The Mathematical Organisation of Social Wasps

Eliot Azar, Andrew Habelrih, Malak Zakaria

Abstract

This paper explores the nest structure, complex behaviours, social structures, and resource optimization of social wasps. Social wasp nests are structurally adapted to minimize the use of resources and maximize energy conservation. For independent founding wasps, the growth of the colony’s population starts by following an exponential growth mathematical model and switches to a logistic growth mathematical model when resources start becoming less available. Population numbers also vary depending on prey populations. Wasps also manage their limited resources efficiently to maximize the wellbeing of the colony by prioritizing important tasks whilst still following the law of diminishing returns. Their behaviour when encountering wasps of other colonies follows the principles of game theory, once again maximizing the wellbeing of the colony. Social wasps’ behaviour also involves hierarchal dominance. Adult wasps establish ranks which influence task roles, reproduction and nesting behaviours.

Introduction

Several insect species have developed fascinating interactions among members of their own colony as well as with other insects or other species. Indeed, interactions are crucial for most living organisms, especially for social species such as wasps. Social wasps must interact strongly with their peers to ensure species survival, but they also interact on a larger scale with other species, such as predator ants or prey, like caterpillars. Interactions with their environment, natural resources and materials are also an integrating part of social colonies’ lives. All these social phenomena can be analyzed through mathematical models to shed light on the intricate natural, yet organized, mechanisms through which colonies organize themselves and their environment. Social wasps’ nests often adopt the most energy and material efficient structures, which can be determined through geometrical analysis of various structures and derivation of relationships, similarly to Euler’s relation for polyhedrons. Wasps also often prioritize collaborative relationships over competitive ones with other colonies. For instance, they often share various resources in the same territory. Game theory helps to elucidate these interactions under a mathematical lens. Furthermore, wasp populations grow, and decrease based on many factors, may they be related to predation or natural factors, such as resource availability. Differential equations describing exponential and logistic growth are helpful to provide an accurate model of the growth of wasp populations.

Nests

Colonies of social wasps build nests for several purposes, and these complex structures exhibit many fascinating architectural features granting them numerous advantages. Nest structures are often optimized for environmental features. For example, tropical wasps are significantly exposed to ant predation. Their nest structure reflects an evolutionary adaptation to this menace. Indeed, their nests are elongated rather than spherical, which grants the ant fewer entry points and makes the nest easier to guard by the wasps (Jones & Oldroyd, 2006). Some nest structures in more extreme climates are optimized to be energy efficient and conserve heat (Jones & Oldroyd, 2006).

Among the many adaptive structures of social wasp nests, one criterion that is always of primordial importance is the economy of the nesting material. Indeed, wasp nests are made of an agglomeration of hollow cells, and not all arrangements of a certain number of cells requires the same quantity of material, as walls from one cell can also be used as a wall for another cell simultaneously. While it is obvious that building the first cell will require the most material as it has no other cell to share a wall with, it is not an easy task to determine the quantity of material to build the n^th cell, as many different arrangements are possible and offer different material efficiencies. By assimilating the cells from a wasp nest to regular hexagons, we can create a model to quantify the efficiency of a certain arrangement and find the most efficient structure for a given number of cells.

Fig. 1: Two different structures of social wasp nests (Klein, 2003).

The quantity of nest material used in a given configuration can be quantified by the total number of cell walls in the nest. This is not equivalent to multiplying the number of cells by six, as some cells share walls with their neighbours. By separating the walls of the nest into two categories, exterior walls (w_e) and interior walls (w_i), it becomes clear that exterior walls are used once, while the interior walls contribute to two different cells and are thus equivalent to two regular walls. Since the total number of potential regular walls is 6n, where n is the total number of cells, we get:

(1) $6n = \omega_e + 2 \omega_i$

Using the fact that w_e + w_i = w, we get (Jennifer C. Klein, 2003):

(2) $\omega = 6n - \omega_i = 3n + \frac{\omega_e}{2}$

While this formula provides an interesting quantification of the nest material used for a certain arrangement, it is far from ideal, as it still rests on the count of interior or exterior walls. This can be a very tedious task for complex nest structures, and a more efficient formula must be derived in order to express the quantity of nest material used in terms of simpler quantities.

