From Individual to Swarm Dynamics of Motile Photokinetic Organisms

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—, , , and —governed by two key dimensionless parameters: the Péclet number ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mtext>Pe</a:mtext> </a:math> ), representing motility persistence, and the vertical swimming asymmetry ratio ( <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>W</b:mi> <b:mo>=</b:mo> <b:msub> <b:mi>w</b:mi> <b:mtext>down</b:mtext> </b:msub> <b:mo>/</b:mo> <b:msub> <b:mi>w</b:mi> <b:mtext>up</b:mtext> </b:msub> </b:mrow> </b:math> ), 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 ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi>W</c:mi> <c:mo>&lt;</c:mo> <c:mn>1</c:mn> </c:mrow> </c:math> ) promotes surface aggregation, while excessive downward bias ( <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>W</d:mi> <d:mo>&gt;</d:mo> <d:mn>1</d:mn> </d:mrow> </d:math> ) 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.