## ABSTRACT

The collective transport of massive food items by ant teams is a striking example of biological cooperation, but it remains unclear how these decentralized teams coordinate to overcome the various challenges associated with transport. Previous research has focused on transport across horizontal surfaces and very shallow inclines, disregarding the complexity of natural foraging environments. In the ant *Oecophylla smaragdina*, prey are routinely carried up tree trunks to arboreal nests. Using this species, we induced collective transport over a variety of angled surfaces with varying prey masses to investigate how ants respond to inclines. We found that weight and incline pose qualitatively different challenges during transport. Prey were carried over vertical and inclined surfaces faster than across horizontal surfaces, even though inclines were associated with longer routes and a higher probability of dropping the load. This additional speed was associated with more transporters being allocated to loads on steeper inclines and not with the persistence of individual ants. Ant teams also regulated a stable prey delivery rate (rate of return per transporter) across all treatments. Our proposed constrained optimization model suggests a possible explanation for these results; theoretically, prey intake rate at the colony level is maximized when the allocation of transporters yields a similar prey delivery rate across loads.

## INTRODUCTION

Coordination allows groups of organisms to perform tasks that are insurmountable for individuals. For example, in some species of ants, workers will cooperatively transport food items that are too large for a single forager to carry. Cooperative transport requires that teams contend with many different challenges. Loads of highly variable size, shape and weight must be retrieved and carried around obstacles and over complex terrain (McCreery and Breed, 2014). Team members must take on appropriate roles and be prepared to change their behavior rapidly as conditions change (for example, when the load rotates or slips, or other ants join or leave the group). The team must solve these challenges without direction by a well-informed leader, instead relying on decentralized mechanisms of coordination that remain poorly understood.

Insight into the ants' behavior can be gained by examining how their transport performance scales with load size. Some group-raiding species (marauder ants and army ants) become more efficient at larger load sizes (Franks, 1986; Moffett, 1988a; Franks et al., 1999). This ‘super-efficiency’ can be quantified by the prey delivery rate (PDR), which is calculated as the load's mass multiplied by its velocity and divided by the number of active transporters. That is, PDR is an index of the average individual's contribution to the mass flux in a cooperative-transport team (Buffin and Pratt, 2016). Larger PDR values indicate that fewer ants are needed to carry the same object at the same speed. For army ants, at least, rising PDR with load size is attributable to their distinctive transport behavior, in which porters straddle loads and face in the same direction, resulting in reduced conflict and rotational forces (Franks et al., 2001). There may be a similar super-efficient characteristic for other ants with distinctive linear and branched cooperative-transport assemblages, such as some chain-forming ants of the ponerine genus *Leptogenys* (Peeters and De Greef, 2015; Mizuno et al., 2022), but this has not yet been studied. For most ants, however, PDR declines with load size (Buffin and Pratt, 2016), and teams (which tend to encircle loads) carrying larger loads move more slowly, involve more transporters, and may take a more sinuous path back to the nest (Detrain, 1990; Cerdá et al., 2009; Buffin and Pratt, 2016; McCreery et al., 2019). These deficiencies may reflect constraints imposed by how they carry loads and coordinate with one another, but the ways in which load size affects transport behavior remain poorly understood.

Even less attention has been paid to how the transport environment affects group performance. Most research has been conducted over artificial, featureless surfaces (Moffett, 1988b; Czaczkes et al., 2011; McCreery, 2017; Buffin et al., 2018). Several studies have examined how ant teams navigate around discrete obstacles (McCreery et al., 2016b; Gelblum et al., 2020), but these investigations have still been restricted to flat, horizontal planes (but see Qin et al., 2019, which focuses on how ants disassemble artificial loads on trees). In nature, ants are often seen transporting through leaf litter, across vines and up tree trunks. Transport teams in some species regularly scale sheer vertical surfaces (Wojtusiak et al., 1995) and even carry objects upside down along the underside of horizontal surfaces. Such inclines present several unique challenges – including supporting the load's mass against gravity and overcoming inevitable mistakes that can reverse progress when not quickly corrected. Cooperative-transport teams in many ant species ascend vertical surfaces in ways that show little obvious difference from transport across horizontal trajectories despite these challenges, and yet the coordinating mechanisms that allow them to do this remain unknown.

