Buzzing Through the Numbers 

Tzu-Yu Hung, Alysha Kutuzyan, Katarina Simanic, Steven Yang 

Abstract 

Exploring the mathematical principles underlying the behaviors of bees reveals a world of complexity and efficiency. By studying honeycomb optimization, reproductive patterns and everyday division of labor, we uncover the adaptive strategies bees use for success and survival. Bees construct their hives to maximize storage space while minimizing the use of wax, following patterns that align with mathematical principles. The haplodiploid reproductive system, which mirrors the Fibonacci sequence, showcases and provides fascinating insight on how different mathematical patterns are integrated into the colonies. Bee’s temporal polyethism, which involves the division of labor based on age, shows how bees optimize their productivity through efficient role allocation.These behaviors offer valuable insights into the conditions necessary for a colony’s success, as well as the risks—such as fluctuating food supplies and elevated forager mortality—that can contribute to its collapseUsing mathematical models and simulations, we can observe the balance within a hive and understand how instability can arise. Bees exemplify how mathematical concepts are integrated into an organism’s most fundamental processes and collective behavior. Studying these systems highlights the brilliance behind their behaviour and allows us to gain thorough insight into one of nature’s efficient and complex superorganisms.   

Introduction

The world of bees is renowned for its intricate integration of mathematical concepts into many aspects of their lives, from daily behavior and social structure to hive architecture. Each component of the superorganism operates on underlying mathematical principles, enabling the efficient coordination of thriving, large colonies. These principles go beyond basic mathematics and involve ideas such as the Fibonacci sequence, hexagonal tiling, and complex differential equations.   

One of the most fascinating examples is the Honeycomb Conjecture, which demonstrates how the hexagonal honeycomb structure is built to optimize the physical and biological needs of the colony. This design allows for efficient food storage and safe brood development while minimizing the use of wax (Hales, 2001). Bees can adapt the construction of the combs to suit their surrounding environments.  

Bee-yond hive construction, bees exhibit an intriguing mathematical reproductive pattern. Like other eusocial species in the order Hymenoptera, bees live in large colonies and based on different reproductive roles, each bee colony consists of drones, a queen and workers. In the family tree of male bees (drones), the number of ancestors in successive generations follows the Fibonacci sequence (Hartono & Pham, 2024). This pattern reflects the haplodiploid reproductive system of bees, where male drones have one parent while workers and queens have two.   

Even the everyday activities of bees follow underlying mathematical frameworks. For instance, worker bees divide labor based on age to improve the colony’s productivity and those that specialize in a particular labor division are more likely to stay there (Schmickl & Crailsheim, 2008). Equations can model these behaviors and show how environmental conditions can disrupt the system. Furthermore, mathematical simulations can be used to predict colony collapse disorder, which occurs when a colony experiences a quick population decline (Myerscough et al. 2017).  

By examining the mathematics behind the lifestyle of bees, we can see the incredible strategies they use to increase efficiency and survival. These processes give us a closer understanding of the optimization of a colony and showcase the complexity behind their behaviour. 

Hexagonal Harmony

Hexagons, the Best-agons

To maintain the superorganism, bees need to efficiently store their large amounts of provisions (in the form of honey and bee bread) in the hive. They also need safe places for their larvae to develop into workers and queens for the future of the colony. For these reasons, bees secrete beeswax and mold it into a host of tiny jars (big enough for the bees to fit into), where each cell provides a suitable environment to either store food or raise brood. But the production of beeswax is costly: bees consume about 20 g of honey to create a mere 1 g of wax.

It is due to these circumstances that famed philosophers and scientists of the past (including Darwin) have justified the bee-autiful hexagonal tessellation withinthe hive (Dutta, 2021; Hales, 2001). Indeed, the mathematical Honeycomb Conjecture stems from the idea that the hexagonal honeycomb tiling is an optimal partitioning of a plane that maximizes the stored area within the hexagons while minimizing the perimeter of the shapes. In bee terms, forming beeswax cells into hexagons maximizes the amount of food that can be stored with minimal use of costly beeswax (Hales, 2001). There exist mathematical proofs of the conjecture, here is one of them: 

Let C represent the collection of infinite planar clusters defined by a 1-dimensional set S and a selection of infinitely many, not necessarily connected, disjoint, unit-area components of  . For any closed ball B, define a perimeter-to-area ratio.  