Nest structures can be divided in two categories: thin and thick nests (Jennifer C. Klein, 2003). Thin nests are nest structures which have sections of one-cell thickness, as shown in Figure 2.

Fig. 2: Examples of thin nest structures (Jennifer C. Klein, 2003).

Inversely, thick nests do not have such segments. If considering only thick nests, we can see that as an exterior cell is added, the number of exterior walls increases by two, as shown tree in Figure 3.

Fig. 3: The addition of an exterior cell to a thick nest structure (Jennifer C. Klein, 2003).

Here, while the two red walls are now excluded from the exterior wall count, the four green walls are added, for a total of two additional exterior walls for each additional exterior cell. Considering the six initial walls of the first cell of a nest, this would give:

(3) $\omega_e = 2n_e + 6$

By combining this with the equation (2) in order to remove the dependency on the number of exterior walls, the formula becomes (Jennifer C. Klein, 2003):

(4) $\omega = 3n + n_e + 3$

This new formula provides an insightful way of quantifying the nest material used for a given thick nest structure while avoiding laborious counting tasks. However, thin nests are not negligible. Many wasp species build nest with thin portions, as they can provide many advantages. For nests with a small number of cells especially, adopting a thin linear structure provides an advantage relative to predation, as the nest is less visible to predators (Jeanne, 1975). It would thus be interesting to generalize Equation (4) to include thin nests. This can be done by replacing n_e by a more general quantity, c_e. This value is measured by choosing an exterior cell, and counting every exterior cell encountered while circling all around the nest (Jennifer C. Klein, 2003). This process will ensure that cells in thin sections are counted more than once. The formula for thick and thin nests thus becomes:

(5) $\omega = 3n + c_e + 3$

Another interesting structural irregularity to take into account in Figure 4.

Fig. 4: Examples of nest structures with holes.

These nest structures cannot be described by the above equations, as each hole removes a number of walls from the complete filled structure. The number of walls that must be removed for each hole is simply given by the number of interior walls of a nest with the shape of the hole in question. This problem is simpler and can be solved with the equations for regular nests. Indeed, considering m to be the number of cells missing in a hole, and combining Equations (2) and (5), the number of walls w* missing from a hole is given by:

(6) $w^*=3m-c_{e\mathrm{~hole}}-3$

If h is the number of holes in a nest and N is the total number of cells if the holes are filled in:

(7) $N=n+\sum_{i=1}^hm_i$

Using equations (5) and (6), the number of walls can be found:

(8) $W = 3n + c_e + 3 + 3h + \sum{c_{j}}$

This formula is valid for thin or thick nest structures, with or without holes (Jennifer C. Klein, 2003). Interestingly, the structure of this formula is like that of a similar problem, the generalization of Euler’s formula for polyhedrons. Indeed, the Swiss mathematician Simon L’Huillier found a generalization for this formula to consider holes:

(9) $F - E + V = 2 - 2H$

where F is the number of faces, E is the number of edges, V is the number of vertices and H is the number of holes in the solid (Watkins). The presence of the 2H term is equivalent to the 3h term in equation (8) for the number of walls in a wasp nest structure.

The most efficient structure for a nest with n cells is the one for which w is minimized. By analyzing each term in equation (8), it is obvious that a nest without holes is more efficient, as the two last terms will disappear. Next, c_e was used to replace n_e to consider thin nests, thus counting cells in thin regions more than once. c_e is thus always greater than n_e, so the most efficient structure does not have thin regions. However, it is important to note that even if these factors are ignored and are not present in the most material-efficient structure, they are still present in nature. Indeed, these structures grant other advantages that sometimes surpass material efficiency, such as predation or thermoregulation. Taking into account these observations, the function to optimize becomes equation (4). As n is constant, the only way to minimize the nest material used is to minimize the number of exterior cells. This shape happens to be a regular hexagon. Indeed, starting with two cells sharing one wall, the most efficient way to place the third cell is by using two preexisting walls. This applies for the fourth cell and the fifth cell, and until a full ring is created around the first cell.