The differences between vertical and horizontal collective transport, as well as the implications of said differences on ant behavior, are not immediately clear. Previous studies present somewhat conflicting evidence about possible energetic costs. Among leaf-cutter ants, individual transport while ascending vertical surfaces seems to be more energy intensive than horizontal transport (Lewis et al., 2008), but in unladen carpenter ants, vertical locomotion is not more metabolically costly (Lipp et al., 2005).

Additionally, Endlein and Federle (2015) have shown that weaver ants employ different biomechanics when walking over various inclines, using either their arolia or tarsal claws. However, these studies only consider individual walking ants, and it is not clear whether these findings would translate to individuals coordinating motion in transport groups. Furthermore, vertical transport presents not only an energetic challenge but also a logistic one as mistakes (e.g. dropping loads) that would be minor on horizontal surfaces can be catastrophic on steep inclines. These mistakes may occur more frequently with teams, as the objects they carry could be significantly heavier than any one ant (or even a subset of the team) can carry. Moreover, other challenges such as increasing load mass may be exacerbated or compounded when transporting over vertical surfaces.

To better understand how ants that are effective at cooperative transport overcome the likely challenges unique to steep inclines, we focused our study on green weaver ants, *Oecophylla smaragdina* Fabricius 1775, a highly polydomous, arboreal-nesting species with very large colonies. *Oecophylla* are voracious hunters that can scavenge upon small lizards, mice and birds (Wojtusiak et al., 1995). They forage both among the trees and along the ground, and so they must regularly engage in both vertical and horizontal transport of these relatively large prey items. Furthermore, although these ants form conspicuous chain assemblages out of their bodies in other contexts (Carlesso et al., 2023; Carlesso and Reid, 2023; Lioni et al., 2001), they tend to cooperatively transport by encircling loads much like many other cooperatively transporting ant taxa. Given their ability to transport across inclines and collect prey of vastly different masses, *O. smaragdina* represents an ideal model of how transport efficiency is affected by the challenges posed by load mass, incline and their potential interaction. By presenting multiple challenges simultaneously, we can also better tease apart how the various subcomponent measures of group performance interact. Additionally, we can test whether vertical collective transport poses a logistical challenge; under what circumstances might slipping or falling backwards occur, and do the ants employ mechanisms to mitigate this risk? To investigate these questions, we offered field colonies loads of various mass across different inclines while capturing video of cooperative-transport bouts and measuring team performance. We then quantified the effects of incline, weight and their interaction on multiple aspects of transport performance.

## MATERIALS AND METHODS

### Brief summary

We induced weaver ant colonies to establish trails over experimental platforms with angles of inclination that were 0, 45 or 90 deg from horizontal. We presented these colonies with prey items that differed in mass, with treatments having nominal masses of 0.25, 1.25 or 2.25 g. We video recorded the platforms as ant teams collectively transported the loads toward their nests (where all recorded ant teams encircled loads; formation of pulling chains for load transport was not observed). From these video data, we tracked the position of the load and the number of ants engaged with the load over time. We also measured the duration for which a subset of randomly chosen individual ants remained engaged with the load. We then calculated multiple metrics of transport performance, including speed, path straightness and prey delivery rate, and we examined the effects of incline and load mass on these factors. Specific details of these methods are given below.

### Field site

Experiments were conducted during midday (10:00–15:00 h) between 16 October and 5 November 2018, in Kirwan, a suburb of Townsville in Queensland, Australia. We studied three polydomous colonies of *O. smaragdina*, with each possessing nests in multiple nearby trees (although data from one colony were excluded from final analysis; see below). We determined colony identity by manually swapping workers between trees and monitoring them for aggressive interactions. Colonies were approximately 200 m apart from one another at their closest points. The activity level of each nest – determined by visually assessing the rate of foraging trail traffic – was found to be approximately the same. All cooperative-transport events observed involved ants encircling loads; no chain formation was observed for pulling loads.