Then the hexagonal honeycomb H is (up to translation and sets of measure 0) the unique C∈∁ which for each B minimizes 𝜌(C. B) in competition with all C’ ∈∁  which agree with C outside B. Moreover, H minimizes →∞𝜌(C, 𝐁(0, r))  in competition with all C’∈∁ (Morgan, 1999). 

To begin a more intuitive explanation, the optimal shape to enclose the largest single area with minimum perimeter is the circle. Unfortunately, multiple circles do not pack efficiently, leaving small triagonal gaps between them (Fig. 1). There are only three equilateral geometric shapes that fill a two-dimensional space without leaving any gaps: equilateral triangles, squares, or hexagons (Fig. 2). The area (S) of a regular hexagon having side length is  ; for a triangle, it is ; for a square of the same side length, it is . The corresponding perimeters (P) of each shape are 6a, 3a, and 4a, respectively. Their perimeter-to-area ratios are therefore

  for the regular hexagon, triangle and square, respectively. From these equations, we can see that hexagons have the smallest perimeter to area ratio, so they are the optimal solution for the storage problem of bees (Dutta, 2021).  

Fig. 1 Example of packing of circles on a plane. There is wasted space in between the circles (Dutta, 2021). 

Fig. 2 Tiling of a plane space with (a) squares, (b) equilateral triangles, and (c) regular hexagons. No other equilateral shape can fill the space without leaving gaps (Dutta, 2021)

Bees are not 2D creatures

The Honeycomb Conjecture may work in 2D, but it does not necessarily apply to the 3D honeycomb structures that it was inspired from. Indeed, only the openings of honeycombs (or their prismatic base) show a hexagonal pattern. Their actual geometry consists of two layers of congruent cells, each with a hexagonal opening and a non-flat bottom formed by three rhombi (Fig. 3) (Räz, 2017). The question arises as to whether the bee honeycomb is an optimal solution in a 3D world to minimize wax surface area for a given cell volume that can be filled full of honey. To replicate the hive conditions of honeycomb sheets (and make sure bees can access the cells), any better mathematical solution should have the limitations that the cells should be arranged between two parallel planes, congruent, fill the space between the planes without overlap, and have an opening in exactly one of the two planes. Fejes Tóth designed a comb structure superior to the bees’ design in this regard (Tóth, 1964). Fejes Tóth’s structure differs from the bees’ actual combs in the way that the cell bottoms in his structure are made of two hexagons and two rhombi each (Fig. 3). The calculated saving in surface area of his structure is less than 0.35% the area of a cell opening, which is quite marginal. In addition, real cells are not perfectly regular and the gain from Tóth’s structure might not be worth the construction effort, so the optimization conditions behind the comb structure of bees may simply not be well understood. Interestingly, Weaire and Phelan reproduced honeycomb-like structures with soap bubbles of a liquid solution enclosed between two glass plates in such a way that the bubbles formed a double array of cells with hexagonal openings (Weaire & Phelan, 1994). The mathematics of honeycomb formation can be approximated with the formation of soap bubbles since both rely on similar optimization processes: the amount of wax in honeycombs is minimized for an evolutionary edge while soap bubbles minimize surface area to attain a lower energy level. The main finding in Weaire and Phelan’s experiment is that the bubbles formed by a low amount of soap liquid correspond to Tóth’s structure, but when adding more liquid in a way that thickens the bubble walls, the Tóth structure becomes unstable and suddenly switches to the natural bee honeycomb array (Fig. 4). To conclude, the Tóth structure is optimized for thin walls while the actual honeycomb formation is optimized for thicker walls (Räz, 2017). 

Fig. 3 (a) Fejes Tóth comb structure. (b) Actual honeycomb structure. Figures below the structures show the pattern overlap between two comb layers of each structure (Weaire & Phelan, 1994). 

