Social inhibition maintains adaptivity and consensus of honeybees foraging in dynamic environments.

To effectively forage in natural environments, organisms must adapt to changes in the quality and yield of food sources across multiple timescales. Individuals foraging in groups act based on both their private observations and the opinions of their neighbours. How do these information sources interact in changing environments? We address this problem in the context of honeybee colonies whose inhibitory social interactions promote adaptivity and consensus needed for effective foraging. Individual and social interactions within a mathematical model of collective decisions shape the nutrition yield of a group foraging from feeders with temporally switching quality. Social interactions improve foraging from a single feeder if temporal switching is fast or feeder quality is low. When the colony chooses from multiple feeders, the most beneficial form of social interaction is direct switching, whereby bees flip the opinion of nest-mates foraging at lower-yielding feeders. Model linearization shows that effective social interactions increase the fraction of the colony at the correct feeder (consensus) and the rate at which bees reach that feeder (adaptivity). Our mathematical framework allows us to compare a suite of social inhibition mechanisms, suggesting experimental protocols for revealing effective colony foraging strategies in dynamic environments.

Introduction

Social insects forage in groups, scouting food sources and sharing information with their neighbours [1-3]. The emergent global perspective of animal collectives helps them adapt to dynamic and competitive environments in which food sources’ quality and location can vary [4]. Importantly, decisions made by groups involve nonlinear interactions between individuals, temporally integrating information received from neighbours [5]. For example, honeybees waggle dance[1] to inform nest-mates of profitable nectar sources [6,7], and use stop signalling[2] to dissuade them from perilous food sources [8] or curb recruitment to overexploited sources [9]. While waggle dancing rouses bees from indecision, stop signalling prevents decision deadlock and builds consensus when two choices are of similar quality [10]. Thus, both positive and negative feedback interactions within the group are important for regulating collective decisions and foraging [11,12].

Honeybee colonies live in dynamic environments, in which the best adjacent nest or foraging sites can vary across time [13,14]. Bees adapt to change by abandoning less-profitable nectar sources for those with higher yields [15,16], and by modifying the number of foragers [17,18]. Prior studies focused on how waggle dance recruitment or the division of individual bee roles shape colony adaptivity [19,20]. Inhibitory social interactions, whereby bees stop each other from foraging, have been mostly overlooked as a communication mechanism for facilitating collective adaptation to change [21,22]. We propose that inhibitory social interactions are important for foraging groups to adapt to change in a fluid world.

To study how social inhibition shapes foraging yields, we focus on a task in which the nectar quality of feeders is switched periodically. Related situations probably occur in nature due to the dynamics of competitor and predator prevalence, crowding by nest-mates, and weather fluctuations [23-25]. Precisely periodic dynamics do not occur naturally but can be generated in controlled experiments [16,20]. There are important distinctions between the goals of colonies in foraging as opposed to those searching for a new home site. Once a colony establishes a permanent nest site, this is the starting and ending point for each food foraging excursion. The colony does not need to reach consensus to obtain nutrition from foraging, since food is brought to the nest regardless of how many foraging sites the group is split between [25]. By contrast, when a honeybee swarm looks for a nest, it must reach consensus for all bees and the queen to fly to the selected site. If not, their transition to a permanent nest site will be delayed, or the swarm might split. Bees use stop signals to obtain this needed consensus when house-hunting, especially when two potential sites are of similar quality [26]. Consensus is not essential when foraging for food, but, as we will show, increasing the fraction of the colony at the best foraging site increases foraging yields.

Foraging colonies appear to be able to adapt to change. In prior studies [15,16,20], colony foraging targets shifted in response to food quality switches, suggesting bee collectives can detect such changes. Uncommitted inspector bees can lead bees away from feeders whose nectar quality has dropped [20], and recruitment via waggle dancing appears to be unimportant for effective foraging in changing environments (see also [27]). Here, we also find recruitment can be detrimental, but social inhibition can rapidly pull bees from low- to high-yielding feeders. This, paired with ‘abandonment’ whereby bees spontaneously stop foraging, facilitates temporal discounting of prior evidence. By contrast, strong positive feedback via recruitment causes bees to congregate at feeders even after food quality has dropped, biasing a colony’s behaviour based on past states of the world.

