SIAM News Blog

Simulating the Mechanics of Honeybee Swarm Behavior

By Lina Sorg

In the spring and early summer, honeybee colonies expand quite rapidly in both population and physical size. This expansion crowds the existing hive and limits available storage and living space. As a result, colonies swarm — both as a natural means of reproduction and a solution to overcrowding. Swarming occurs when the queen leaves her colony to form a new one, taking up to 60 percent of worker bees with her. While the separated colony seeks out a new location, the swarm—sometimes comprised of tens of thousands of bees—temporarily stops on tree branches or other objects, forming a hanging pendant. 

While in this configuration, the bees are completely exposed and vulnerable to the surrounding environment. Therefore, they must individually respond to unexpected perturbations while accounting for static gravity and maintaining collective stability. The swarms themselves are dynamic, adapting to branch shape, shrinking to preserve warmth in response to ambient temperature change, moving flexibly in heavy wind, and forming a shingle-type structure for protection from the rain. However, the exact mechanics of the way in which bees maintain stability are still unknown. During a minisymposium presentation at the 2018 SIAM Conference on the Life Sciences, which took place last month in Minneapolis, Minn., Orit Peleg of the BioFrontier Institute at the University of Colorado Boulder used a combination of experiments and simulations to model mechanical swarm stability. “This is already complicated enough, but there’s a hard mechanical problem under all of this,” Peleg said. “Given thermoregulation, branch geometry, and gravity, how does the swarm ensure mechanical stability by using local information? We’re trying to understand the local rules that each bee is following, and how they use their local mechanical environment to maintain collective stability.”

She and team began with experiments involving real honeybees. Working with Harvard University’s Concord Field Station, Peleg created artificial swarms by caging a queen; because other bees could still sense the queen’s pheromone through the cage, they followed suit. “The idea was to perturb the system and conduct reverse-engineering to see what kind of rules the bees are following,” she said. Peleg situated the hanging swarm on a board controlled by a motor, which repetitively shook the swarm horizontally back and forth for a half hour. Upon speeding up the resulting video footage, she discovered that the bees moved around to collectively change the swarm’s shape in response to the perturbation. “They start with a column-like structure, then spread out and increase the height and base area,” Peleg said. “But when we stop the perturbation, they begin retracting slowly back to their steady-state shape.” 

Holding acceleration constant and changing the frequency yielded a similar result; while higher frequencies caused more drastic spread, the swarm continued returning to its steady-state when movement ceased. The same was true of heightened acceleration. When she plotted this data as a function of perturbation events rather than time, Peleg realized that all of the curves fell on a universal curve rather than reaching plateau values. “This was an indication that there is a kind of universal response and memory to each of these perturbation events,” she said. 

To cover their bases, Peleg and her team conducted the same experiment with a vertical up-and-down perturbation rather than horizontal shaking. Instead of spreading out, the swarm reacted minimally and broke apart upon reaching a certain level of aggression. This begged the following question: why do bees respond to horizontal but not vertical perturbations?

To investigate the mechanical properties of this phenomenon, Peleg created a model of a swarm structure’s normal modes, with 1,000 particles connected by linear springs representing bees. The springs connect the “bees” to their nearest neighbors, reminiscent of the small hairs on bees’ legs that create friction when attaching to each other in a swarm. The corresponding set of equations—which Peleg solved numerically—include spring potential and force (a small deformation). She found that the most probable motions for a perturbed swarm are a pendulum mode (that swings right to left) and a spring mode (that moves up and down), indicative of her earlier experiments. A gap between the graphs of the two modes reveals that when perturbed by a propogating wave, the swarms unsurprisingly move more easily and overtly in pendulum mode. This indicates that the bees are likely holding on to and remembering the local deformations that horizontally change their swarm’s shape.

Peleg then modeled local deformations as a function on time, focusing specifically on dynamics. Her model accounts for gravity and spring connections, and showed that local deformations for vertical perturbation are smaller than for horizontal perturbation. She also examined aspect ratio’s effect on local deformation when a swarm spreads in the horizontal direction. “We have a stopping condition for an agent-based model,” Peleg said. “They incorporate local deformations, and bees stop their response when those deformations are low enough.”

She concluded with a discussion of agent-based simulations. “This was more of a proof of concept that we can reproduce the experimental results,” she said. “We shake the swarm one period right or left or up and down, and integrate the local signals that each of these bees are experiencing for that period.” Unlike her previous simulations, arrows on the agent-based models point up the gradient and represent the addition of a local threshold. “We wanted to create a local rule that the bees might be following, but we weren’t entirely sure they would know which direction was up when in a congested swarm,” Peleg said. “Once they cross the threshold, they would start going up the gradient of the deformation.” This occurred more frequently with horizontal perturbation, because the bees were more likely to become agitated and react.

Ultimately, Peleg provided a concrete example of an active, functioning super-organism’s adaptive response to dynamic mechanical loading in order to strengthen collective capability. She demonstrated how swarms regulate their shape to become less susceptible to mechanical perturbations while reacting to frequency, force, and directionality. “What we think is going on is that they modulate their aspect ratio to minimize local deformations,” she said.

This work was recently published in Nature Physics. 

Acknowledgments: Orit Peleg performed this work as part of her postdoctoral studies at Harvard University in L. Mahadevan's group, together with graduate students Jacob Peters and Mary Salcedo.

 Lina Sorg is the associate editor of SIAM News.
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