Fig. 4 Soap bubble structures in Weaire and Phelan’s experiment. (a) Interface of Tóth structure in soap viewed at a slight angle. (b) View of the surface of the soap bubbles (as a Tóth structure) before and (c), after transition to the actual honeycomb structure (Weaire & Phelan, 1994)

Tiny Architects, Not Automatons

While the honeycomb is often praised for its efficient use of space, the architectural genius of bees reveals itself in many other equally astonishing feats. It was previously mentioned that bees do not create perfectly symmetrical hexagonal cells. A major reason for this seeming imperfection is that they adapt their comb architecture in response to environmental stimuli (Bader et al., 2022). Workers start building comb independently and simultaneously in multiple locations of the future hive (whether it is a tree crevice or an apiculturist’s Langstroth hive). Eventually, the independent pieces grow, and the workers merge them (imperfectly, but as best as they can) into a single comb (Bader et al., 2022; Smith et al., 2021). Because natural nests have irregular shapes, their comb construction must conform to the space and surfaces available (Smith et al. 2021). Typically, combs are constructed as double-sided sheets of cells that are parallel to each other to allow the cells to approach their theoretical maximum packing density while still allowing efficient bee navigation between the comb sheets (Bader et al., 2022). Since the crevices wherein the combs are constructed may not always allow for perfectly parallel sheets, honeycomb sheets can curve following the contour of the hive walls to roughly conserve parallelism (Fig. 5). Also, honeybees tend to establish a build direction in line with the force of gravity, with individual comb cells being built at an average upward angle of 13 degree from an interface where the basal side of two cells meet (Fig. 5) which may help better retain honey within the cells and/or provide additional structural reinforcement. Finally, cell orientation typically depends on the orientation of the substrate—the initial attachment point of the comb—with one vertex of each cell (i.e., a corner of the hexagonal shape) directed toward the substrate.  Since comb construction follows general consistent guidelines, computational geometry tools can be used to characterize comb-building behavior. By applying Gaussian curvature analysis, geodesic distance measuring, Laplacian and frequency analysis, as well as morphological and Boolean operations, researchers can infer the build sequence and variations in material properties throughout the comb. These inferences are drawn from micro-CT scans of hives that were periodically rotated to introduce construction irregularities.. The direction of gravity during the building of a particular comb section can be deduced from the tilt of the cells. Using the geodesic distance field from the initial substrate attachment point and the gravity vector of individual cells, moving back along the geodesic build direction and evaluating gravity direction at each comb branch intersection enables the reverse tracing of the comb build order (Fig. 6) (Bader et al., 2022). 

Fig. 5 General characteristics of A. mellifera comb. (a) feral comb containing multiple quasi-parallel sheets. (d) cross-section of a comb sheet containing 2 cell layers, with the cell tilt of 13 degrees to counteract gravity being shown (Bader et al., 2022). 

Fig. 6 Simulation of comb construction predicated from geometric measurements. “An example prediction of a comb’s construction and morphological evolution over time, using geometric data gathered by determining (a) the medial axis, (b) the geodesic field from the substrate, and (c) the gravity vector of each cell.” For each time-point, the next constructed comb section is depicted with reduced opacity (Bader et al., 2022). 

A Royal Headache

Another reason behind the imperfection of honeycombs stems from the fact that not all larvae can fit in a standard sized worker cell. Royalty (future drone bees and queen bees) need larger cells to properly develop. In particular, the merging of drone cells next to smaller worker cells creates new efficiency problems for the comb architecture, preventing worker bees from building perfect hexagons (Fig. 7) (Smith et al., 2021). Smith et al. have shown that bees use a variety of techniques to solve this issue. First, bees cluster different cell types with others of the same type, minimizing the contact length between cells of different sizes which would distort the hexagonal lattice. Generally, bees build worker cells before building drone cells. Worker bees build intermediate-sized worker cells to transition between cell types, approaching the transition towards the drone cells with gradually increased cell wall lengths. Although some of the irregularly sized cells remain viable for brood rearing or food storage, tiny cells are generally filled with wax and drone cells have an upper size limit. When bridging between two drone comb cells, the bridging drone cells are already at the upper size limit of viability for drone rearing, so bees build smaller cells which can still be used for worker rearing despite being in a drone rearing area (Fig. 7). The above demonstrates that, even when forced to build suboptimal cells, bees make the most use of the available space. Also, it is more difficult to merge cells with different tilt directions since their walls and vertices will collide. It was found that the proportion of non-hexagonal cells increased at higher tilt differences between comb cells. To minimize their building hardship, in addition to gradually pairing the different cell sizes, workers gradually change the cell tilt to match both comb cell tilts together. All the adjustments that workers make for comb transitions and merging occur over an area 1 to 2 cells wide, since the workers build in a dark hive and likely use their sense of touch to understand local comb architecture. Continuing with irregular forms, 4.4% of cells in the comb transition interface observed by Smith et al. had non-hexagonal shapes. The cells ranged from 4 to 9 sides, but these cells were found in motifs. Bees prefer using 5- and 7-sided cells paired together (Fig. 8) (Smith et al., 2021). In addition to being a convergent solution between multiple hive-building species of insects, this pairing of pentagons with heptagons is also found in graphene grains at the merge line between two sheets of hexagonal carbon lattice, which increases the material strength of graphene. The geometric parallel between the two worlds is interesting for an optimization perspective (Smith et al., 2021; Smith et al., 2023). 