We quantify the contribution of these positive and negative feedback interactions within a mathematical model of a foraging colony. Our study focuses on four potential inhibitory social interactions—discriminate and indiscriminate stop signalling [8,26], direct switching [28,29] and self-inhibition—by which foraging bees alter the behaviour of other foraging bees. Self-inhibition has not been reported in honeybee foraging experiments, but we consider its effects as a potential social inhibitory mechanism, claiming it could be observable in behavioural assays for which it is advantageous (e.g. single switching feeder). Strategies are compared by measuring the rate of foraging yield over the timescale of feeder quality switches. When bees have a single feeder, social interactions are less important unless temporal switching is fast and food quality is low, but in the case of two feeders the performance of different forms of social interactions is clearly delineated. Direct switching, by which a bee converts another forager to their own preference, is the most effective means for a colony to adapt to feeder quality changes. Also, foraging yields are most sensitive to changes in group-wide interactions in rapidly changing environments with lower food quality. Model linearizations allow us to calculate a correspondence between social interaction parameters and the consensus (steady-state fraction of bees at the high-yielding feeder) and adaptivity (the rate of switching from low- to high-yielding feeders). This provides a clear means of determining the impact of social interactions on a colony’s foraging efficacy.

Results

The mathematical model of bee colony foraging decisions assumes potential foragers may be uncommitted or committed to one of the possible feeders [29]. Uncommitted bees spontaneously commit by observing a feeder or by being recruited by another currently foraging bee. Committed bees may spontaneously abandon their chosen feeder, or may be influenced to stop foraging or switch their foraging target based on inhibitory social interactions [26,29]. A population-level model emerges in the limit of large groups. Stochastic effects of the finite system do not qualitatively change our results in most cases (see appendix C(f)).

We mostly focus on two-feeder (A and B) systems, in which the fraction of the foragers committed to either feeder is described by a pair of nonlinear differential equations in the limit of large groups (see figure 1a for a schematic) and where α(t) are time-dependent food qualities at feeders A, B (see figure 1b for examples); β min−1 is the rate bees recruit nest-mates to their feeder via waggle dancing; γ min−1 is the rate bees spontaneously abandon a feeder[3]; and is a nonlinear function describing inhibitory social interactions (e.g. stop signalling or direct switching as described in appendix B(a)). Since commitment fractions are bounded within the simplex 0 ≤ u ≤ 1 and 0 ≤ u + u ≤ 1, the commitment (α) and recruitment (β) provide positive feedback and the abandonment (γ) and inhibition () provide negative feedback.

Figure 1.

(a) Schematic of colony foraging model with two feeders (e.g. flowers or feeder boxes), equation (2.1). Bees move along arrows between different opinions (uncommitted or committed); arrow labels indicate interactions that provoke those opinion switches. (b) Example feeder quality time series α(t), which switch with period T min.

We assume feeders are large enough to accommodate all the bees in the colony and hence we neglect the effects of crowding. Foraging efficacy is thus quantified by the group reward rate (RR), assuming net nutrition is proportional to the fraction of the colony at a feeder u times the current quality of that feeder minus the foraging cost c (e.g. energy required to forage), α(t) − c. Integrating this product and scaling by time yields the effective RR. Given a food quality switching schedule α(t) and total foraging time T, colonies with more efficient foraging strategies have higher RRs J.

Before studying how social inhibition shapes a bee colony foraging in two-feeder environments, we analyse the single-feeder model, finding that commitment and negative feedback from either abandonment or inhibition are usually sufficient for the group to rapidly adapt to feeder quality switches.

Shaping colony adaptivity and consensus for single feeders

Inhibitory social interactions in a single-feeder model can only take the form of self-inhibition, by which a foraging bee stops another based on a detected change in food quality (figure 2a). Since transit from the hive to the feeder takes time, we incorporate a delay of τ min, so the fraction of foraging bees u evolves as where α(t) is the food quality schedule of the feeder that switches at time intervals T (min) between α(t) = 0 and [16,20] (figure 2b), and are the recruitment and abandonment rates, and is the rate of self-inhibition.

Figure 2.