Fig. 5: Disposition of the first cells to form a hexagon ring.

In Figure 5, the red cell represents the central cell, and the green cell is the cell added at each step. In steps two through five, the cell is added at the most efficient spot, using two preexisting walls. The last cell uses three preexisting walls and closes the first ring around the central cell. In step 5, for example, it would have been possible to place the cell on top of the structure, thus still using two preexisting walls but not creating a ring. However, this would not have allowed the next cell to use three preexisting walls. The structure in which the cells form rings around the central cell is thus the most efficient.

Fig. 6: An example of a wasp nest with the material-efficient structure (Richerman, 2010).

In accordance with this result, many social wasp nests in nature adopt a similar hexagonal or circular structure. This structure is the one that requires the least material, and thus the less energy to build. Driven by evolutionary patterns, social wasp colonies have developed into building the mathematically optimized structure of nests, showcasing the dazzling complexity of evolution.

Population Dynamics

There are two types of wasps when it comes to establishing a new colony. The first kind is the swarm-founding wasp. These wasps gather a large group of workers and one or more queens to migrate and build a new nest. This allows swarm-founding wasps to quickly expand after establishing a new colony by bypassing the first stages of the development of a new colony. They are also better capable of defending themselves from threats due to their large numbers (Jeanne, 2020). The second kind is independent-founding wasps. The establishment of a new colony relies on a solitary queen. This causes colonies to expand less quickly since the queen must overcome the first few stages by herself. This causes independent-founding wasp colonies to grow from a single individual, and that growth can be examined using mathematical equations (Jeanne, 2020). The growth of such a colony can be split into two parts, the first being exponential growth:

(10) $N(t)=N_0e^{rt}$

Equation (10) is the formula representing the exponential growth of wasp colonies in the early stages. $N(t)$ represents the new population size we are trying to find at a given time $t$ , $N_0$ is the initial population size and $r$ is the intrinsic growth rate of the population measured by considering reproduction and survival rates. Exponential growth is due to the presence of a surplus of resources available to the wasps (Manu & Shikaa, 2023). Because of the low competition of these resources, the colony can focus heavily on reproduction and expand its nest and its territory. This behaviour continues until the colony encounters external pressures like an increase in the competition for nearby resources, territorial disputes, and limited food. When the population of the colony reaches a certain threshold and starts encountering these issues, the growth of the colony switches to the second stage which is logistic growth:

(11) $N(t)=\frac{K}{1+\left(\frac{K-N_0}{N_0}\right)e^{-rt}}$

Equation (11) is the formula representing the logistic growth of a wasp colony after it starts encountering constraints. The new variable that appears is 𝐾 and it represents the maximum population size that the environment can support for a given colony. This mathematical model shows that as the population of the colony ( $N_0$ ) approaches the maximum value 𝐾, the growth rate slows down and reaches an equilibrium. At 𝐾, the rates of births and deaths balance out. Logistic growth explains why colonies do not grow indefinitely (Manu & Shikaa, 2023). Using these two models, researchers can also evaluate how social wasps react to environmental changes. A visual representation of these two mathematical models is shown in the Figure 7 below.

Fig. 7: A comparison of the graphs of exponential growth (a) and logistic growth (b). The exponential growth has no restrictions on the total amount of population growth. The logistic growth has a plateau over which the population can not surpass (GeeksforGeeks, 2024).

The population of social wasps also depends on the population of prey in its environment. The Lotka-Volterra differential equations are a pair of equations which evaluate the rate of change of the populations of a predator and a prey. These equations can be combined to the previous equation for logistic growth to create a more accurate mathematical model to represent the population of wasps (Musacchio, 1998). The new differential equations for logistic growth of wasps and prey are:

(12) $\frac{dN}{dt}=rN{\left(1-\frac{N}{K}\right)}-\alpha NP$

(13) $\frac{dP}{dt}=\beta NP\left(1-\frac{P}{K_p}\right)-qP$

In these two equations, the population of prey is represented by 𝑁 and the population of wasps is represented by the letter 𝑃. The equation (12) explains the rate of change of the population of prey over time ( $\frac{dN}{dt}$ ) and the equation (13) explains the rate of change of wasp populations over time ( $\frac{dP}{dt}$ ). The variable 𝛼 is a coefficient which explains the effectiveness of the predator population at consuming the prey. The variable 𝛽 is the efficiency for which the consumed prey affects the population growth of the predators (Musacchio, 1998). The variable 𝑞 is a coefficient indicating the natural death rate of the wasp. In both equations (3) (4), the rate of death of each population is subtracted to the logistic growth rate. These new equations also help explaining cyclical population dynamics found in many different environments since the population of predatory social wasps and the population of prey are both linked. As the population of predators increase, the prey gets hunted more often which causes a decline in its overall population. This causes the population of predators to also decrease since there is not enough food to support the population (Musacchio, 1998). A visual representation is shown in the Figure (8) below.

Fig. 8: Graph of the Lotka-Volterra System demonstrating a cyclic tendency of the variation in the population of prey and predators over time. (Musacchio, 1998)

Resource optimization in colonies

Social wasps have complex resource allocation behaviours that optimize the survival and growth of their colonies. Unlike solitary wasps, their societal structure relies on the efficient distribution of resources such as food, energy, and labor. This adaptability enables them to respond effectively to environmental changes through robust strategies. With limited resources, these wasps must balance every activity to ensure no aspect of colony life is neglected. Such activities include foraging, defense, nest maintenance, and caring for larvae. To achieve this balance, tasks are dynamically managed by reassigning workers based on the colony’s current needs, driven by environmental conditions or internal pressures. This decentralized decision-making is carried out by individual worker wasps, who assess local conditions to determine their actions (Johnson et al., 2009). Predicting these behaviours is also possible due to the establishment of mathematical models. The most common approach to solve and understand these type of optimization problems is by using a linear programming model:

(14) $Z=\sum_{i=1}^mc_ix_i=c_1x_1+c_2x_2+c_3x_3+…$

In this equation (14), $Z$ is the objective to be maximized by the wasp colony. An example of such an objective is the overall wellbeing of the colony. The value $x_i$ is the possible decisions that can be made by the wasps and $c_i$ is the impact of each of these decisions to the current objective. The variable $c_i$ is dynamic since it changes based on the needs of the wasp colonies. If at a certain time foraging is required more than maintaining the nest, a higher importance will be attributed to this action. Maximizing the wellbeing of the being of the colony can be achieved by attributing more wasps to tasks of higher importance whilst still maintaining a certain amount to satisfy other needs. The principle of diminishing returns prevents allocating all nearby workers to the task of highest importance because in colonies, when more energy or more resources are attributed to a certain activity, the benefits derived start decreasing (Fantino & Abarca, 2013).

This formula must also satisfy a certain set of conditions which represent limitations on the available resources or the objectives which must be met. The main constraint is represented by the following sum (15):

(15) $\sum_{j=1}^ma_{ij}x_j\leq b_i$

This signifies that the sum of the resources used by the different activities cannot exceed the maximum number of resources available to the colony which is represented by $b_i$ . The variable $a_{ij}$ is a coefficient which indicates how much of the resource is consumed by the activity j for each number of units dedicated to a certain activity ( $x_j$ ) (Fantino & Abarca, 2013). An example of this is if 20 wasps ( $x_j$ ) are assigned to defending the colony and this action uses 2 units of energy per wasp ( $a_{ij}$ ), the amount of energy depleted would be 40 units. As such, if only 35 units of energy were available, this distribution would not be possible.

Game theory in wasp interactions

Energy optimization in social wasps involves finding efficient foraging paths. They excel at exploiting environmental resources and identifying high-quality food sources that offer the greatest benefits (Taylor et al., 2010). Key components of their foraging success include the use of pheromone trails for communicating food locations, visual memory for navigating complex environments, and group foraging strategies. Additionally, interactions with other colonies and competition with other species can influence their foraging behaviour (Steinmetz & Schmolz, 2004). The effects of these encounters can be analyzed by the mathematical framework of game theory where decisions are taken based off other players. This enables social wasps maximize resource gathering whilst minimizing energy losses from conflicts. An example of such a choice to be made by wasps is when food resources are low. In competitive foraging, wasps may defend a food source aggressively and deter other wasp colonies from exploiting it, but this may use up a lot of energy. However, in cooperative foraging wasps may share information and work together with other colonies to reduce conflict and improve resource exploitation (Mora-Kepfer, 2013). Game theory helps to model these different interactions to evaluate the potential payoffs of every action (Cressman et al., 2014).

