## Inferring individual rules from collective behavior

###### Associated People:

Leah Keshet (University of British Columbia)

Ryan Lukeman (Saint Francis Xavier University)

###### Associated Sites:
PIMS University of British Columbia
###### Associated PIMS Programs:

IGTC in Mathematical Biology

Research can be challenging and arduous, but sometimes a discovery arises by serendipity. Such was the case with a project carried out by former IGTC PhD student Ryan Lukeman and his co-supervisors (LEK and Yue Xian Li). Ryan had been writing a PhD thesis on mathematical models for schools and flocks. He derived conditions under which such flocks form perfect structures, with equally-spaced individuals, all moving coherently. He had already found interesting results and had material ready for a thesis.

It was on one winter day in 2008 that his advisor happened to take a stroll along the boardwalk of Canada Place, a popular tourist destination in Vancouver BC. She noticed flocks of surf scoters in strangely regular formations, diving to forage along the water’s edge. She suggested that Ryan have a look. Using his own camera and tripod, and much patience and ingenuity, Ryan reinvented himself as a “field biologist”. He waited patiently for the birds day after day, recorded their movements, designed and calibrated the set-up, and then used Matlab to digitize and quantify the positions and velocities of each bird. By pooling together images for many hundreds of birds, he was able to produce density maps such as the one shown below. Quantifying the interactions in moving social animals is notoriously challenging. This project represents one of the first to track individuals within a flock numbering tens to hundreds of members.

The data revealed interesting structure: Each duck had a repulsion zone (in which neighbors were excluded), and an attraction zone, further out. Alignment was found to be strongest along the sides (not shown). But even more than this, each bird was seen to clearly prefer its nearest neighbor to be just in front or back.

Using the data and results of his analysis, Ryan was able to test a sequence of mathematical models finding one that was best able to describe what he saw. He found that mutual repulsion is the strongest social force, followed by attraction, alignment, and interaction with one frontal neighbor. The project has been published in the Proceedings of the Academy of Sciences USA
http://www.pnas.org/content/107/28/12576.full and made a cover story of the AMS Notices (April 2011). It was also featured in an editorial in Nature 466: 163.

Ryan Lukeman has since graduated with a PhD from the University of British Columbia and is now an assistant professor in mathematics at St Francis Xavier University in the Maritimes.