this framework)	ODE system	Yes	Strong	Yes	Number-conservative; parameter derivation

BST is based on rigorous ODE systems, enabling stability and bifurcation analysis, and does not require the infinite-size limit inherent to kinetic theories. It improves over classical Cucker–Smale models by permitting the coupling between discrete, dynamic behavioral states and physical motion.

6. Open Problems and Future Directions

The paper identifies key avenues for further development:

• Non-conservative extensions: Incorporating birth-death processes, state transitions (differentiation), and irreversible events that are central to biological/social systems.
• Derivation of kernels: Grounding the forms of $\eta_{ij}, \chi_{ij}, \psi_{ij}$ in first-principles or minimal-axiom schemes, rather than empirical/phenomenological selection, remains an essential challenge for both explanatory power and predictive accuracy.
• Interfacing with multi-scale and machine learning: The ODE-centric structure of BST is amenable to coupling with mean-field or continuum models and to embedding in physics-informed neural network schemes—potentially enabling real-time adaptation and control in engineered swarms or for inference in biological collectives.
• Applicability to new domains: BST's formalism is broadly extensible to problems in economics, epidemiology, cell biology, etc., especially where agent-level adaptation and behavioral feedback are primary collective drivers.

7. Synthesis and Theoretical Significance

Behavioural swarm theory achieves a formal unification of adaptive agency and collective mechanics. By endogenizing internal states as dynamic variables, the framework captures key features of living systems: heterogeneity, adaptability, context-sensitivity, and emergence of complex macro-scale structure from agent-level behavioral rules. Empirically relevant phenomena—such as price consensus in decentralized economies, panic propagation in crowds, and spontaneous group-level coordination—all arise naturally within this mathematically tractable framework. BST thus offers a foundation for both analysis and design of collectives where agency and adaptation, not just local mechanics, are fundamental (Fabregas et al., 16 Aug 2025).