Fig. 7 An example of naturally built honeycomb, with arrows showing the five locations where workers had simultaneously initiated building; full sample sizes listed below (A). Bees build two types of combs: small-celled worker comb (B, i) and large-celled drone comb (B, ii). Histograms of cell areas (B, iii) and wall lengths (B, iv) show that the two cell types form distinct distributions (worker cells, red; drone cells, blue). Bees face two key challenges during comb building: transitioning from worker to drone-sized cells in a continuous piece of comb (C) and merging independent pieces of comb (D–F) (boxes in A; Insets in C–F, i). Histogram of cell areas across the transition/merge (gray in C–F, i) as compared to perfect worker and drone cells (red and blue, respectively; random subsets from B, iii). Bees adjust cell areas (C–F, ii) and wall lengths (C–F, iii) as they build across the transition/merge (vertical dotted line in C–F, i–iii). The black line and shading in C–F, ii and iii show mean ± SD and the data point colors denote number of walls in C–F, ii” (Smith et al., 2021). 

Fig. 8 Observed transition region between worker cells (blue) and drone cells (red) of A. florea, where cells of unorthodox sizes are represented in a lighter color. Pentagonal and hexagonal cells are colored brown (adapted from Smith et al., 2023). 

Researchers built a model to understand how non-hexagonal cells would arise in an optimal transition patch between worker and drone comb cells. Based on studies of A. mellifera, the transition patch can be adequately modeled by a Voronoi partition of polygons, since the number of walls for each cell depends on the position of optimally placed cell centers. From a model of optimal placement of cell centers, the Voronoi partition decomposes the plane into Voronoi polygons (all the points inside a polygon are closest to the cell center that is inside the partition and all the edges are equidistant from two cell centers) (Fig. 9). The graph generated by connecting cell centers whose Voronoi partition share an edge is a Delaunay triangulation, a graph where any circumscribing circle of a triangle contains no other vertices (Fig. 10). The Delaunay triangulation makes the connection topology of cell centers easier to analyze since one Voronoi edge perpendicularly bisects each vertex of the Delaunay triangle. Also, bounds can be placed that enforce the triangulation. Analyzing cell topology according to the Delaunay condition permits the computation of non-hexagonal cells along the transition line. Assuming cell centers on either side of the transition are aligned with each other according to the Delaunay triangulation, because of the size difference between the cell types, regular connection patterns between the two sides are impossible. By counting worker cells along the transition line and pairing two worker cells with one drone cell to form a Delaunay triangle, at some point along the transition line, the size mismatch between the worker and drone combs misaligns the cell centers between the two sides in such a way that the circumscribing circle of a trio of two worker cells and  one drone cell will envelop the drone cell of the previous trio, thereby violating the Delaunay rule. The two worker cells therefore pair solely with the drone center of the previous trio, which breaks the triangle zigzag pattern. Because of this dislocation from the regular connection pattern, a worker cell loses an edge to become 5-sided, and a reproductive cell gains an edge to become 7-sided, which creates a 5-7 pair as observed in real comb formations (Fig. 11) (Smith et al., 2023). 

Fig. 9 Voronoi diagram of 11 points in a cyclic plane. Voronoi patterns can be found in nature, like on the skin of giraffes (Aurenhammer & Klein, 2000). 

