Compartmental Model Evaluates Generalized Stressors on Social Bee Colonies

By Lina Sorg

“Social bees” are part of the superfamily Apoidea and live together in colonies, working communally to maintain nests or hives. This classification includes bumblebees, stingless bees, and honeybees, the latter of which is one of the most economically valuable pollinators of crop monocultures in the world. According to the U.S. Department of Agriculture, honeybees contribute more than $15 billion annually to the country’s crop economy. Yet bees’ value extends well beyond agriculture and economics to ecology as well. Previous studies revealed that individual bumblebees tend to specialize on a specific species of flower, collect pollen from only that species, and largely deposit pollen on the same flower type — ultimately leading to higher seed quality yield. Therefore, declines in bee populations strongly influence plant reproduction and ecosystem function.

Such declines have been occurring globally for most of the present century. Roughly 24 percent of bumblebee species are currently threatened and 50 percent of Apidae species are either declining, vulnerable, or endangered. In addition, two-thirds of U.S. beekeepers lose 40 to 50 percent of their colonies each year. “Throughout the world, a nontrivial proportion of bee species are threatened in some way,” David Elzinga of the University of Tennessee, Knoxville said. 

During a minisymposium at the 2022 SIAM Conference on the Life Sciences, which took place earlier this month in Pittsburgh, Pa., in conjunction with the 2022 SIAM Annual Meeting, Elzinga used a traditional disease modeling framework to study the detrimental effects of environmental stressors on social bee colonies. One can think of these stressors as a matrix, in that they all interact with and influence each other (see Figure 1). “One practicality perspective is that there are limited floral resources,” Elzinga said. “Whatever stressors bees face, they can’t just choose to avoid them because there are only certain environments with which they can interact.” Unfortunately, fungicides and insecticides like pyrethroids and neonicotinoids negatively impact bees’ environments. These influencers can cause direct harm, impair bee function, and even result in death. The added presence of parasites and pathogens compound chemical stressors and produce the same damaging results.

Given the ongoing threats to bee populations, many researchers have tried to model the offending stressors over the last decade. Elzinga referenced a 2011 paper by David Khoury, Mary Myerscough, and Andrew Barron that examined social inhibition as an inherent structural property [3]. The authors stratified the population based on a bee’s status as either a hive bee or forager bee. They then analyzed the resulting emergent structural properties, including precocious foraging (wherein “underage” bees are required to forage), labor destabilization, and ultimately colony collapse disorder.

Two years later, a 2013 paper stratified the bee population according to impairment status—susceptible versus impaired bees—rather than caste [2]. Susceptible bees become impaired based on exposure within their environments. In this model, sublethal stress was the inherent structural property and colony collapse disorder was the resulting emergent property. “While the stress can directly kill [the bees], it also impacts their ability to function,” Elzinga said.

A 2017 study attempted to marry the themes from the aforementioned two papers by once again stratifying by caste and dividing the population into hive bees and forager bees [1]. Here, the inherent structural properties—social inhibition and sublethal stressors—yielded two emergent properties: (i) bistability of extinction and persistence and (ii) colony collapse disorder. 

Because these models were all developed in quick succession, Elzinga posed the following two questions to drive his own work:

1. Can the mechanisms and emergent behavior of these concurrently developed models coexist under a single model framework?
2. Can we think of a stressor more broadly? Do various attributes of stressors matter? How much?

In response to these queries, he set out to develop an analytically tractable, mechanistic model of stressors in social bee colonies. Elzinga wanted to be able to validate his model through biological emergent properties and also generalize the attributes/meanings of stressors. He shared the resulting compartmental generalized stressor model with the audience (see Figure 2). “This is our attempt to stratify not only based on caste but also based on impairment,” he said. The model accounts for hive bees \((H)\), unimpaired forager bees \((F_U)\), and impaired forager bees \((F_I)\), and social inhibition allows hive bees to transition to unimpaired forager bees. Unimpaired forager bees can of course become impaired, and impaired forager bees that recover can regain their unimpaired status. Depending on the stressor in question, the populations of forager bees can die at different rates. Finally, Elzinga assumes that hive mortality is negligible. “The really cool thing is that you can do a lot of analytics on this model,” he said. 

Elzinga analyzed the model’s three equilibria—\(\textrm{E}_1\), \(\textrm{E}_2\), and \(\textrm{E}_3\)—and noted the occurrence of class balancing. \(\textrm{E}_1\) is an extinction equilibrium and \(\textrm{E}_2\) and \(\textrm{E}_3\) are persistence equilibria. “The extinction equilibrium is always an asymptotically stable node,” Elzinga said. “And the persistence equilibria are biologically feasible if and only if two conditions hold.” One condition pertains to growth rate, in that the growth rate must be larger than an established lower bound. “It’s basically saying that you have to achieve some rate just to be able to survive, regardless of what the stressors are like in your environment,” Elzinga continued. “If you can’t reproduce fast enough, you’re going to die off regardless of the stressors you face.” The second condition is a critical stress threshold, meaning that the infection rate must be sufficiently small to prevent the population from going extinct.

One can also investigate stressors and their boundaries by considering mortality, in addition to the infection and impairment rates. Elzinga demonstrated that a high infection rate combined with high mortality generates extinction in all cases. However, low infection and mortality rates—or even just a “low enough” infection rate—can enable persistence in all cases. “Where you separate persistence to extinction with a stressor depends on the amount of sublethal stress,” he said. “In this case, that’s the proportion of impaired bees that can function.” When no bees can function \((p=0)\), the persistence regime is quite small. But when \(p=1\), a much bigger range of values lead to persistence.

Finally, Elzinga considered the application of stressors on an annual basis over the course of a 180-day growing season (see Figure 3). “If you apply stressors for the entire duration—starting at 0 days and ending at 180 days—you get a much smaller five-year population than if you apply the stressors for a shorter period of time,” he said. “But you can also see asymmetrical effects if you apply [the stressor] for 30 days at the beginning, 30 days at the end, or move it around. So this is kind of just a quick ability of our model to start indicating stressor time. It’s something that we really think is a future direction.”

In summation, Elzinga incorporated four inherent structural properties—social inhibition, sublethal stressor, caste and impairment stratification, and generalized stressors—in an effort to yield all of the emergent properties from the previous models [1-3]: bistability of extinction and persistence, precocious foraging, labor destabilization, and colony collapse disorder. He found that particular stressors that cross the critical stress threshold typically cause precocious foraging, labor destabilization, and ultimately colony abandonment or collapse. “This is all indicating that stressor characteristics are vital,” Elzinga said.

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References
[1] Booton, R.D., Iwasa, Y., Marshall, J.A.R., & Childs, D.Z. (2017). Stress-mediated Allee effects can cause the sudden collapse of honey bee colonies. J. Theor. Biol., 420, 213-219.
[2] Bryden, J., Gill, R.J., Mitton, R.A.A., Raine, N.E., & Jansen, V.A.A. (2013). Chronic sublethal stress causes bee colony failure. Ecol. Lett., 16(12), 1463-1469.
[3] Khoury, D.S., Myerscough, M.R., & Barron, A.B. (2011). A quantitative model of honey bee colony population dynamics. PLoS One, 6(4), e18491.

Lina Sorg is the managing editor of SIAM News.