🌀 nonlinear sciences

From agent-based dynamics to a kinetic theory of jellyfish swarms

This paper derives a continuous kinetic theory and hydrodynamic closure from an agent-based model to bridge the gap between individual jellyfish behaviors and large-scale swarm dynamics, enabling the prediction of massive jellyfish swarm formation and evolution within realistic ocean currents.

Original authors: Nicolas Perez, Erik Gengel, Zafrir Kuplik, Eyal Heifetz

Published 2026-09-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Nicolas Perez, Erik Gengel, Zafrir Kuplik, Eyal Heifetz

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

The ocean is rarely empty, even when it looks that way. In many parts of the sea, life does not just drift; it gathers. Massive clouds of jellyfish can stretch for tens of kilometers, containing millions of individuals moving as a single, shifting entity. For a long time, scientists viewed these swarms as passive drifters, carried along by the wind and waves like leaves on a stream. This view suggested that if you knew how the ocean currents moved, you could predict where the jellyfish would end up. However, recent observations have challenged this simple picture. Certain species, such as the nomadic jellyfish found in the Mediterranean, are not just floating; they are powerful swimmers. They actively push against the current, often moving in the opposite direction of the water flow. This ability to swim against the tide, known as positive rheotaxis, hints that the jellyfish are making decisions, responding to invisible cues in their environment, and perhaps communicating with one another. Understanding how these creatures move is not just a matter of biological curiosity; it is crucial for predicting when and where these massive swarms will form, which can impact coastal ecosystems, fisheries, and tourism.

A team of researchers at Tel Aviv University has taken a significant step toward solving this puzzle by building a new mathematical framework that connects the behavior of a single jellyfish to the movement of the entire swarm. Their work moves away from tracking every individual animal, a method that becomes impossible when dealing with millions of creatures, and instead treats the swarm as a continuous fluid. They started with a model that describes how a single jellyfish reacts to its surroundings: it swims at a speed that changes with the current, it turns randomly due to turbulence or its own unpredictability, and it steers itself toward favorable conditions. These favorable conditions might include swimming against the flow, avoiding areas of turbulent water, or following a chemical signal released by other jellyfish. The researchers realized that while these individual actions happen quickly, the overall movement of the swarm evolves much more slowly, driven by the large-scale ocean currents.

By separating these fast individual reactions from the slow movement of the group, the team derived a set of equations that describe the swarm's density and direction without needing to simulate every single jellyfish. This approach, known as a kinetic theory, allows them to calculate how the swarm will spread, concentrate, or disperse over time. A key finding of their work is that the swarm's behavior is not just a result of the ocean pushing them around. Instead, the jellyfish actively shape their own distribution. When they swim against the current or follow a chemical trail, they can form dense patches that persist even in weak currents. The researchers also incorporated a mechanism for chemical signaling, suggesting that jellyfish release a substance into the water that attracts others, creating a feedback loop that helps the swarm stay together. This chemical signal is not a permanent marker; it diffuses and decays, meaning the swarm must constantly renew its cohesion through active swimming and sensing.

The paper explicitly rules out the idea that these swarms are formed solely by passive advection, where the ocean simply concentrates drifting organisms. The authors argue that without the jellyfish's active ability to swim against the flow and respond to environmental cues, the large, coherent structures observed in the real world would not be possible. They also clarify that while jellyfish do collide with one another, these collisions are mostly passive bumps that do not drive the swarm's organization; rather, the organization comes from how each jellyfish reacts to the water and chemical signals. The model suggests that the balance between the jellyfish's drive to swim in a specific direction and the random jostling of collisions or turbulence determines how tightly the swarm is packed. If the drive to swim together is strong, the swarm becomes a tight, directed group; if the random movements dominate, the swarm spreads out.

This new framework provides a practical tool for oceanographers. Because the equations are simplified to focus on the large-scale movement, they can be plugged into existing computer models that simulate ocean currents. This means that in the future, scientists could potentially forecast the formation of jellyfish swarms with greater accuracy, much like weather forecasting. The researchers acknowledge that their model relies on several assumptions that still need to be tested against real-world data. For instance, the exact chemical signal the jellyfish use is still hypothetical, and the precise strength of their response to different stimuli needs to be measured in controlled experiments. The team proposes a roadmap for the future, which includes using drones to track swarms in the wild and conducting tank experiments to see how jellyfish react to specific currents and chemicals. By combining these observations with their mathematical model, they hope to refine the parameters and eventually create a reliable system for predicting these massive biological events.

The work represents a bridge between the microscopic world of individual animal behavior and the macroscopic world of ocean dynamics. It shifts the perspective of jellyfish from passive victims of the ocean's motion to active participants that shape their own destiny. While the model is currently a theoretical construct, it offers a clear path forward. It suggests that the mystery of the jellyfish swarm is not just about where the water goes, but about how the jellyfish decide where to go. By understanding these decisions, scientists can begin to see the ocean not just as a fluid that moves things, but as a complex environment where life actively navigates, communicates, and organizes itself on a grand scale.