### Experimental details

Prior to any experimental manipulation, we chose trees with active ant trails within the territory of each colony. Adapting methods from Romeu-Dalmau et al. (2010), we wrapped the lower portion of the tree trunks in plastic sheets dusted with talcum powder. This disrupted the existing trails as ants would not cross the coated plastic. We then attached smooth cardboard bridges across these obstacles, and the ants' trails were quickly reestablished along these bridges.

The following day, we placed the 65 cm×35 cm experimental platforms (made from smooth cardboard layered over particle board panels) at the base of these trees at inclines of 0, 45 or 90 deg (*n*=36, 36 and 34, respectively) from horizontal (Fig. 1). We determined the exact angles using spirit levels and the iPhone 6s gyroscope. The cardboard bridges were adjusted so that the active trails would continue down the center of the platform and so that ants could not cross the plastic barrier without walking over the platform. The Australian and African species of *Oecophylla* can employ short- and long-range recruitment, and this setup encouraged ants to form trails over the platforms, thus minimizing recruitment latency (Hölldobler and Wilson, 1978; Hölldobler, 1983). We also used fresh cardboard platforms for each colony, each day. To control for differences in the initial ramp-up phases of recruitment and foraging, we placed previously freeze-killed crickets (∼0.25 g) at the far end of each platform before the experiment began. Experiments did not commence until the initial cricket had been transported to the nest and the trail had been thoroughly established.

We recorded transport in Cinematic 4K video with Panasonic GH4 cameras aimed and positioned orthogonally to the incline of the platforms at a distance of approximately 1 m, which rendered identical viewing angles regardless of platform angle. Our prey items were freeze-killed crickets of similar size, thawed and delimbed, and either unadorned or with one or two lead weights attached via hot glue as standard loads, similar to a previous study of collective transport in this genus (Lioni, 2000). This method allowed us to increase prey mass without inflating prey surface area, which could confound the experimental design by introducing additional ant attachment sites. The mass of these light, medium and heavy mass classes was approximately 0.25 g (±0.05 g), 1.25 g (±0.06 g) and 2.25 g (±0.07 g), respectively; we recorded the exact mass of each load before presenting it to the ants. Sample video clips of their transport behavior (for both horizontal and vertical surfaces) can be seen in Movies 1 and 2.

For each colony, we selected three inhabited trees with active foraging trails for our experiment. To remove the possible confounding effect of a particular tree, we presented each tree with each angled platform over three successive days. Each day of recording, we presented all trees with all three prey mass classes, randomly ordered, twice. After each load was transported to the end of the platform, we removed the attached ants from the load and from the experiment and then immediately returned the load for a second run. After its second run, the load was replaced with a new load from a different mass class, and the process was repeated. Details on the number of instances of transport recorded can be found in Table S1. Unfortunately, unexpected water damage warped our experimental platforms during 2 days of filming, so we did not include data from the affected colony in the final analysis.

### Data extraction

We extracted load trajectories and data on individual transporters from the videos. We converted the transport videos to 5 frames s^{−1} and imported them into Fiji v1.52 (Schindelin et al., 2012), where the position of the cricket load's head was manually tracked every 60 frames (12 s) (Fig. S1). No smoothing or filtering was applied to the displacement data. To control for differences in initial recruitment dynamics, we only analyzed data collected after the load had moved 5 cm. We considered runs as completed once the load reached the end of the platform nearest the nest.

#### Average displacement speed and average instantaneous speed

From the load trajectory data, we calculated two alternative measures of transportation speed. ‘Average displacement speed’ is the straight-line distance between the load's initial and final positions divided by the total time it took to move between them. ‘Average instantaneous speed’ represents the magnitude of the load's velocity in any direction between successive frames (tangent to the load's trajectory), averaged over the entire run. Average displacement speed and average instantaneous speed would be identical if a load were to move directly toward its ending position, but average displacement speed would be lower than average instantaneous speed if a load were to take a more circuitous path.

