Birds fly in flocks, fish swim in schools. What bonds them together, without dispersal or collision? Mathematicians from New York University (NYU) explained the method to the madness: the animals move in tandem, like an orchestra with a maestro leading them. A bird’s flapping wings, for instance, produces “vortex wakes” (or travelling waves) with each guided by others’ wakes, the researchers explained in a new paper published in the journal Physical Review Fluids.

The study likens these clustering creatures behaviour to “soft crystals” with regularly spaced “atoms”. Their positioning is, however, susceptible to deformations and dynamical instabilities such as temperature fluctuations or physical force by a predator; the “atomic” organisation frays, but rapidly reconnects and synchronises into a moving crystal-like organisation. 

Elastic birds

“[The] 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,” said Christiana Mavroyiakoumou, a researcher at NYU’s Courant Institute School of Mathematics, Computing, and Data Science (now a fellow at Oxford University’s Mathematical Institute), in a release.

Each bird has a specific form of flapping and this determines the flyer’s self-propulsion and also leaves behind a “wake flow signal” in its trail, the paper explained. Experiments, says the paper, have shown that when driven with identical flapping kinematics (the motion of objects), the “follower takes up one of several discrete, stable positions within the leader’s wake, and the two travel together at the same speed.”

As for fish, their uncoordinated milling at low swimming speeds may transform into orderly “lattices” of strongly aligned members that are forced into position by the strong movements generated at high speeds, said the paper, adding that this phenomenon is similar to atoms “that transition from a disordered gas phase to an orderly crystalline solid”.

The researchers used a mathematical model to understand how these animals collectively and interactively move, which they say can also be scaled up to study larger groups. They built on previous models, which suggested that each bird flies along a line while emitting a wake flow and also interacting with the wake flows other birds emit.

Cohesion at well-separated positions allows birds to move en masse. And the flight speed is determined solely by the leader. “The leader is in fact unaffected by the followers since the interactions are one way… as dictated by the wake flows left by one member and encountered by the next,” said the paper. The smallest disturbance would be amplified to other members in the group, suggesting their sensing of predators, prey, or flow disturbances.

Informing robotics

The authors suggested that the hydrodynamic and aerodynamic interactions birds and fish exhibit could inform aerospace and automotive engineering, robotics and energy harvesting. Several robophysical experiments that have been carried out over the last decade have aattempted to explain the interplay between physical and behavioral interactions in formation movement.

There is a similarity of this phenomenon to the formation of building blocks of materials, which “opens new avenues for analyzing—and potentially manipulating—how these components interact,” said Leif Ristroph, director of NYU’s Applied Mathematics Laboratory, where the research was conducted.