Collision frequency of birds, insects, bats and whales is predicted by body mass and wing area alone

A single universal equation can approximate the frequency of wing beats and string strokes made by birds, insects, bats and whales, despite their different body sizes and wing shapes, report Jens Højgaard Jensen and colleagues from Roskilde University in Denmark in a new study published in PLOS ONE on June 5.

The ability to fly has evolved independently in many different groups of animals. To minimize the energy required to fly, biologists expect that the frequency at which animals flap their wings should be determined by the wing’s natural resonance frequency. However, finding a universal mathematical description of bumpy flight has been difficult.

The researchers used dimensional analysis to calculate an equation that describes the frequency of the wing beats of flying birds, insects and bats, and the fin beats of diving animals, including penguins and whales.

They found that flying and diving animals beat their wings or feathers at a frequency that is proportional to the square root of their body mass divided by the area of ​​their wings. They tested the accuracy of the equation by plotting its predictions against published data on wingbeat frequencies for bees, moths, dragonflies, beetles, mosquitoes, bats and birds ranging in size from hummingbirds to swans.

The researchers also compared the equation’s predictions against published data on stranding frequencies for penguins and several species of whales, including humpbacks and northern bottlenose whales.

The relationship between body mass, wing area and wing beat frequency shows little difference between flying and diving animals, despite large differences in their body size, wing shape and evolutionary history, they found.

Finally, they estimated that an extinct pterosaur (Quetzalcoatlus nothropi)—the largest known flying animal—beat its wings 10 square meters at a frequency of 0.7 hertz.

The study shows that despite vast physical differences, animals as distinct as butterflies and bats have evolved a relatively constant relationship between body mass, wing area and wingbeat frequency.

The researchers note that they did not find publications with all the required information for swimming animals; data from different publications were pooled together to make comparisons, and in some cases animal densities were estimated based on other information.

Furthermore, extremely small animals—smaller than any yet discovered—would likely not fit the equation because the physics of fluid dynamics changes on such a small scale. This could have future implications for flying nanobots.

The authors say the equation is the simplest mathematical explanation that accurately describes wing beats and string kicks in the animal kingdom.

The authors add, “By varying by almost a factor of 10,000 in wing/fin beat frequency, data for 414 animals from blue whales to mosquitoes fall in line. As physicists, we were surprised to see how well our prediction was The simple wing-beating formula works for such a diverse collection of animals.”