Fig. 10 A Delaunay triangulation in a plane with circumcircles shown. A Delaunay triangulation maximizes the minimum angle among all triangle angles, thereby reducing the occurrence of sliver triangles (Wikipedia contributors, 2024).  

Fig. 11 Modeling of bee cell arrangement through a Delaunay connection pattern for transition between worker (blue) and drone (red) cells, where the center of each drone cell is offset by

∆,∆, the mismatch between worker and drone cells. (a) Lateral displacement of cell centers increasing left to right. The left gray disk shows the circumscribing circle for a triplet of cell centers that would be connected. The right disk violates the Delaunay condition, since it contains four, not three, cell centers. (b) The lattice irregularity introduces a 5-7 cell pair in the Voronoi partition (Smith et al., 2023). 

Fibonacci Sequence in Bee Reproduction 

Reproductive Roles

Drones are the male bees that mainly hatch from haploid unfertilized eggs, carrying only one set of chromosomes (Zhao et al., 2021). They are assigned with exclusive reproductive role to mate with queens from other colonies during nuptial flights (Asaah, 2023), and they are incapable of engaging in foraging behavior due to lack of food collection body parts, such as honey stomach and pollen basket. In turn, drones are produced only when needed, which is usually during the reproductive season, when a sufficient population of attending workers and food resources are available (Zhao et al., 2021). They often die after mating as they have fulfilled their reproductive role (Asaah, 2023). Queens are the primary reproductive female bees developed from diploid fertilized eggs, carrying two sets of chromosomes. In every hive, there is one healthy fertilized queen who lays all the eggs, ensuring the robustness of the colony’s population. A virgin queen undergoes one brief mating period in lifetime, and this happens throughout the first 1-2 weeks of her adult life, during which she exits the hive and goes on multiple mating flights with drones. The mated queen collects and stores the sperms in the spermatheca, which are then used throughout her lifetime to fertilize eggs (Zhao et al., 2021). If an egg is not fertilized, it eventually hatches into a drone, whereas diploid fertilized egg produces a queen or a worker, depending on the specific conditions and nutrition provided [hyperlink to Bee Chemistry Essay – Jelly for brood section]. This feedback loop ensures that if enough sperms are provided by drones to fertilize the queen, then less unfertilized eggs are laid and less drones are hatched, and vice versa. Unlike the queen, workers are sterile female bees, and they are responsible for foraging and hive maintenance tasks, including caring for larvae and defending the hive (Asaah, 2023).  

Family Tree

The differentiation between haploid drones and diploid female bees (queens and workers) is fundamental in the unique bee reproduction pattern. Fig. 12 illustrates the family tree of a drone in a simple ancestry model. Starting with a single drone in the 1^st generation, the haploid male only has one parent, usually the queen, represented by the female symbol in the 2^nd generation (Hartono & Pham, 2024). Though workers do not lay eggs under normal circumstances, under conditions which the queen is absent in the colony or which the queen has compromised egg-laying capacity, they can produce drones, ensuring continual production of males in the colony (Asaah, 2023). On the other hand, diploid queens and workers both have two parents (a queen and a drone) as they hatch from fertilized eggs, and thus, two bees are illustrated in the 3^rd generation. Similarly, the male in the 3^rd generation has only one parent while the female has two, and so three bees in the 4^th generation and the pattern continues in the same way. 

Interestingly, the family tree of a drone exhibits a fascinating relationship with the Fibonacci sequence. The Fibonacci sequence is a series of numbers, where each number is the sum of the two preceding ones. The sequence begins with 0 and 1, progresses as 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …, and extends indefinitely. As explained, the total number of bee ancestors in each successive generation follows the sequence 1, 1, 2, 3, 5, 8, 13, … and this parallels the Fibonacci numbers (Hartono & Pham, 2024). Let B_n be the total number of bees in n^th generation, F_n be that of females and M_n be that of males. Every bee has one female parent, so the total number of bees in the (n-1)^th generation is equal to the number of female bees in their parents’ generation, which is the n^th generation (eqn. 1). 

