By Lina Sorg
Honeybees are social insects. They live in colonies comprised of thousands of individuals and engage in group decision-making when foraging for food or searching for new places to settle. Bees regularly make decisions about swarming patterns, future colonies, foraging locations, reproduction, food storage, and overall life in the hive. This collective decision-making begins on the individual level. When making shared decisions, certain bees leave the hive to glean local information from both their neighbors and the surrounding environment. The resulting global perspective aids the decision-making process. While most existing studies observe collective decision-making in static environments, nearly all natural environments are dynamic. As a result, bees must process new evidence and alter their decisions as their environment changes around them — making mathematical modeling decidedly more complicated.
During a minisymposium presentation at the 2018 SIAM Conference on the Life Sciences, currently taking place in Minneapolis, Minn., Subekshya Bidari of the University of Colorado Boulder examined decision-making in honeybees foraging in a dynamic environment. Flowering plants only bloom for a limited amount of time, so foraging bees must strategically account for seasonal variations, physical obstacles, and potential predators. Bidari identified four different interaction mechanisms that occur during foraging. The first is commitment. “Once they come upon a food source, they can decide to commit to that source,” Bidari said. The quality of an individual bee’s choice correlates positively with the rapidity at which it commits to its decision; a better-quality source yields higher commitment.
The second mechanism is forgetting, or spontaneous abandonment. Sometimes bees abandon their existing opinions and begin accumulating completely new data. “This lets them reevaluate their commitment at any point and helps them make better decisions in a changing environment,” Bidari said. Recruitment is yet another mechanism. When an individual bee finds and commits to a prospective food source, it returns to the hive to convince other bees of its discovery via a waggle dance — a figure-eight pattern of movement that indicates positive feedback. The duration of the waggle indicates the food’s distance from the hive, and the angle points other bees in the right direction. In opposition to the waggle dance, stop signals—which convey negative feedback—serve as the final foraging mechanism. When dedicated to opposing commitments, bees head-butt each other in an attempt to inspire a change of opinion.
Bidari then posed the following research questions:
To begin addressing these questions, she summarized an experiment by Boris Granovskiy’s research group proving that bees are optimal foragers. Granovskiy presented the insects with two feeders, one of which had more sucralose solution and was thus a better food source. He switched the source qualities every 105 minutes, but the bees continued to find the more desirable feeder. “At any point, there was the highest proportion of bees feeding at the best quality feeder,” Bidari said. Granovskiy’s team also conducted a control experiment that eliminated bees’ foraging-based communication during the waggle dance. The bees were able to identify the superior feeder even without a functional dance.
Bidari borrowed her foraging model—which illustrates the interaction of bee populations with the four aforementioned mechanisms—from an existing Science paper. Rather than holding all rates constant, she changed alpha to a piecewise constant function; such a function is easy to analyze and consistent with Granovskiy’s experiments. A constant alpha yields two steady-state solutions, and an alpha level greater than zero produces one positive and one negative steady-state solution. When alpha is equal to zero and no food supply is available, foraging dynamics are dependent on forgetting and recruitment. If recruitment is stronger than forgetting, bees still go out and forage despite the obvious lack of accessible food. Solving the corresponding equation generates a periodic function of sorts.
Next, Bidari examined foraging yield and created a reward rate function that accounts for the cost of foraging, such as risk of predation. “There are parameter values for alpha and beta that maximize the foraging yield,” she said. “We’re not really interested in numbers, we’re interested in the fact that reward rate is maximized when the value of alpha and beta are roughly equal.” Larger colonies also maximize the reward rate because more foraging bees increases the chance of finding an optimal food source.
Upon conducting phase plane analysis, Bidari found that switching the optimal food source in two steady-state solutions prevented the bees from following the new ideal. The bistability of the system and subsequent unstable manifold are responsible for this blockage. However, bees are able to follow alternating optimal food sources in systems with monostability.
Bidari was also particularly interested in the speed at which bees must abandon their commitments based on the rate of environmental change. Rapidly-changing environments demand high levels of forgetting, while less dynamic settings tolerate lower levels. Because bee colonies are of finite size, finite size becomes the model’s source of stochasticity.
Ultimately, Bidari’s collective decision-making model reveals valuable truths about the role of forgetting in optimal foraging. When individual bees abandon a previous opinion at a rate that correlates with dynamic environmental change, colonies can more effectively move to increasingly lucrative foraging sites.