Colony dynamics in the single-feeder model. (a) Schematic of group foraging single feeder, equation (2.3). (b) Food availability α(t) switches on and off 0 at time intervals T (min). (c) Phase line plots: equilibria of equation (2.3) within each food quality epoch are marked as dots. Dynamic increases/decreases of the foraging fraction are indicated by right/left arrows. Bees forage when food becomes available ( for t ∈ [0, T)) and is stable and abandon the feeder once food is removed (α → 0 for t ∈ [T, 2T)) if recruitment is weaker than abandonment (β < γ). (d) The fraction of bees foraging u(t) tracks environmental changes. Higher/lower consensus is obtained by changing the balance of recruitment β and abandonment γ. (e) Reward rate maximizing strategies vary with feeder quality () and switching interval (T). Each coloured region denotes a different optimal strategy given the environment . The best strategies exclude recruitment (β = 0). Boldness of letters γ and ρ denote the strength of colony behaviours that best adapt to the given environment. In rapid (short T) or low-quality (low ) environments (white region), strong inhibition ρ and weak abandonment γ is best, whereas in slow or high-quality environments, inhibition ρ can be weak. We take τ = T/10 min throughout. See appendix A(c) for optimization methods.

Colony’s adaptivity and consensus is shaped by both individual behaviour changes (commitment α(t) and abandonment γ) and interactions (recruitment β and inhibition ρ) [26]. Periodic solutions to equation (2.3) can be found explicitly, allowing us to compute a colony’s RR (see appendix A(b)). Adaptive colonies rapidly return to the hive when no food is available and quickly populate the feeder when there is food (figure 2c,d). Equation (2.3) admits one stable equilibrium in each time interval: when no food is available (α(t) = 0) the non-foraging () equilibrium is stable as long as recruitment is not stronger than abandonment (β < γ). When food becomes available () the stable fraction of foragers increases with food quality (see figure 2c and appendix A(a)). This fraction corresponds to the consensus of the group [30], and the rate the group responds to change we deem its adaptivity.

Robust foraging should adapt to the environmental conditions

The performance of a colony’s interaction strategies strongly depends on the feeder quality and switching time T. Colonies with stronger rates of abandonment γ and self-inhibition ρ more quickly leave the feeder once there is no food (), but have limited consensus when food becomes available (). Increasing the recruitment rate β, on the other hand, boosts consensus but can slow the rate at which the group abandons an empty feeder (figure 2d).

To quantify the effect of abandonment γ, recruitment β and self-inhibition ρ, we compute the long-term RR of the colony, measuring the foraging yield over a single period (2T min) once the group equilibrates to its periodic switching behaviour (see appendix A(b)) where is the cost of foraging and is the quality of the feeder.

For each feeder quality level, , there is an optimal foraging strategy (abandonment γ, recruitment β and stop signalling ρ) within our set of possible strategies (see appendix A(c)) that maximizes the RR J(γ, β, ρ) (figure 2e). Here, private information is sufficient for individual bees to commit to foraging (quality sensing α(t)), and recruitment does not benefit the colony (β = 0). Reinforcing the majority opinion via recruitment is detrimental once the environment changes, as opposed to static environments [26,31,32]. In rapid (small T) or low food quality ( low) environments, stronger inhibition (large ρ) is needed to swap group commitment when the environment changes (white region, figure 2e). This nonlinear mechanism increases the adaptivity of the group, but tempers the initial stage of consensus after the feeder is switched on (see appendix A(a) for details). On the other hand, when food is plentiful (high ) (brown regions, figure 2e), inhibition should be weak (small ρ). In intermediate environments, the best strategies interpolate these extremes.

Linearizing solutions to the model equation (2.3) provides us with a closer look at how group dynamics impact foraging yields. In sufficiently slow environments (large T) with small delays (τ → 0), we can linearly approximate the evolving foraging fraction (see appendix A(d)) where is the foraging fraction (consensus) and λon/off are the rates the group arrives/departs the feeder once food is switched on/off. Plugging equation (2.5) into equation (2.4), we estimate the RR It can be shown that ∂ J > 0 for λ = λon/off, so the RR increases with the rates at which the group switches behaviours. These rates increase as abandonment γ and social inhibition ρ are strengthened (appendix A(a)). Clearly, J increases with since more bees forage when food is available. Increasing abandonment γ tends to decrease consensus, so the most robust foraging strategies cannot use abandonment that is too rapid (appendix A(a)).