Payoff matrices are used in game theory to represent the different possible outcomes of different strategies employed by players. In this case, the players are wasps. Every possible outcome also has a value associated for each party. The higher the number, the more beneficial the outcome is and the lower the number, the more disadvantageous the outcome (Cressman et al., 2014). If we consider a scenario where a wasp A and a wasp B compete for a food source during a period of scarcity, their possible options would be to defend, to share or to leave. The payoff matrix of such an encounter is the following:

		Wasp B strategy
		Defend	Share	Leave
Wasp A strategy	Defend	(-1, -1)	(2, 0)	(1, 0)
	Share	(0, 2)	(3, 3)	(1, 1)
	Leave	(0, 1)	(1, 1)	(0, 0)

Fig. 9: Payoff matrix demonstrating the game theory of a possible encounter between two wasps at a food source

In this payoff matrix, the numerical values associated to each of the strategies picked by the wasp depend on the other player. In this case, the worst possible payoff for both wasps is the outcome when both wasps choose to defend the food source. This is due to injuries that may be sustained and the use of the wasps’ energy on fighting instead of foraging. The best possible payoff for both wasps is when they both share the food source since the wasps can help each other effectively extract the resource and can offer each other security due to their increased numbers. In game theory, retreating can also count as a potential gain since the energy that could be spent on fighting can be spent on the search for other resources. This explains the equal gain for both wasps when one shares and the other leaves (Cressman et al., 2014).

Dominance

There are many factors that influence the linear hierarchical dominance in social wasp species. Amongst them are the body size of a wasp, its status badge and the order of arrival or rather, emergence, of wasps within a colony. Typically, dominant alpha wasps called queens or foundresses have a larger size than regular worker wasps and a higher fat composition. Similarly, female wasps with higher hierarchical positions (beta wasps) resemble the queen in both size, fat composition and fighting abilities. The status badge refers to the wasp’s morphology, serving as an indicator of its social position. For instance, specific coloration of the clypeus region in wasps has been reported to express dominance as seen in Figure 10 (Tibbetts & Dale, 2004). As for the order of emergence of wasps, the first emerged brood usually has similar dominance to that of the queen, this is one place where beta wasps come into play. Queens spend most of their time on the nest and invest their energy in reproductive performance and colony control rather than foraging. They feed the larvae, inspect the cells and specifically pay attention to higher hierarchical female wasp reproduction to ensure that reproduction dynamics are in their favor. While beta subordinates spend equally as much time on the nest, they focus on increasing their reproductive potential while taking care of the brood alongside the queen. (Murakami et al., 2013). Both the dominant wasp (queen) and higher hierarchical position wasps (subordinates) are responsible for internal nest tasks. The interest of the subordinates also lies in potentially substituting for the queen in her absence.

Fig 10: Diversity in facial patterns of nine foundresses of P. Dominulus. Clypeus spotting is strongly correlated with overall body size, which is in turn associated with social rank. (Tibbetts & Dale, 2004)

In primitively eusocial wasps of genus Polistes, several related queens cooperate in nest building to form a new colony. The most dominant of them, called the foundress or (alpha) queen, produces the majority of offspring while regulating the reproduction of her (beta) subordinate co-foundresses by eating approximately 1/3 of their eggs. At times, there are conflicts of dominance between the queen and her subordinates, which explains their complex relationship. If allowing the beta subordinates to reproduce increases the chances of the beta to cooperate peacefully with the queen instead of leaving and forming her own colony or fighting for complete reproductive dominance, the queen may allow her to reproduce. This means that if a subordinate interferes with the queen’s egg production, the queen retaliates aggressively. Additionally, beta subordinates occupy a higher rank in the colony, as they share many similarities with the queen. Larger subordinates may challenge the queen as they can match her fighting abilities and have a similar body size. Thus, the optimal skew model adjusts to consider that these stronger and larger beta subordinates have more reproduction rights (can reproduce more than regular subordinates). For reference, optimal skew models predict that cooperative breeding occurs as a results of dominant wasps conceding reproductive benefits to their subordinates to mitigate aggression within a colony and minimize the cost of potentially losing a subordinate.

