Emergent collective dynamics from motile photokinetic organisms

Abstract: The day-night cycle drives the largest biomass migration on Earth: the diel vertical migration (DVM) of aquatic organisms. Here, we present a three-dimensional agent-based model that incorporates photokinesis, gyrotaxis, and stochastic reorientation to explore how individual-level swimming behaviors give rise to population-scale DVM patterns. By solving Langevin equations for swarms of swimmers, we identify four distinct regimes -- Surface Accumulation, Shallow DVM, Deep DVM, and Sinking -- governed by two key dimensionless parameters: the Peclet number (Pe), representing motility persistence, and the vertical swimming asymmetry ratio (W=wdown/wup), encoding photokinetic bias. These regimes emerge from nonlinear interactions between light-driven navigation and active noise, diagnosed through topological and statistical features of vertical distributions. A critical feedback is uncovered: upward-biased swimming (W<1) promotes surface aggregation, while excessive downward bias (W>1) leads to irreversible sinking. Analytical estimates link regime boundaries to gyrotactic alignment and velocity reversals. Together, our results provide a mechanistic framework to interpret DVM diversity and emphasize the central role of light gradients-beyond absolute intensity-in shaping ecological self-organization.

• The paper presents a novel agent-based model that links individual photokinetic and gyrotactic behaviors to the emergence of distinct diel migration regimes.
• It employs Langevin dynamics and key parameters like the Péclet number and vertical speed ratio to identify regimes such as surface accumulation, deep DVM, and sinking.
• The findings highlight how subtle changes in motility bias and noise can predictively shape observed migration patterns and inform synthetic active matter design.

Emergent Collective Dynamics from Motile Photokinetic Organisms

The study presents a comprehensive agent-based modeling (ABM) framework that elucidates the mechanistic origins and population-level consequences of diel vertical migration (DVM) in motile, photokinetic aquatic organisms. By synthesizing advances in active matter physics and ecological modeling, the work addresses the multiscale feedbacks underlying the largest synchronized migration event in aquatic ecosystems.

Model Architecture and Theoretical Innovations

Each organism is modeled as a self-propelled, gyrotactic, photokinetic Brownian particle inhabiting a three-dimensional water column with a dynamically modulated light field. The agents sense local light intensity and adjust their vertical velocity accordingly, employing a bidirectional strategy governed by an isolume— the depth corresponding to a preferred light niche. The model explicitly incorporates:

• Photokinesis: Modulation of swimming speed and direction as a direct response to local light intensity gradients.
• Gyrotaxis: Alignment torque that biases motion against gravity, capturing realistic reorientation observed in planktonic systems.
• Rotational Diffusion: Stochastic fluctuations in orientation, parameterized by a rotational diffusion constant $D_R$.
• Inter-agent Exclusion: Short-range interactions are encoded via the Weeks-Chandler-Andersen (WCA) potential to enforce non-overlapping constraints.

The governing Langevin equations integrate these effects and are numerically solved using explicit Euler integration. Two central dimensionless parameters organize the system’s dynamics:

• Péclet number ($Pe = u_0 / (D_R \sigma)$): Ratio of advective to diffusive timescales, controlling persistence of directed motion.
• Vertical Speed Ratio ($W = w_\text{down}/w_\text{up}$): Quantifies asymmetry in upward versus downward motility, directly related to photokinetic bias.

Numerical Results and Identified Regimes

Extensive simulations— systematically varying $(Pe, W)$— reveal four robust collective regimes, each mapping onto qualitatively distinct ecological behaviors:

1. Surface Accumulation: High persistence and strong upward swimming bias ($Pe \gg 1$, $W \ll 1$) drive confinement near the upper water column. This regime is reinforced by gyrotactic reorientation and non-penetrating surface boundary, resulting in sharply peaked distributions at shallow depth.
2. Shallow DVM: Emerges for very asymmetric motility with limited swimming capacity ($W \ll 1$), leading to highly localized, thin-layered aggregation close to the nocturnal niche with minimal vertical range.
3. Deep DVM: Balanced vertical motility ($W \approx 1$) and moderate to large $Pe$ allow organisms to synchronize their movement with the diel isolume, reproducing canonical DVM behavior. Concentration fields display phase-locked migration patterns with pronounced excursions spanning the water column.
4. Sinking: Excessive downward bias ($W > 1$), regardless of $Pe$, results in a loss of coupling with the light field and progressive, irreversible descent below the niche isolume.

Statistical diagnostics— specifically, the normalized mean depth and the excess kurtosis of the vertical probability density— are computed for each regime, providing a quantitative phase diagram of DVM as a function of agent-level parameters.

Key Quantitative and Contradictory Findings

• Sharp Regime Mapping: The boundary between deep DVM and sinking is tightly constrained around $W \approx 1$, showing that only a narrowly balanced motility permits stable migration. This contradicts models that attribute DVM solely to environmental gradients by demonstrating its sensitive dependence on motility asymmetry and persistence.
• Role of Noise and Directionality: At $Pe < 1$, diffusive noise dominates, yet collective migration regimes are predominantly shaped by the vertical speed ratio, not by persistence.
• Gyrotactic Feedback: Upward swimming and gyrotactic alignment interact nonlinearly; models without gyrotaxis fail to recover the surface accumulation regime, underscoring the necessity of including physical orientation mechanisms.

Practical and Theoretical Implications

The ABM framework bridges individual photokinetic behavior and population-scale migration, providing a predictive theory for DVM diversity across a range of aquatic taxa. Practically, the results enable:

• Interpreting Acoustic/Optical Observations: The modeled vertical profiles can inform the analysis of field data from ADCP or bioacoustic surveys, supporting the identification of ecological regimes in situ.
• Design of Synthetic Active Matter: The generalized feedback between light environment and agent motility offers guidance for engineering artificial microswimmers with tunable aggregation or migration properties in structured light fields.
• Ecohydrodynamics and Biogeochemical Fluxes: Quantitative phase diagrams support large-eddy and global ocean models that require mechanistic parameterization of organism-induced mixing and carbon sequestration processes.

Theoretically, the findings highlight the centrality of local sensory-motor rules in the emergence of macroscale ecological phenomena, reinforcing the view that collective migration is a nontrivial function of both intrinsic agent properties and external environmental structure.

Future Directions

There is considerable potential to extend the present framework:

• Hydrodynamic Coupling: Incorporating explicit fluid flows, turbulence, or collective-induced mixing would enhance biological realism and connect to physical bioconvection phenomena.
• Multispecies and Evolutionary Dynamics: Future models could examine interactions between taxa differing in photokinetic properties, or evolutionary adaptation of $Pe$ and $W$ under environmental change.
• Adaptive and Non-Markovian Control: Agents with memory or learning—modulating $W$ in response to persistent environmental trends—could capture the plasticity observed in natural populations.

Conclusion

This work establishes a minimal, mechanistic framework linking agent-level motility and environmental sensing to emergent, population-scale DVM. The identification and mapping of distinct migration regimes in $(Pe, W)$ space provide both theoretical insights and practical tools for interpreting and predicting ecological self-organization in light-structured environments. The approach is extensible to broader classes of active matter beyond aquatic ecosystems, offering a fundamental building block for future eco-physical modeling efforts in both natural and synthetic systems.