We conclude that the volatility (1/T) and profitability () of the environment dictate the colony interactions that yield efficient foraging strategies. One important caveat is that we bounded the interaction parameters, so group communication cannot be arbitrarily fast. This biological bound may be lower in practice, explaining slow adaptation of colonies to feeder changes in experiments [15,16,20]. Our qualitative finding, that social inhibition is more effective in slow and high-quality environments, should be robust to even tighter bounds. We have also shown that when social inhibition is not present, abandonment must be increased as the speed and quality of the environment is increased (appendix C(a) and figure 7). In the next section, we extend these principles to two-feeder environments, particularly showing how specific forms of social inhibition shape foraging yields.

Figure 7.

Reward rate, equation (2.4), maximizing values of abandonment (γ) parameter for a given food quality (α) and switching period (T) in the single-feeder model. Colony can maximize the reward J by calibrating the level of abandonment with the switching rate and feeder quality, discounting faster as the environment changes more quickly.

Foraging decisions between two dynamic feeders

For a bee colony to effectively decide between two feeders, it must collectively inhibit foraging at the lower quality feeder. Our mean-field model, equation (2.1), generalizes house-hunting swarm models with stop signalling [10,26,32] to a foraging colony in a dynamic environment with different forms of social inhibition (figure 1). How do these inhibitory interactions contribute to foraging efficacy? Honeybees can inhibit nest-mates foraging at potentially perilous or crowded feeders [8,24,25,33], but group-level effects of these mechanisms are not well studied in dynamic environments [34]. As we will show, the specific form of social inhibition can strongly determine how a colony adapts to change.

Forms of social inhibition

Generalizing previous models [26,28], we consider four forms of social inhibition (all parametrized by ρ as before): (a) direct switching: bees foraging at the superior feeder directly switch the preference of opposing foragers to the better feeder; (b) indiscriminate stop signalling: when two foraging bees meet, one will stop foraging; (c) self-inhibition: when two bees foraging at the same feeder meet, one will stop foraging; and (d) discriminate stop signalling: when bees foraging at different feeders meet, one stops foraging. These interactions are visualized in figure 3a–d and their evolution equations are given in appendix B(a) (see also electronic supplementary material of [26]).

Figure 3.

Social inhibition of colony foraging at two feeders (figure 1) and their resultant dynamical behaviours: (a) direct switching: inhibiting bee switches opposing nest-mate to their preference; (b) indiscriminate stop signalling: inhibiting bee may either cause opposing or agreeing nest-mate to become uncommitted; (c) self-inhibition: inhibiting bee causes agreeing nest-mate to become uncommitted; (d) discriminate stop signalling: inhibiting bee causes opposing nest-mate to become uncommitted. Phase portraits: (e) Monostable behaviour arises from direct switching, discriminate stop signalling, and self-inhibition so the group always tends to a single equilibrium foraging fraction given fixed feeder qualities. (f) Bistable behaviour that can emerge for strong discriminate stop signalling.

We can divide these forms of social inhibition into two classes, based on the group foraging dynamics they produce: monostable or bistable consensus behaviours. The first three forms of social inhibition yield groups with monostable consensus behaviours (see appendix B(b) and electronic supplementary material of [26]), tending to a single stable foraging fraction when the feeder qualities are fixed (figure 3e). The colony will thus mostly forage at the higher-yielding feeder. On the other hand, strong discriminant stop signalling can produce colonies with bistable consensus behaviours (figure 3f). Such hysteresis in stop-signalling populations was also identified in [10]. As a result, the group can remain stuck at an unfavourable feeder, after the feeder qualities are switched. This is similar to ‘winner-take-all’ regimes in mutually inhibitory neural networks [29,35]. Inhibition from bees holding the colony’s dominant preference is too strong for bees with the opposing preference to overcome, even with new evidence from the changed environment.

Direct switching leads to most robust foraging

To determine the most robust forms of social inhibition for foraging in dynamic environments, we studied how the rate of reward, equation (2.2), depended on the foraging strategy used. Environments are parametrized by the time between switches T (min) and the better feeder quality and the lower feeder quality  , which periodically switch between feeders A and B. As in the single-feeder case, we tune interactions of each strategy (figure 3a–d) to maximize RR over a discrete set of strategies (see appendix B(c) for details). Comparing each social inhibition strategy type’s RR in different environments (figure 4a), we find direct switching generally yields higher RRs than other strategies. Differences in the effectiveness of distinct strategies are most pronounced at intermediate environmental timescales T. As expected, RRs increase with the maximal feeder quality (figure 4b,c).