#### Backtracking

We occasionally observed loads slipping backwards and falling, and we wanted to determine what factors led to this behavior during transport runs. Because of the difficulty in accurately discriminating between slips caused by gravity and general backwards movement, we use the equivocal term ‘backtracking’ to refer to both. We calculated the amount of backtracking as the sum of distances covered when teams were moving further from the nest (and thus towards the ground) normalized by the total start-to-end distance the load moved during its run.

#### Straightness of the load's path

We calculated path straightness by dividing the distance between the load's initial and final positions by the total distance it traveled. For this calculation, total distance traveled included only motion with a nest-ward component. That is, we did not include segments classified as backtracking, to avoid confounding path straightness with backtracking events. Total distance did include recovery from backtracking, when the load was again carried nest-ward to the point at which it began to slip.

#### Group size and individual persistence

We recorded the number of transporters involved with each load over time. We considered an individual ant to have engaged with the load when the ant's mandibles were visually overlapping with the load for more than two consecutive video frames. We then manually recorded the number of ants engaged with the load every 120 frames (24 s) or, if the video was particularly long, a multiple of 120 frames that resulted in at least five records. Because transporter numbers tended to increase initially before plateauing, we used the median number of transporters over time for each run.

To determine how long individual ants remained engaged with the load, we first chose a random frame between one-quarter and three-quarters of the way through each video. A randomly chosen individual engaged with the load at this time-point was then manually followed frame by frame until the individual let go of the load and remained disengaged for two consecutive frames. This duration was recorded for at least one ant for each of the 106 recorded transport trials, but for nine of the runs every ant engaged with the load at the randomly chosen time-point was tracked, for a total of *n*=194 ants tracked. To account for this lack of independence, the trial was included as a random mixed effect in the model.

### Statistical analysis

All statistical analyses were completed using R version 3.6.1 (http://www.R-project.org/). Unless explicitly stated otherwise, all regression coefficients are standardized, as indicated with *.

#### Question 1: how do physical challenges affect transport efficiency?

The efficiency or ability of ant teams to collect massive food objects is often measured by PDR in ant collective transport studies (Traniello and Beshers, 1991; Buffin and Pratt, 2016). PDR is usually defined as the mass of the load being retrieved multiplied by its average displacement speed and divided by the number of ants transporting it. In large ant colonies with hundreds to thousands of foragers simultaneously retrieving prey items, the net intake rate per ant likely represents an ecologically relevant metric, and it allows for comparison across species as well as across load and group sizes.

Previous work has shown that the challenge of increasing load mass often decreases transport efficiency and that larger team sizes are associated with this decrease (Buffin and Pratt, 2016). Thus, we examined the effect of three potential predictor variables (incline angle, load mass and the median group size) on the PDR using linear mixed-effect models (LMM) with the random effects of date nested within colony ID. We applied a square root transform to the PDR values to make them more normally distributed, as confirmed by *Q*–*Q* plots and the Shapiro–Wilk test. We compared the full model containing all three predictors and their interactions with all simpler sub-models. Additionally, because we did not find any predictor to have a significant effect on PDR in any of the models, we also used a multi-model averaging approach to quantify possible effect sizes, employing the methods used in Camacho et al. (2018).

#### Question 2: how do more granular measures of group performance such as transportation speed, path straightness, degree of backtracking and group size respond to changes in substrate incline and load mass?

We used generalized linear mixed-effect models to analyze the effects of incline and load mass on average displacement speed, average instantaneous speed, the straightness of the path taken by the load and the median number of ants engaged with the load over each run. For each dependent variable, we used stepwise regression via backward elimination and likelihood ratio tests to determine the model which best matched the observations.