(1)  

While every female bee has one male parent, so the total number of females in the (n-1)^th generation is equal to the number of males in the n^th generation and is also equal to the total number of bees in the (n-2)^th generation (eqn. 2). 

(2)  

Since the total number of bees in any generation is equal to the sum of males and females (eqn. 3), substitution of (1) and (2) into (3) will generate equation 4, which is the recursive formula for the Fibonacci sequence. Notice that F_n and M_n also follows the Fibonacci numbers (Gross, 2019). 

(3)  

(4)  

Fig. 12 Family tree of a drone in a simple ancestry model. Mars symbolizes a male, while Venus symbolizes a female (Hartono & Pham, 2024). 

Moreover, computation of the ratio B_n:B_n-1 will give the factor by which the number of ancestors increases in every generation. For bees, it can be observed that for sufficiently large n, the ratio approaches 1.618034 (Table 1). This ratio can be symbolized by the Greek letter  (phi) and is known as the golden ratio (Gross, 2019). 

Table 1 Computation of the ratio between the total number of bees in n^th generation and that in (n-1)^th generation. 

Biological Mechanisms

The Fibonacci sequence is a great representation of the unique bee reproduction pattern, and this demonstrates a clear mathematical sequence emerging from biological processes. There are three key mechanisms employed by the bee colony that contribute to the pattern. Although, as mentioned, the specialized mating role and haploid status of drones limit their contribution to the daily hive activities, their mating behavior is essential for colony propagation and genetic diversity and impacts population dynamics of the subsequent generation (Asaah, 2023). In addition, haploid drones are produced through an asexual reproductive process known as parthenogenesis, which involves the development of embryos without fertilization (Hartono & Pham, 2024). Both queens and unmated queen-less worker bees can utilize this reproduction method to introduce drones into the colony. This directly affects drone population and results in an overall population distribution that adheres to Fibonacci sequence. Lastly, under rare circumstances in which diploid drones are produced, they are generally sterile, and their larvae are cannibalized by workers to ensure only viable haploid drones remain in the colony (Asaah, 2023). Workers recognize diploid drone larvae through their distinctive pattern of cuticle secretions (Wang et al., 2021) [hyperlink to Bee Chemistry Essay – Chemical detection section]. This removal of diploid drones sustains the balance of colony population, which will then continue to align with the Fibonacci numbers (Asaah, 2023). 

The observation of the Fibonacci sequence and golden ratio in bee reproductive pattern demonstrates how natural systems can exhibit recurring and predictable patterns. This can allow engineers to generate predictive modelling for population dynamics and lineage tracing. This also provides insights into evolutionary strategies, reflecting nature’s optimization of reproductive success and resource allocation (Gross, 2019). 

Extra-ordinary division of labor  

Temporal polyethism is a key mechanism behind how tens of thousands of worker bees can self-organise and collaborate Fig. 13) (Schmickl & Crailsheim, 2008). Once an adult worker bee emerges from her larval cell, she spends the first stage of her life as a “nurse” bee, performing storage and brood-rearing tasks around the hive (Khoury et al, 2013). After approximately two to three weeks, the worker will become a “foraging” bee, leaving the safety of the hive to collect nectar stores for the colony. These age thresholds are flexible; using chemical and behavioral signals, bees respond to fluctuating environmental conditions and shift the age they switch labor divisions [hyperlink to Bee Chemistry Essay – Communication by pheromones section] (Schmickl & Crailsheim, 2008). For example, young worker bees will prematurely assume foraging positions should old foragers suddenly die off; similarly, old forager bees can revert to nursing roles to accommodate a rise in the demands of the brood. Such extraordinary coordination can be better understood through a set of ordinary differential equations. 

Fig. 13 The four divisions of worker bee labor as described by the Schmickl and Crailsheim algorithms. Each labor division contains multiple tasks for a bee to complete; arrows indicate pathways between labor divisions. This model does not distinguish guard bees from nurse bees (Schmickl & Crailsheim, 2008). 