Problem Statement
Massive jellyfish swarms, which can span tens of kilometers and contain millions of individuals, exhibit complex large-scale dynamics that are poorly understood. While agent-based models (Lagrangian approaches) effectively describe individual jellyfish responses to ocean currents and environmental cues, they become computationally prohibitive when scaled to the spatial and population sizes relevant to ocean circulation. Conversely, existing models often treat jellyfish as passive drifters, failing to capture the active swimming behaviors—such as positive rheotaxis (swimming against currents) and responses to shear or chemical gradients—observed in species like Rhopilema nomadica and Rhizostoma pulmo. There is a need for a continuum framework that bridges individual behavioral mechanisms with large-scale swarm dynamics to enable numerical prediction within ocean-current models.

Methodology
The authors derive a continuous kinetic theory from a previously proposed agent-based model (Gengel et al. [21, 22]) where jellyfish are treated as two-dimensional active Brownian particles. The methodology proceeds through the following steps:

Kinetic Formulation: The individual dynamics, comprising passive transport by flow $U$ , active swimming with speed $V$ , and orientation changes driven by deterministic steering (response to stimuli) and stochastic torques (self-diffusion and collisions), are reformulated into a Fokker-Planck equation for the distribution function $f(x, \theta, t)$ .
Collision Modeling: Instead of complex scattering integrals, pairwise collisions are modeled as a density-dependent angular diffusivity, $\delta(\rho)$ , which increases disorientation in dense regions without altering swimming speed.
Chemical Signalling: A chemoattractant field $\phi$ is coupled to the system via an advection-diffusion equation with a source term proportional to jellyfish density, representing chemical release during bell cycles.
Hydrodynamic Closure: Exploiting the separation of time scales between fast individual orientational dynamics (seconds to minutes) and slow swarm spatial evolution (days), the authors derive a hydrodynamic closure. They assume the angular distribution rapidly relaxes to a local quasi-equilibrium (von Mises distribution) determined by the alignment parameter $\kappa = \Psi/D(\rho)$ .
Derivation of Reduced Equations: By integrating the Fokker-Planck equation over orientation, the authors derive a continuity equation for the jellyfish density $\rho$ that includes both passive transport and active directional swimming terms dependent on the steering field $\Psi$ and the alignment parameter $\kappa$ .

Key Contributions and Results

Derivation of a Kinetic Framework: The paper presents a Fokker-Planck framework that incorporates transport by prescribed currents, stochastic reorientation, direct interactions (modeled as density-dependent diffusion), and stimulated steering. This allows chemical signalling to be represented through a coupled field.
Hydrodynamic Closure: The authors derive a reduced density equation suitable for implementation in ocean-current models. This equation is valid for all values of the alignment parameter $\kappa$ , bridging the gap between weak-stimuli (diffusive) and strong-stimuli (directed) regimes.
Coupled System: The resulting model consists of a closed system of partial differential equations:
- A continuity equation for jellyfish density $\rho$ , driven by flow advection and active swimming velocity proportional to $V \frac{I_1(\kappa)}{I_0(\kappa)} \mathbf{e}_{\theta_\Psi}$ .
- An advection-diffusion equation for the chemoattractant concentration $\phi$ , coupled to $\rho$ via a source-sink term.
- Definitions for the steering field $\Psi$ based on positive rheotaxis, shear avoidance, and chemotaxis.
Analysis of Limits: The study analyzes the behavior of the system in the limits of small and large $\kappa$ . In the limit of small $\kappa$ , the model recovers a form similar to the Keller-Segel model with an additional passive transport term. In the general case, the model accounts for non-linear effects where the angular distribution is sharply peaked.
Roadmap for Implementation: The paper outlines a protocol for implementing these equations in ocean-current models (e.g., MITgcm) and proposes a roadmap for validation using biological data, controlled tank experiments, and in-situ swarm tracking.

Significance
The paper establishes a theoretical foundation for the numerical prediction of large jellyfish swarm formation and evolution in realistic ocean flows. By providing a route from individual behavioral mechanisms to continuum descriptions, the framework allows researchers to:

Investigate the relative contributions of passive transport versus active swimming in the formation and persistence of aggregations.
Study how competing mechanisms (e.g., rheotaxis vs. chemotaxis) interact in heterogeneous environments like fronts and shear zones.
Constrain model parameters using observations and in-situ measurements.
Move beyond the assumption of jellyfish as passive tracers, enabling the incorporation of active swimming and sensory responses into large-scale ecological and oceanographic models.

The authors emphasize that while the model provides a workable theoretical framework, the specific parameters and the existence of certain mechanisms (like specific chemical signalling) require validation through future experimental data and observations.

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