We log-transformed velocity and speed data to improve normality and reduce heteroscedasticity. After transformation, the *Q*–*Q* plots showed very slight deviations from normality, but because of the relatively large sample size (*n*=106), the increased homogeneity of variance among the residuals and the fact that Gaussian models are fairly robust to violations of normality (Knief and Forstmeier, 2021), we used the log-transformed speed and velocity data in the generalized linear mixed-effect models. Given that path straightness as a value is bounded between 0 and 1, we applied an arcsine square-root transform to the straightness data before analysis (Sokal and Rohlf, 1995).

To analyze how load mass and incline influence backtracking during transport, we used a hurdle mixed model from the R package *GLMMadaptive.* Specifically, a hurdle (or ‘two part’) model uses a standard linear mixed model for the log-transformed backtracking distance of the non-zero responses, as well as a logistic regression (LR) for whether any backtracking occurred at all. We included both incline and load mass as possible predictor variables for the linear and logistic components, and we used colony identity as a random effect in both components as well. We used stepwise regression via backward elimination and likelihood ratio tests to determine the model which best described the observed results.

#### Question 3: how do individual persistence and group size affect instantaneous movement speed?

After finding a statistically significant, positive association between steeper inclines and increased instantaneous movement speed, we used our data to test the validity of several hypotheses that could explain this counter-intuitive result. Specifically, we tested the (not mutually exclusive) hypotheses that the quality, i.e. persistence, of individual transporters and/or their quantity were responsible for this association.

##### Survival analysis

We examined how long individual ants remained locally engaged with loads via survival analysis. We used a mixed-effect Cox regression from the R package *coxme*, with incline and load mass as our predictor variables, and included the random effects of trial nested within date within colony identity. We used stepwise regression with likelihood ratio tests and backward elimination to compare a full model with both predictors and the set of simpler models.

##### Mediation analysis

To explore the underlying causal relationship between steeper slopes and load speed, we also employed a mediation model. Mediation analysis, a common tool in the social sciences, tests whether a third variable can explain (or mediate) the apparent effect of an independent variable on a dependent variable. We wanted to test whether incline's effect on load speed was mediated through transporter team size, which was larger on steeper slopes.

To model the relationship of group size, load mass and incline on average instantaneous load speed, we ran a full linear regression using all three predictors and then all simpler sub-models, determining the best predictor variable(s) based on the Bayesian information criterion (BIC). We tested the significance of this indirect effect using quasi-Bayesian procedures from the R package *mediation.* Unstandardized indirect effects were computed for the 1000 simulation samples.

## RESULTS

### Prey delivery rate

We found that none of our predictors – in any of our sub-models – had a significant effect on PDR (Fig. 2A). Furthermore, the potential effect size of each of our predictors was less than the effect on PDR of adding a single, theoretical lazy ant to the largest transport teams carrying the heaviest loads. That is, while transporting the heaviest loads, ants would continuously be engaging and disengaging from the load. We can calculate the maximum decrease in PDR that would result from the addition of a single, theoretical lazy ant to these already large (15+ ants) transport teams. This unhelpful individual's presence would decrease the PDR (which is calculated per capita) by 0.00161 g cm s^{−1} per ant, which is larger than the model-averaged effect sizes for any of our predictors. For example, the predicted difference in PDR between loads carried over vertical and horizontal surfaces is only 0.00154 g cm s^{−1} per ant. So, load mass, incline and the number of transporters have, at most, a biologically negligible effect on PDR, with such an effect being practically equivalent to zero.

### Component measures of group performance

#### Average displacement speed

Increasing load mass decreased the average displacement speed of transportation (GLMM, *=−0.33, *P<*0.001, *t*_{99}*=*−12.98, *R*^{2}*=*0.576), with medium loads moving at nearly half the speed of light loads (Fig. 3). On average, loads were transported from the beginning to the end of our platforms at velocities of 0.420, 0.125 and 0.0876 cm s^{−1} for the light, medium and heavy loads, respectively. However, the angle of inclination did not have a significant effect (Fig. 3). Likewise, there was no significant interaction between load mass and the angle of the experimental platforms on the average displacement speed.

