Long-tailed tits are what most internet users would affectionately label “birbs.” They’re tiny, round, and cute, which fits the Audobon Society’s rigorous scientific standards for birbdom, whose existence may be the best thing about 2020.  Looking at these little round birds, most people don’t think of differential equations.

Mathematician Natasha Ellison and her colleagues at the University of Sheffield in the UK, however, recently used a set of equations called advection-diffusion equations to explain how flocks of long-tailed tits decide where to live, based on where and how they’ve encountered other flocks. It worked thanks in part to a mathematical theory developed by cryptanalyst and mathematician Alan Turing – best known for cracking the Nazis’ Enigma codes during World War II.

“In my master’s course, I studied Turing’s famous 1952 paper for my dissertation in detail and became very interested in mathematical biology,” said Ellison in an email.

In that paper, Turing used equations called reaction-diffusion equations to describe the behavior of two chemicals, which spread across a surface and reached with each other any time they met. That model helped explain how color patterns like spots and stripes form in some animals.

“If you could have placed a bet, before the Second World War, on who might have asked the question that Turing posed, you would not have expected it to come from this mathematician,” wrote biologist Phillip Ball in the Proceedings of the Royal SocietB in 2015.

Reaction-diffusion equations were around long before Turing’s day. The difference was how he used them. “The important part of his work was that he developed a mathematical theory to give the conditions under which the patterns will form,” explained Ellison in her email. “Our research uses Turing’s pattern formation theory to understand when behaviors (and their associated parameters) will produce patterns.”

After her master’s course, while teaching mathematics at a high school in Sheffield, Ellison learned about the work of Jonathan Potts, a University of Sheffield mathematician who uses math to answer questions about ecology. When Ellison learned that Potts was about to launch a project with the aptly named University of Sheffield ornithologist Ben Hatchwell, she joined the university as a PhD student to get in on the research.

In the Rivelin Valley near Sheffield, ecologists and ornithologists had noticed that each flock of long-tailed tits tends to settle into its own patch of the available habitat. At face value, that doesn’t make much sense; in most parts of the Rivelin Valley, there’s enough food to support several flocks at a time, and long-tailed tits don’t usually bother defending territory against other members of their own species.

It seemed as if the birds should all flock to the best feeding grounds in the valley, but instead they split up, and each flock occupied its own patch of land. There was a pattern in how the flocks were sorting themselves out across the landscape, but the ecologists couldn’t see what it was without crunching the numbers.

“Without the help of these mathematical models, these behaviors wouldn’t have been discovered,” said Ellison in a press release.

Advection-diffusion equations describe how animals move around in response to their environments. For the long-tailed tits, the most important factor in their environment seemed to be memories – in particular, memories of where they had encountered other flocks and what those interactions were like.

Ellison and her colleagues came up with the idea after field ecologists Sarah Biddiscombe and Clare Napper spent several months trekking through Sheffield on foot, listening for bird calls and peering through binoculars. “We hypothesized that flocks may be avoiding each other after interacting (via sight or sound),” said Ellison in her email.

With the advection-diffusion equations, Ellison and her colleagues described how likely flocks were to avoid or occupy an area after meeting another flock there. It turned out that long-tailed tits were most likely to avoid places where they interacted with larger flocks or flocks made up of birds they weren’t related to. Flocks of relatives seemed to be more tolerable, but still created a slight tendency to avoid the area in the future.

Once Ellison and her colleagues factored in the tits’ preference for areas near the center of woodlands, “we found a home range model that fit the data well and was built up from underlying behaviors,” she said in the email. They published their results in the Journal of Animal Ecology.

It’s not the first time Turing’s pattern formation theory has helped explain how animals move around in space; previous studies have helped predict interactions between predators and prey, or the settlement patterns of very territorial animals like coyotes, meerkats, and even people. What Ellison and her colleagues did for the first time, however, was model the behavior of an animal that could technically go anywhere it wanted in the available habitat – an animal that wasn’t territorial and didn’t have a den to anchor it to a particular spot and limit its range.

“Mathematical models help us understand nature in an extraordinary amount of ways, and our study is a fantastic example of this,” said Ellison in a press release.