A study by H. K. Reeve and P. Nonacs evaluated the aggression levels of alpha and beta wasps in a colony of Polistes fuscatus when they manually removed reproductive-destined eggs from the nest. Results show that beta’s aggression rates significantly increase the more her eggs are removed. The magnitude of the aggression also depends on the size of the beta subordinate in comparison to the queen. The relation between the beta’s size compared to the queen’s and overall change of rate of aggression is represented by the partial Spearman correlation. This is a statistical calculation that measures the relationship between two continuous variables while controlling for the effect of another continuous variable. In this study, the correlation between the size difference in beta and alpha wasps and the overall aggression rate was negative (with the number of removed reproductive eggs controlled). The negative correlation signifies that as the aggression rates increase, the size difference between the wasps decreases. Similarly, the correlation between the number of reproductive eggs removed and the overall rate of aggression (with size difference controlled) was positive, indicating that the more of their reproductive-destined eggs are removed, the more aggressive the beta wasps become. These correlations show that, indeed, beta subordinates retaliate aggressively when reproductive eggs are removed, while the magnitude of their change in aggressive behaviour also increases when the beta’s size is similar to that of queen’s (Figure 11.). Additionally, contrary to the expected behaviour of the alpha queen in response to the removal of her eggs, she did not showcase any aggressive retaliation. It is also important to note that the behaviour of subordinates did not change when worker-destined eggs were removed, showcasing the link between their reproductive dominance and the number of reproductive-destined eggs laid. (Reeve & Nonacs, 1992)

Fig 11: Change in the rate of aggression between control observations (0% of reproductive eggs removed) and treatment observations (50% of reproductive eggs removed) in alpha and beta wasps from 16 different colonies of P. fuscatus. Data shows that the more reproductive eggs are removed, the more beta wasp aggression rates increase. (Reeve & Nonacs, 1992)

Beyond reproductive monopolization

So far, the complex relationship between a queen and her subordinates has been explored. The first brood of the colony, whether built by one foundress or multiple (queen and subordinates), also happens to be regulated by the queen. Another study by Reeve, Nonacs, and colleagues, Starks and Peters, suggests that the foundress also regulates the nutrition of the first-emerging brood. This is to ensure that female workers are not too large as to challenge her reproductive dominance and to lower their fighting abilities, considering that the newly emerged females typically have hierarchical dominance due to their birth order being the first. As a way of protecting her reproductive monopolization, the queen’s calculative self manages the nutrition of the 1^st brood to ensure that the worker females emerge smaller than her, effectively reducing the potential threat to her dominance. Consequently, other wasps from the same brood are significantly smaller than both the queen and their beta sisters. Moreover, the larger worker wasps benefit from remaining at the nest site as they have the future ability to challenge the queen and increase their reproductive right within the native colony. (Reeve et al., 1998)

Disappearance of worker wasps

Strangely enough, it has been observed that some female worker wasps in the same eusocial species, P. fuscatus, disappear within a few days of their emergence. Why is that? Quite frankly, their decision depends on the state of the colony and their social rank. If the number of pupae in the colony is sufficiently large, then worker wasps will want to remain on the nest to help with either internal tasks or foraging. The larger wasps as well as the ones born first will likely decide to stay as they see more benefits in helping the colony than leaving. On the contrary, smaller worker wasps are more likely to leave the colony in search for another if the nest does not have sufficient pupae. This is because they understand that there will likely be a decrease in the number of opportunities for them to work or help in the development of colony, considering both the low number of pupae and their low rank. Thus, the most important factor in determining whether a worker wasp decides to stay and help, or leave is the proportion of pupae in the native nest. This relation is demonstrated by a partial regression analysis where the change in the dependent variable (the decision) associated with a unit increase in the independent variable (number of pupae) is measured while all other independent variables, such as the rank of the wasp, its size and order of emergence, are held constant. This also suggests that colonies with a singular founding queen are more likely to experience the dispersal of their worker wasps due to their lower pupae proportion. Additionally, leaving wasps tend to have a smaller mean size (called wing-length) while staying wasps are larger, with a higher wing-length as shown in Figure 12. (Reeve et al., 1998)