#### Group size

Both steeper slopes (GLMM, *=0.30, *P<*0.001, *t*_{98}*=*6.89) and heavier loads (GLMM, *=0.82, *P<*0.001, *t*_{98}*=*18.6) increased the median number of transporters engaged with a load (Fig. 4; adjusted *R*^{2}=0.746 for best-fitting model), and there was not a significant interaction between the two factors. All things being equal, group size would be expected to increase from ∼10 ants on a horizontal platform to ∼13 on a vertical surface and from ∼7 ants on a 0.25 g load to ∼17 on a 2.25 g load.

#### Backtracking and slipping

Backtracking was more likely to happen when loads were heavier (zero-inflated LR component, *_{zero-inflated}=1.45, *P<*0.001, *z=*−4.52), and the distance backtracked was greater on steeper slopes (GLMM component, *=0.67, *P<*0.001, *z=*3.38). The mass of the load predicted the likelihood of whether backtracking occurred during transportation, with lighter loads being less likely to backtrack (adjusted *R*^{2}*=*0.187). However, the distance backtracked was significantly affected only by the inclination of the experimental platform; steeper slopes resulted in longer distances backtracked (Fig. 5).

#### Path straightness

Teams took straighter paths when transporting lighter loads (GLMM, *=−0.31, *P<*0.001, *t*_{98}*=*−3.46) and when crossing platforms that were less steep (GLMM, *=−0.30, *P<*0.005, *t*_{98}=−3.37) (see Fig. S1). However, we found no significant interaction between load mass and incline (Fig. 6; adjusted *R*^{2}*=*0.169 for best-fitting model). All things being equal, path straightness would be expected to decrease from ∼0.93 on a horizontal platform to ∼0.89 on a vertical surface and from ∼0.94 with a 0.25 g load to ∼0.88 with a 2.25 g load.

#### Instantaneous speed

To assess the speed teams can attain independent from the straightness of their path, we also calculated the average instantaneous speed – the moment-to-moment speed of the load in any direction – excluding backtracking events. Ant teams moved faster on inclines (GLMM, *=0.06, *P<*0.005, *t*_{98}*=*3.10) and slower while carrying heavier loads (GLMM, *=−0.34, *P<*0.001, *t*_{98}*=*−14.8), and we found no significant interaction between these factors (Fig. 7; adjusted *R*^{2}=0.661). However, because of the backtracking and longer paths taken on vertical surfaces, this increase in average instantaneous speed did not translate into a significantly faster average displacement speed from start to finish. All things being equal, average instantaneous speed would be expected to increase from ∼0.16 cm s^{−1} on a horizontal platform to ∼0.22 cm s^{−1} on a vertical surface and to decrease from ∼0.4 cm s^{−1} with a 0.25 g load to ∼0.1 cm s^{−1} with a 2.25 g load.

#### Quality versus quantity

The increase in instantaneous speed over steeper slopes could be explained by a change in the behavior of individual ants, a change in the group composition (e.g. a greater number of ants), or a combination of the two. Individual ants might remain engaged with the load longer on vertical surfaces. Theoretical and comparative work suggests that individual transporters' persistence with loads can lead to greater success during collective transportation (McCreery et al., 2016a; McCreery, 2017), and the weaver ants might grip the load tighter while on vertical surfaces, thus being better suited to maintain higher speeds. Although persistence is usually defined as a worker's fidelity to a particular direction while pulling on an immobile load, in our case, loads were being actively transported. Because the directional contribution of individual transporters could not be accurately estimated, we recorded how long they remained engaged with the load. Our measure of engagement is consequently analogous to a short-term version of McCreery's (2017) ‘total engagement effort’ measure.

Alternatively (or in addition), changes in velocity may be due to changes in team size; the slight increase in group size on steeper slopes could explain the faster movement. Thus, we used measurements of ant persistence and group size to test both of these two hypotheses, which are not incompatible with each other.