A key criterion of how an individual worker bee identifies which tasks it should perform is nectar flow. The honeybee model developed by Schmickl and Crailsheim describes how, at a given time t, the colony’s nectar can be found in one of three states: an adult nectar load, a brood nectar load, or a storage cell. The rate of change of an individual adult bee’s nectar load, N, is based on several variables, including collection rate, metabolism, etc. shown below: 

The nectar load increases when a bee collects flower nectar, receives nectar through trophallaxis, and drinks nectar from storage cells. The load decreases when a bee metabolizes, donates nectar through trophallaxis, feeds the brood, or deposits nectar in storage cells. Depending on the labor division of the bee it describes, the magnitude of each term will vary. For example, metabolism(t) for a foraging bee will be approximately 2.5 times greater than bees performing in-hive tasks, due to the energetic demands of flight. Meanwhile, nursing bees will have a larger feed(t) term but collect zero nectar from foraging. The nectar load of a brood can be represented by an altogether condensed form, where the metabolic term is always low.   

The nectar load of an individual worker bee constantly fluctuates and can be predictive of which task they perform within their labor division. The Schmickl and Crailsheim model includes the approximate threshold amounts of nectar at which bees will change tasks (Fig. 14). Here, one full load corresponds to approximately 66µl. A foraging bee’s load generally stays between the maximum collect threshold, which triggers a return to the hive, and the minimum startforaging threshold, which occurs after they have consumed, shared, or stored most of their load. Meanwhile, nursing bees oscillate between high, which prompts brood feeding, and refill, at which point they will replenish their loads using nectar stored in cells. Brood bees are fed to stay between high and low levels. 

Fig. 14 The various threshold levels of a bee nectar load (Schmickl & Crailsheim, 2008). 

The movement of a bee between labor divisions – that is, movement between storing, nursing, and foraging roles – involves physiological changes which primarily depend on age. However, every bee also has its own individual probability, P(t), of assuming a labor division, m, for non-age-related reasons.

The term s(t) indicates the strength of an external stimulus, which can range from pheromones to waggle and tremble dances. Clearly, as the strength of the external stimulus increases, the probability of assuming that labor division increases as well. However, the relationship between probability and stimulus strength is nonlinear due to the 𝜃(𝑡) term, which represents the individual bee’s “threshold” for a labor division.   

T_{m, engaged} represents the period a bee agent has spent in labor division m, whereas T_{m, disengaged} denotes the period a bee has not spent in that division. At the start of a bee’s lifespan, it has an equal threshold for each division,

𝜃0 . However, as a bee gains experience in one labor division, its threshold for that division slowly decreases, while its threshold for all others increases. Lower thresholds mean the bee has a higher probability of assuming that labor division (Fig. 15). In other words, bees who spend more time and specialize in a labor division are more likely to stay in their current labor division in the presence of a stimulus. This serves to optimize the colony’s workforce efficiency.   

Fig. 15 The probability curve, P(t), for different threshold levels, 𝜃(t). The more specialized a bee, the lower its threshold, and thus the lower the stimulus required for the bee to participate in that labor division (Schmickl & Crailsheim, 2008). 

Calculating colony collapse 

Though resilient, the hive is not invulnerable. Arguably the most catastrophic of bee fates is colony collapse disorder (CCD), wherein once healthy hives are suddenly abandoned by adult bees, in most cases leaving behind viable food stores and brood (Myerscough et al. 2017). The hive’s susceptibility to collapse is reflected in the mathematical intricacies of beehive models.   

Myerscough and colleagues demonstrate how strain on the forager population affects the population of the entire hive with another set of differential equations. Fluctuating food stores f(t) are modelled at the colony level, increasing as foragers collect nectar and decreasing as the hive metabolizes it. Here, F, B, and H denote the population of forager, brood, and hive bees¹, respectively. The constant c represents the average weight of nectar each forager collects per day, while the variable 𝛾 indicates the average weight of nectar brood and adult bees consume, respectively. Notably, this model is similar to the above-mentioned Schmickl and Crailsheim model; however, it represents the hive’s net accumulation of nectar and so discounts the internal exchange of food between workers, the brood, and honeycomb cells. 

The function S shows the fraction of brood bees which survive into adulthood. This depends on the food stores, f, the population of brood-rearing hive bees, H, and rate parameters b and v.   

Function S allows us to model the change in the brood population more generally. More specifically, the population increases as the queen lays eggs at rate L and decreases at the brood pupates into adults at rate 𝜙.  