Fig 12: The correlation between the size of wasps (wing-length in cm) and the number of leaving and staying female workers. (Reeve et al., 1998)

Combining the two notions of the reduced size of female wasps in the first brood and the number of wasps leaving the nest after their emergence, we get an intricate equation describing the minimal worker size of a wasp that will stay and help the colony. The degree to which the foundress may reduce the size of the female brood is bounded by the tendency of smaller female workers to leave the nest altogether.

(16) $p\left(\frac{Sw}{Sf}\right)=\frac{x-r(k-1)}{k(1-r)}for\left(r<1\right)$

In this equation, p is the probability a worker’s direct reproduction (reminder: beta workers can reproduce) as a function of the ratio of the worker’s size (Sw) and the foundress’ size (Sf). The numerator represents the mean relatedness, r, of the worker to the queen’s offspring, while the denominator represents the mean relatedness of the worker to its own offspring. x is the expected success of a leaving female, and k is the colony success if the worker remains in the nest. If the female successfully disperses, then r =1. The interpretation of the equation is as follows. When x is less than (k-1), it is beneficial for the queen to retain the female worker. If the queen’s optimal ratio is $\frac{Sw}{Sf}$ , then the size of the worker wasp’s brood should theoretically be $Sw = ASf$ , meaning that the size of the first-emerged female workers is linearly related to the size of the foundress. In other words, the foundress will manage their nutrition bearing in mind that they should emerge smaller than her. Additionally, considering colonies with multiple foundresses, the relatedness of the worker will be lower than in single-foundress colonies, which would affect the relative data computed out of this model equation. (Reeve et al., 1998)

Cooperative Drifting

On a similar note, when the worker-to-brood ratio is too high, this leads to worker wasps leaving the colony but for different reasons. The colony no longer significantly benefits from the efforts of the worker wasp. Instead, the wasp will leave its colony in which it has a high relatedness and look for related colonies that need more help to care for their brood. These wasps are called drifting workers as they drift from their native nest to cooperate and assist wasps of another colony. (Kennedy et al., 2021)

Conclusion

Wasps face several critical challenges in their survival and reproductive success, including minimizing energy expenditure when building nests, optimizing the nest shape for their needs, strategically navigating their environments to find foraging areas, maintaining dominance within their social hierarchies and much more. Depending on the species, wasps will determine the most energy efficient building patterns to support their survival needs. The colony’s population is determined by exponential growth and logistic growth models, taking into account the varying wasp and prey population amongst different species. Over time, they have also evolved highly specialized behavioral and physiological adaptations to address dominance issues. One of the primary challenges in eusocial species is balancing reproduction between dominant queens and subordinate females. This competition could lead to conflicts that destabilize the colony. To mitigate this, wasps have developed a system of hierarchical dominance reinforced through physical size, status badges and strategic behaviours. These mechanisms maintain order and reduce costly conflicts. Mathematical models, such as the optimal skew model, shed light on how wasps distribute reproductive rights within their colonies. The model predicts that dominant individuals, like alpha queens, will allow subordinates limited reproduction to minimize aggression and encourage cooperation. This balance optimizes the colony’s overall fitness by ensuring the subordinates’ reproductive interests align with the queen’s goals. Wasps are strategic in many ways, using energy-efficient designs and mathematically optimized behaviors to address challenges ranging from survival to social organization and beyond.

References

Cressman, R., & Tao, Y. (2014). The replicator equation and other game dynamics. Proceedings of the National Academy of Sciences, 111(Supplement 3), 10810–10817. https://doi.org/10.1073/pnas.1400823111

Fantino, E., & Abarca, N. (2013). Choice and optimal foraging: Tests of the delay-reduction hypothesis and the optimal diet model: Edmund Fantino, Nureya Abarca, and Masato Ito. Foraging, 196–222. https://doi.org/10.4324/9780203781685-18

Jeanne, R. L. (1975). The Adaptiveness of Social Wasp Nest Architecture. The Quarterly Review of Biology, 50.