##### Individual engagement with the load

We found that ants did not stay engaged with the load longer on steeper inclines (Fig. 8A), suggesting that individual persistence is not one of the factors contributing to the observed increase in instantaneous speed on steeper slopes. However, the best model indicated that individual ants did remain engaged longer with lighter loads (Cox *=0.246, *P<*0.01, *z=*2.8), which may suggest that among these ants, persistence is impeded by physical challenges as opposed to a response to them. We found a 35.4% increase in the expected probability of disengaging relative to a 1 g increase in load mass (Fig. 8B; *R ^{2}=*0.0399).

##### Team size and load speed

Our models indicated that the incline of the experimental platform had a significant statistical effect on the average instantaneous speed during transport. However, this may not represent an immediate, causal link between the two factors. Incline also affected the number of transporters engaged with the load. Thus, the number of transporters could potentially be acting as a mediator, and the incline would only indirectly affect the load's speed via the number of ants involved.

As shown earlier, the regression of incline on the logarithm of average instantaneous speed, ignoring the mediator (number of transporters), was significant (GLMM, =0.00330, *P<*0.005, *t*_{98}*=*3.10). Likewise, the regression of incline on the mediator variable (number of transporters) was also significant (GLMM, =0.0376, *P<*0.001, *t*_{98}*=*6.89). Additionally, the mediation process showed that the mediator variable had a significant effect on the logged average instantaneous speed (GLMM, =0.0446, *P<*0.001, *t*_{97}*=*2.40). Finally, the mediation analyses revealed that, after controlling for the mediator (number of transporters), incline was not a significant predictor of average instantaneous speed (GLMM, =0.00164, *P=*0.199, *t*_{97}*=*1.29).

We tested the significance of this indirect effect using quasi-Bayesian procedures. Unstandardized indirect effects were computed for each of 1000 simulated samples, and the 95% confidence interval was computed by determining the indirect effects at the 2.5th and 97.5th percentiles. The bootstrapped unstandardized indirect effect was 0.0016, and the 95% confidence interval ranged from 0.000550 to 0.00348 (*P<*0.001). Thus, we found that the number of transporters fully mediated the relationship between incline and average instantaneous speed (Fig. 9).

## DISCUSSION

We investigated how weaver ant transporter teams responded to loads of varying mass across inclines of varying angles. Our goal was to understand how transport efficiency (PDR) changed when teams were faced with these challenges. We also wanted to know how other more granular measures of group performance – such as transport speed, path straightness, slipping and backtracking, and group size – interact to ultimately constitute the PDR.

### Constant prey delivery rate: mechanisms and function

Given previous studies on non-nomadic ant species, we expected that heavier loads (and likely steeper inclines) would decrease transport team efficiency. However, we found that, on average, weaver-ant transport teams maintained a nearly constant per-capita rate of return across our entire range of scenarios. From light loads carried over a flat plane to crickets weighing 9 times as much and being pulled up a shear wall, the differences in mean PDR were biologically negligible. This pattern of constant mean PDR can be visualized in the inversely proportional relationship between the average displacement speed of loads and their median mass per transporter (Fig. 2B), where the constant mean PDR is reflected by the constant area of a rectangle inscribed between the origin on the lower left and the inverse relationship on the upper right. Consequently, the higher the mass per ant (the greater the ‘width’ of the inscribed rectangle), the slower the load travels (the lower the ‘height’ of the inscribed rectangle). So, a constant PDR on average indicates a possible fundamental trade-off between these two quantities. We are not aware of any other work that has identified a similar trade-off, and better understanding the mechanisms and possible functions underlying this trade-off is a target for future investigations. In the meantime, we can only speculate on some possibilities for its significance.