Figure 4.

Optimal reward rates across forms of social inhibition. (a) Reward rate (RR) increases as the interval between feeder quality switches T increases for all social inhibition strategies. Across most parameter sets, direct switching is the most robust strategy, yielding the highest RRs. In rapid environments, self-inhibition can be slightly better. (b) For T = 5 min fixed and maximal food quality varied, RRs for direct switching and self-inhibition are separated from discriminate and indiscriminate stop signalling at lower food quality levels . (c) For T = 100 min, direct switching, discriminate stop signalling and self-inhibition yield similar RRs, whereas indiscriminate stop signalling is notably worse. These curves are fit nearly perfectly by a linear function (R2 = 1.0000).

Direct switching is probably a superior strategy because it allows for continual foraging (figure 3a), as opposed to other strategies which interrupt foraging with an uncommitted stage (figure 3b–d) and rely on recruitment β to restart foraging. To study how interactions should be balanced to yield effective foraging, we examined how to optimally tune (β, γ, ρ) across environments in the direct switching model (figure 5). Analyses of other models are shown in figures 8 and 9 of appendix C(b).

Figure 5.

Tuning (a) recruitment β; (b) abandonment γ; and (c) social inhibition ρ to maximize RR in the direct switching model (figure 3a). See appendix B(c) for methods. (a) The best tunings of recruitment vary considerably for rapid (low T) and low-quality environments, but recruitment appears to be less essential for slow (high T) and high-quality environments. (b) The rate of abandonment that best suits the environment decreases as the environment becomes slower and higher quality. (c) Generally, the direct switching rate should be made as strong as possible, except for very fast, low-quality environments.

Figure 8.

Tuning (a) recruitment β; (b) abandonment γ; and (c) social inhibition ρ to maximize the reward rate (RR) (equation (2.2)), in the discriminate stop-signalling model. (a) Recruitment and (b) abandonment should be made weak whereas (c) social inhibition should be made strong except for in slow (high T) and high-quality environments.

Figure 9.

Tuning (a) recruitment β; (b) abandonment γ; and (c) social inhibition ρ to maximize the reward rate (RR) in the indiscriminate stop-signalling model. The best tunings of parameters vary considerably with the recruitment being mostly low.

As in the single-feeder environments, we see a delineation between strategies optimized to slow/high-quality environments as opposed to rapid/low-quality environments. Weak recruitment β (figure 5a) and abandonment γ (figure 5b), and strong direct switching (figure 5c) yield the highest RRs in slow (large T) and high-quality (large ) environments. Recruitment β may be inessential since the food quality signals and are significantly different. Also, direct switching ρ provides strong adaptation to change. In fact, for virtually all environments, we found it was best for ρ to be as strong as possible. The strategy changes significantly when the environment is fast (small T) and low quality (small ), in which case abandonment γ should be strong, and in extreme cases direct switching ρ should be made weak (figure 5b,c). Changes in the optimal recruitment strength are less systematic, and there are stratified regions in which the best β can change significantly for small shifts in environmental parameters. Overall, a mixture of abandonment and direct switching is more effective in more difficult environments (lower T and ).

Direct switching does underperform self-inhibition in rapid environments (figure 4a), since the colony can forage more efficiently by keeping some bees uncommitted, and not risking the cost of foraging at the lower-yielding feeder. Strong self-inhibition ρ keeps more bees from foraging. Overall, both direct switching and self-inhibition can perform similarly, as recruitment interactions can be strengthened in self-inhibiting colonies, so more bees return to foraging after such inhibitory encounters (figure 4). This balances adaptivity, so the colony’s preferences change with the environment, and consensus, so the colony mostly builds up to forage at the better feeder given sufficient time. We now study this balance in each model using linearization techniques. Overall, these measures can account for discrepancies between the RR yields of colonies using different social inhibition strategies.

Linearization reveals strategy adaptivity and consensus

Each interaction mechanism differentially shapes both the fraction of bees that forage at the better feeder in the long-time limit (consensus ) and the rate at which this bound is approached (adaptivity λ). Focusing specifically on these measures, we demonstrate both how they shape foraging efficiency and how they distinguish each social inhibition strategy.