Jeanne, R. L. (2020). The evolution of swarm founding in the wasps: Possible scenarios. Neotropical Social Wasps, 23–46. https://doi.org/10.1007/978-3-030-53510-0_2

Jennifer C. Klein, T. Q. S. (2003). Taking the Sting out of Wasp Nests: A Dialogue on Modeling in Mathematical Biology. The College Mathematics Journal, 34(3), 207-215.

Johnson, E. L., Cunningham, T. W., Marriner, S. M., Kovacs, J. L., Hunt, B. G., Bhakta, D. B., & Goodisman, M. A. (2009). Resource allocation in a social wasp: Effects of breeding system and life cycle on reproductive decisions. Molecular Ecology, 18(13), 2908–2920. https://doi.org/10.1111/j.1365-294x.2009.04240.x

Jones, J. C., & Oldroyd, B. P. (2006). Nest Thermoregulation in Social Insects. In S. J. Simpson (Ed.), Advances in Insect Physiology (Vol. 33, pp. 153-191). Academic Press. https://doi.org/https://doi.org/10.1016/S0065-2806(06)33003-2

Manu, S., & Shikaa, S. (2023). Mathematical Modeling of Taraba State Population Growth Using Exponential and Logistic Models. https://doi.org/10.2139/ssrn.4428621

Mora-Kepfer, F. (2013). Context-dependent acceptance of non-nestmates in a primitively eusocial insect. Behavioral Ecology and Sociobiology, 68(3), 363–371. https://doi.org/10.1007/s00265-013-1650-2

Musacchio, F. Lotka–Volterra equations for predator–prey systems. (1998). Evolutionary Games and Population Dynamics, 11–21. https://doi.org/10.1017/cbo9781139173179.006

Richerman. (2010). Structure of a Common wasp nest (Vespula vulgaris). In.

Steinmetz, I., & Schmolz, E. (2004). Influence of illuminance and forager experience on use of orientation cues in social wasps (Vespinae). Journal of Insect Behavior, 17(5), 599–612. https://doi.org/10.1023/b:joir.0000042543.64957.b9

Taylor, B. J., Schalk, D. R., & Jeanne, R. L. (2010). Yellowjackets use nest-based cues to differentially exploit higher-quality resources. Naturwissenschaften, 97(12), 1041–1046. https://doi.org/10.1007/s00114-010-0724-5

Tibbetts, E. A., & Dale, J. (2004). A socially enforced signal of quality in a paper wasp. Nature, 432(7014), 218-222. https://doi.org/10.1038/nature02949

Watkins, T. The Euler-Poincaré Characteristic of Polyhedral Solids with a Cavity. San José State University.

Reeve, H. K., & Nonacs, P. (1992). Social contracts in wasp societies. Nature, 359(6398), 823-825. https://doi.org/10.1038/359823a0

Murakami, A. S. N., Desuó, I. C., & Shima, S. N. (2013). Division of labor in stable social hierarchy of the independent-founding wasp Mischocyttarus (Monocyttarus) cassununga, von Ihering (Hymenoptera, Vespidae). Sociobiology, 60(1), 114-122. https://doi.org/10.13102/sociobiology.v60i1.114-122

Kennedy, P., Sumner, S., Botha, P., Welton, N. J., Higginson, A. D., & Radford, A. N. (2021). Diminishing returns drive altruists to help extended family. Nature Ecology and Evolution, 5(4), 468-479. https://doi.org/10.1038/s41559-020-01382-z

Reeve, H. K., Peters, J. M., Nonacs, P., & Starks, P. T. (1998). Dispersal of first “workers” in social wasps: Causes and implications of an alternative reproductive strategy. Proceedings of the National Academy of Sciences of the United States of America, 95(23), 13737-13742. https://doi.org/10.1073/pnas.95.23.13737

GeeksforGeeks. (2024, April 23). Exponential and logistic growth. https://www.geeksforgeeks.org/exponential-logistic-population-growth/