How do flocking birds and schools of fish move? New research offers crystal-clear answer

Flocking birds and schools of fish are a familiar sight. While previous research has uncovered the broad dynamics driving these movements, their underlying intricacies remain a mystery. 

A study by a team of New York University mathematicians offers some new insights into these phenomena. It reveals that flocks and schools behave in ways that are similar to a soft crystalline material, with individual birds and fish serving as “atoms” that are evenly spaced in a lattice-like formation. 

The findings, which are reported in the journal Physical Review Fluids, offer detailed insights into the hydrodynamic and aerodynamic interactions crucial in aerospace and automotive engineering, robotics, and energy harvesting.

“Our findings offer a new way to understand how animal collectives coordinate movement and respond to their environment,” says Christiana Mavroyiakoumou, a researcher at NYU’s Courant Institute School of Mathematics, Computing, and Data Science at the time of the study and now a fellow at Oxford University’s Mathematical Institute. “More specifically, lines of birds or fish behave like an elastic material with regularly spaced individuals held together by flexible, or spring-like, bonds—akin to soft crystalline substances in which atoms are arranged in an orderly, repeating pattern.”

“Because these movements are similar to those that form the building blocks of materials, the work opens new avenues for analyzing—and potentially manipulating—how these components interact,” adds Courant Professor Leif Ristroph, director of NYU’s Applied Mathematics Laboratory, where the research was conducted. 

The laboratory previously uncovered how birds and fish move together without colliding and the underlying aerodynamics of these movements. However, the detailed nature of these orchestrated motions had been less clear.

The research team, which also included Jiajie Wu, an NYU undergraduate at the time of the study, proposed a mathematical model to explain these movements—one that was akin to those of soft crystalline materials, or soft crystals. These ordered solid materials can change their properties in response to stimuli, such as temperature or physical force, which make its atomic organization fragile. The researchers, then, saw a connection between crystalline organization and how birds or fish move together while adjusting their movements and formation in response to air or water flows, predators, or objects, such as rocks or buildings. 

“Crystalline organization is inherently fragile as positions are susceptible to deformations and instabilities,” explains Mavroyiakoumou. “In similar ways, birds and fish must sense and respond quickly to other forces in order to maintain long columnar formations. So while soft crystals, flocks of birds, and schools of fish are fragile in their makeup, such fragility may also be advantageous as it can be responsive to its surroundings.”

The study’s authors considered previous experiments to determine if the model matched these experimental results. Among these was an experiment that mimicked  the columnar formations of birds—in which they line up one directly behind the other—using mechanized flappers that act like birds’ wings. The wings were 3D-printed from plastic and driven by motors to flap in water, which captured how air flows around bird wings during flight. This “mock flock” propelled through water at different speeds and could freely arrange itself within a line or queue, as seen in a video of the experiment (caption: A live recording of the experimental apparatus in operation. Five foils are driven to flap up and down in unison, and they freely and interactively propel around a water tank. Courtesy of NYU's Applied Mathematics Laboratory at the Courant Institute School of Mathematics, Computing, and Data Science.).

Overall, the flappers as a group behaved as the researchers had conceptualized—one of several experiments that offered support for their proposed model.

The research was supported by a grant from the National Science Foundation (DMS-1847955).

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Editor’s Note: In November 2025, NYU announced the establishment of the Courant Institute School of Mathematics, Computing, and Data Science. The newly established school recognizes the storied history of the Courant Institute of Mathematical Sciences—and its strengths in both applied and pure mathematics—while encompassing NYU’s Center for Data Science and linking the computer science departments at Courant and the Tandon School of Engineering.