We leverage our approach developed for the single-feeder model, and consider linear approximations of equation (2.1) in the limit of long switching times T (see appendix B(d) and figure 10 in appendix C(c)). In the specific case , we can approximate the RR solely in terms of the consensus (long-term fraction of bees at the better feeder) and adaptivity λ (rate this fraction is approached): The RR J increases with consensus and adaptivity λ (figure 6a). Efficient colonies rapidly recruit a high fraction of the colony to the better feeder. Consensus and adaptivity are approximated in each model using linear stability (appendix B(b)). The impact of varying abandonment γ and social inhibition ρ on and λ is consistent with our optimality analysis of the full nonlinear model (figure 6b,c): social inhibition generates more robust switching between feeders than abandonment. While strengthening abandonment adaptivity γ can increase λ, it decreases consensus since it causes bees to become uncommitted (figure 6b). Such consensus–adaptivity trade-offs do not occur in most models, as social inhibition ρ is strengthened (see also figures 11 and 12 in appendix C(d)). Only indiscriminate stop signalling exhibits this behaviour (figure 6c), but the other three models (direct switching, discriminate stop signalling and self-inhibition) do not. Rather, consensus increases with social inhibition, while adaptivity can vary nonmonotonically (direct switching) or even decrease (self-inhibition). Overall, direct switching colonies attain the highest levels of consensus and adaptivity, consistent with our finding that it is the most robust model (figure 4).

Figure 10.

Linear approximation of the switch induced periodic solutions is generally accurate in (a) the single-feeder choice model (, T = 50 min, γ = 2.8 and β = 3) and (b) two-feeder choice model (direct switching here with model parameters , T = 100 min, γ = 0.01, β = 0.1 and ρ = 10). (c) However, when studying the discriminate stop-signalling model close to the saddle-node bifurcation, nonlinear effects shape the periodic solution of the full model equation (2.1) in ways that are not well approximated by the linearizations. Model parameters are , T = 100 min, γ = 0.01, β = 3.6 and ρ = 1.

Figure 6.

(a) Foraging yield varies with consensus () and adaptivity (λ) when T = 100 min. Trade-off between consensus and adaptivity in the linearized model for (b) abandonment, γ between [0, 20] min−1; (c) social inhibition, ρ between [0, 20] min−1. Switching time T = 100 min, food quality , and other parameters are fixed at their optimal level.

Figure 11.

Consensus and adaptivity λ computed as described in §§B(b) and B(d), as abandonment rate γ is increased between (along the direction of the arrows) for all models. Other parameters are fixed at their optimum level.

Figure 12.

Consensus and adaptivity λ computed as described in §§B(b) and B(d), as social inhibition strength ρ is increased between (along the direction of the arrows) for all models. Other parameters are fixed at their optimal level.

Direct switching sustains both high levels of consensus and adaptivity λ (figure 6c; see also figures 11 and 12 in appendix C(d)). The resulting colonies quickly discard their prior beliefs about the highest-yielding feeder, exhibiting leaky evidence accumulation [36]. On the other hand, strong abandonment γ (figure 6b) or indiscriminate stop signalling (figure 6c) increase adaptivity but limit consensus at the better feeder. Strengthening recruitment β leads to stronger consensus at the expense of adaptivity λ to environmental changes. Bee colonies probably use a combination of social inhibition and abandonment mechanisms [37], which would allow flexibility in managing consensus–adaptivity trade-offs in dynamic environments.

Discussion

Foraging animals constantly encounter temporal and spatial changes in their food supply [38]. The success of foraging animal groups thus depends on how efficiently they communicate and act upon environmental changes [39]. Our bee colony model analysis pinpoints specific social inhibition mechanisms that facilitate adaptation to changes in food availability and consolidate consensus at better feeders. If bees interact by direct switching, they can immediately update their foraging preference without requiring recruitment, keeping foragers active following environmental changes. Recruitment is less important to the foraging success of a colony in dynamic conditions; bees can initiate commitment via their own scouting behaviour. Individuals should balance their social and private information in an environment-dependent way to decide and forage most efficiently [40,41].