The rate at which forager bees are recruited for foraging, R, involves three terms. The first,

, represents the minimum foraging recruitment rate that occurs in conditions of abundant honey. The second term represents how the rate of forager recruitment increases under low food conditions, and the third term represents how the rate of forager recruitment decreases due to a large enough preexisting number of foragers.   

Finally, the forager population models an increase due to forager recruitment and decrease as foragers die at death rate .

By analyzing these differential equations in unison, Myerscough and colleagues were able to determine the effect of the forager death rate 𝜇 on the health of the colony. At low values around 𝜇 = 0.1, the colony is healthy, with large adult and brood populations and food stores which increase unboundedly. As the death rate gradually increases to 𝜇 = 0.43, the hive continues to be viable, but the population of adult bees drops significantly, and food stores eventually cease increasing at all. At any value of 𝜇 above this, colony collapse occurs. Under collapse conditions, as 𝑡→∞, the adult population approaches zero, but food stores do not. This concurs with descriptions of CCD observed in abandoned hives; however, the population decay in this model is slow – approximately 250 days – and irreflective of the sudden, rapid collapse that occurs in nature. To resolve this inconsistency, the model must be expanded to account for the age of forager bees (Myerscough et al. 2017). 

  This new model represents the death rate 𝜇 as the product of 𝑚𝑟, the ratio of death rates of a stressed hive to a healthy hive, and 𝑀(𝑎), the forager death rate which depends on the age of foraging onset, a (Myerscough et al. 2017).  The function T(𝑎) represents the proportion of hive bees which complete the transition to becoming foragers. As a decreases, so does the proportion T(𝑎). When the age of forager recruitment is greater than 20 days, the value of 𝑚_r is low (around 1) and the hive is healthy. However, as the age of foraging onset decreases to 9 days, 𝑚𝑟 increases to 1.91, the death rate increases significantly, and the population declines. Above this threshold, the colony population collapses quickly – in approximately 30 days – again leaving behind stores of food and brood.  

The extended model shows that colony collapse is not just the consequence of high forager death rates, but specifically of high forager death rates associated with the premature recruitment of young worker bee. Likely, this is due to the underdeveloped state of young bee brains and muscles, which increases the likelihood of death during a foraging trip and decreases the number of foraging trips they perform in a day, reducing the hive’s efficiency (Myerscough et al. 2017). These results shine a light on the evolutionary significance of temporal polyethism in bees.  Though the ability of the hive to adjust its division of labor in response to stimuli allows it to continuously adapt to its environment, too young a workforce puts the hive in jeopardy of overall collapse. Thus, bee colonies have developed both need-based and age-based labor division strategies to maximize the survival of the group. 

Conclusion

The everyday behaviors and survival mechanisms of bees are governed by an underlying mathematical foundation. These superorganisms show an extraordinary ability to integrate these principles throughout their colonies, ensuring their success and survival. Bees are highly effective at using optimization techniques to perform tasks within the hive.  

Using the Honeycomb Conjecture, bees construct their hives in hexagonal patterns that maximize storage capacity while using the least amount of wax possible. This shape allows for the largest possible area relative to its perimeter, creating a strong, adaptable structure used for food storage and brood development (Hales, 2001). The reproductive system of bees further shows predictable mathematical patterns. In the family trees of male drones, the number of ancestors in successive generations follows the Fibonacci sequence. This reflects that natural systems can follow predictable patterns and emphasizes the efficiency behind bees’ reproduction strategies and resource allocation (Hartono & Pham, 2024). 

Worker bees also demonstrate mathematical frameworks in their division of labor, which is based on age and optimizes colony productivity (Schmickl & Crailsheim, 2008). Models can be used to demonstrate the response of colonies to environmental changes and stressors to see how they manage labor transitions. These models can also be used to look at weaknesses within colonies and stimulate colony collapse disorder, a phenomenon that occurs when the hive is destabilized and results in a rapid decline in population (Myerscough et al. 2017). Such simulations can show how fragile and intricate the inner workings of the colony are.  

The principles underlying bees’ behaviors offer profound insights into their strategies for efficiency and survival. We believe that the connection between these superorganisms and mathematics provides a deeper understanding of their behavior, adaptations, and response to nature’s challenges. 

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