Efficient group decision-making combines individual private evidence accumulation and information sharing across the cohort [42]. However, in groups where social influence is strong, opinions generated from weak and potentially misleading private evidence can cascade through the collective, resulting in rapid but lower value decisions [10,43,44]. Our analysis makes these notions quantitatively concrete by associating the accuracy of the colony’s foraging decisions with the consensus fraction at the better feeder and the speed of decisions with the adaptivity or rate the colony approaches steady-state consensus (figure 6). The best foraging strategies balance these colony-level measures of decision efficiency. Social insects do appear to balance the speed and accuracy of decision to increase their rate of food intake [45,46], and collective tuning is probably influenced by individuals varying their response to social information.

We find that social recruitment can speed along initial foraging decisions, but it can limit adaptation to change. This is consistent with experimental studies that show a reduction in positive feedback can help collectives steer away from lower value decisions. For example, challenging environmental conditions (e.g. volatile and low food quality) are best managed by honeybee colonies whose individuals do not wait for recruitment but rely on their own individual scouting [27]. Ants encountering crowded environments tend to deposit less pheromone to keep their nest-mates from less-efficient foraging paths [47]. These experimental findings suggest social insects adapt to changing environmental conditions by limiting communication that promotes positive feedback [48]. Foragers must then be proactive in dynamic environments, since they cannot afford to wait for new social information [49]. Thus, the advantages of social learning depend strongly on environmental conditions [50].

In concert with a reduction in recruitment, we predict that honeybee colonies foraging in volatile environments will benefit from strengthening inhibitory mechanisms at the individual and group level. Bees enacting social inhibition dissuade their nest-mates from foraging at opposing feeders. We found the most efficient form of social inhibition is direct switching whereby bees flip the opinion of committed bees to their own opinion. So do honeybees use this mechanism in dynamic environments? Observations of bee colonies making nest site decisions show scouts directly switching their dance allegiance [28,51], but these events seem to be relatively rare in the static environments of typical nest site selection experiments [31]. Other forms of social inhibitory signals, especially stop signalling, appear to be used to promote consensus in nest decisions [10,26] and escaping predators while foraging [8,24]. Thus, the role and prevalence of social inhibition as a means for foraging adaptively in dynamic environments warrants further investigation.

Most work studying the effects of social inhibition on honeybee colony decisions has focused on swarms choosing a place to build a nest from sites whose qualities are fixed in time [26,28]. Social inhibition is needed in this context to promote consensus, generating a consistent opinion across the swarm and preventing the deadlock and group splitting. On the other hand, it is not immediately obvious that social inhibition would improve foraging if it primarily increases consensus, since colonies can obtain and store food even when foragers are split between multiple feeders, though stop signals can reduce crowding [25]. Nonetheless, we found that when the colony can rapidly switch opinion so nearly all bees agree to forage from the most profitable feeder, this does increase the nutrition yield of the colony overall. However, consensus is only advantageous in dynamic environments if it does not come at a cost to adaptivity: the opinion around which consensus is built should change with the environment.

Our simple parametrized model, developed from previously validated house-hunting models [26,28,29,32], is amenable to analysis and could be validated with time-series measurements from dynamic foraging experiments. Past experimental work focused on shorter time windows in which only a few switches in feeder quality occurred [16,20], which may account for the relatively slow adaptation of the colonies to environmental changes. We predict bees will slowly tune their social learning strategies to suit the volatility of the environment, but this could require several switch observations. Foraging tasks conducted within a laboratory could be controlled to track bee interactions over long time periods using newly developed automated monitoring techniques [52]. Our study also identifies key regions in parameter space in which different foraging strategies diverge in their performance, suggesting that placing colonies in rapid environments with relatively low food supplies will help distinguish which social communication mechanisms are being used.

Previous computational modelling studies of honeybee collective decisions primarily focused on groups solving house-hunting problems in static environments [26,29], emphasizing how social interactions shape the speed at which consensus is obtained within a collective. However, less attention has been paid to how such collectives must adapt to change, and how social communication determines group adaptivity. Some previous work has discussed the importance of uncommitted inspector bees in affording group adaptivity [20], but our work is the first to systematically compare how different forms of social communication [8,26,28,29] shape group adaptivity. Social communication by which one bee can switch the foraging preference of another appear to be most effective in providing groups with the ability to both build consensus and adaptive to change. Our findings are fairly robust to considerations of interaction heterogeneity within the colony (see appendix C(e) and figure 13). A colony whose bees have individualized rates of recruitment and abandonment exhibits slight decreases in consensus and adaptivity, but qualitatively the group still remained responsive to change.

Figure 13.

Heterogeneity in the recruitment and forgetting rates can decrease consensus and adaptivity. Consensus slightly decreases when simulating equation (B 1) with (a) weak heterogeneity β ∈ Γ (20, 0.1) and γ ∈ Γ (10, 0.1) (heavy lines) as opposed to (light lines) constant recruitment (β = 2) and forgetting (γ = 1) as in equation (2.1), and (b) consensus decreases more when more dramatic heterogeneity is considered: β ∈ Γ (1, 2) and γ ∈ Γ (0.5, 2). Adaptivity decreases with heterogeneity as well, as we see for (c) weak heterogeneity distribution, individuals still abandon relatively quickly after a change point, but (d) more slowly when rates are drawn from distribution with strong heterogeneity. Distributions are independent, so p(β, γ) = p(β)p(γ); switching period T = 100 min; other parameters α = 2, ρ = 1, and τ = 0.

There are a number of possible extensions of our work here. For instance, one could consider separate populations of scouts and foraging recruits as in some previous modelling studies [1,53]. Our analysis assumes bees can fluidly transition between scouting and foraging behaviour, as documented in several previous studies [54,55]. Overall, a strict and unchanging division of labour within the hive provides an incomplete description of colony organization. For instance, bees may switch to foraging when the environment demands it [56] or when socially signalled to do so [57], and thus a strict caste divide between scouts and recruits may be unrealistic [55]. Honeybees' roles appear to be strongly determined by the changing requirements of the colony, such as the influx or availability of nectar, rather than strictly due to some genetic predisposition [54,58]. Bees that scout and forage tend to be in the same life cycle phase, and as such are more amenable to temporal caste switching [59]. Such flexibility may even be a rule rather than exception to colony labour organization [60]. We could also have considered the effects of crowding at feeders [25], so nutrition yields would scale sublinearly with the fraction of bees at the feeder, possibly reordering the efficacy of social inhibition strategies.

Collective decision strategies and outcomes can depend on group size [61,62], though decision accuracy does not necessarily increase with group size [63]. We approximated bee colony dynamics using a population level model, which is the deterministic mean-field limit of a stochastic agent-based model [26]. Finite group size considerations would result in stochastic models, in which the same conditions can generate different colony dynamics [10]. The qualitative predictions of our mean-field model did not change dramatically when considering stochastic finite-size effects (see appendix C(f) and figure 14). However discriminate stop-signalling colonies exhibit bistable decision dynamics (figure 3d,f), so the stochasticity in the finite-sized model could allow colonies to break free from less-profitable feeders, similar to noise-driven escapes of particles in double potential well models [64]. Fluctuation-induced switching may thus provide an additional mechanism for flexible foraging [65,66], and would be an interesting extension of our present modelling work. Moreover, besides their importance to understanding decisions of biological collectives, our mathematical modelling results could inform efficient strategies for organizing distributed decision-making in inanimate groups, like swarm robotics and artificial communication networks [67,68].

Figure 14.

Mean and standard deviation of stochastic simulations of the single-feeder equation (B 2) and two-feeder equation (B 3) in the case of temporal switching of feeder quality occurring at T = 50 min intervals. In the single-feeder model, for (a) N = 100 and (b) N = 1000, near periodic switching of the mean trajectory of simulations (lines) is not far from the behaviour of the mean-field system equation (2.3). Other model parameters are , and . Standard deviations (shaded regions) decrease as the system size is increased. Similar trends are apparent in the statistics of the two-feeder finite-sized models with discriminate stop signalling (c) N = 100 and (d) N = 1000. Other model parameters are , , , , T = 100 min and τ = 0.

Table 1.

Model parameters for single-feeder equation (2.3) and two-feeder equation (2.1) foraging models.

parameter	description	numerical range	citation
α¯	quality of food source	[0.5, 20] M(mol l−1)	[7,20]
β	recruitment rate	O(10^−1−10^1) min−1	[1,26] supplement
γ	abandonment rate	O(10^−2−10^1) min−1	[26] supplement
ρ	rate of social inhibition	O(10^−1−10^1) min−1	[26] supplement
T	period of environment switch	1–200 min	[20]
τ	time delay for switch to be sensed	0.1×T min	—
c	cost of foraging	α¯2	—