The constant mean PDR that we observed may be an epiphenomenon arising from some biomechanical constraint. For example, if there is a physical upper bound on the power output (energy per unit time) that each ant can supply, an ant burdened by more weight may be unable to sustain high speeds, and so teams of heavily burdened ants will move more slowly. However, we know of no reason to expect that such a physical constraint would result in the perfect compensation of speed and mass per ant that would lead to a constant mean PDR over so many different contexts. In other words, if the dashed curve in Fig. 2B was any other shape (e.g. if it had a linear decline rather than a hyperbolic one), then the mean PDR would vary across contexts. Furthermore, although mean PDR is constant across contexts, the PDR for any particular transport team does vary relative to that of other transport teams, which challenges a constraint-based explanation and suggests a possible role of target PDR seeking instead. Regardless of what the underlying proximate mechanism is, there may be some value in considering that PDR regulation may be adaptive, and this function ultimately is selecting for this behavior.

*n*possible prey items to allocate ants to and each item

*i*has a mass

*m*and an allocation

_{i}*x*of ants to it, causing the prey item to move at speed

_{i}*s*(

_{i}*x*). The value of each prey item is the product of its mass and the speed at which it can be secured within the colony (i.e. the PDR for the particular prey item), and

_{i}*U*models the value across all prey items currently being transported (i.e. the sum of all mass–speed products). Fitness maximization would then be modeled as maximizing

*U*over the different possible allocations of ants to prey items. However, because there is a finite number of ants (or, equivalently, a finite budget of energy or time for foraging), then all allocations must satisfy the constraint:where

*N*is the total number of foragers available to allocate. This constraint creates a trade-off to allocating to one prey item over another, as joining one prey team may make a larger impact on

*U*than joining another. Alternatively, the problem could be posed as minimizing an unlimited number of foraging ants so long as the colony-level utility

*U*meets a specific constraint necessary to meet colony energy or nutrient demands (i.e.

*U*≥

*U*

_{0}). Constrained optimization problems like both of these can be solved with the method of Lagrange multipliers (Beavis and Dobbs, 1990). Coincidentally, the solution to both formulations (i.e. maximizing returns subject to worker constraint or minimizing workers subject to demand constraint) satisfies the equimarginal matching condition that:However, d

*U*/d

*x*is exactly the PDR associated with prey item

_{i}*i*, and so the optimality condition is exactly a statement that the PDR should be equal across prey items at the optimal allocation. Thus, fitness maximization over a finite allocation of collective transporters predicts that PDR should ideally be constant across all transported items if regulation is perfect.

Drawing analogies to classical social foraging theory, this equal-PDR result is akin to the equilibrium solution of an ideal free distribution (IFD; Fretwell and Lucas, 1969) where the PDR plays the same role as patch suitability – ants allocate to different patches (prey items) until there is no strong difference between suitability across those patches. If the suitability of a prey item is high, then it is either moving relatively fast or is relatively massive compared with others with the same number of ants attached. Consequently, high suitability prey items will attract more individuals to join. Alternatively, low suitability prey items are those that are either moving relatively slow or are relatively small compared with other items with the same number of porters. These low suitability items will not attract additional porters, and some of the existing porters may switch to higher suitability items. This process equilibrates to an allocation where all items have the same suitability, which represents a kind of opportunity cost for an individual to make a unilateral change from its current position. In the IFD, these suitability functions represent proxies for rewards to each individual being allocated. Here, the individual-level suitability functions are the marginal returns of the colony-level utility, which is consistent with an inclusive-fitness perspective on IFD allocations.

This result is also analogous to the marginal value theorem (MVT) and thus can be viewed through the lens of optimal foraging theory (OFT) (Charnov, 1976; Stephens and Krebs, 1987). In OFT, the focus is on how a single forager maximizes her rate of gain by allocating a minimal time budget across a portfolio of different patches. Whereas in OFT, an individual can increase the time spent in a good patch (thus taking time away from other patches), the ants can increase the number of individuals allocated to a good prey item (thus taking ants away from other prey items). The MVT states that individuals should continue allocating time to a patch until the marginal return from the next small unit of time equals the marginal return to allocating that same small unit of time to any other patch. Similarly, the equal-PDR result states that more individuals should allocate to a prey item until the marginal value of adding an individual to the prey item is equal to the marginal value of adding an individual to any other prey item.

Both the IFD and MVT interpretations of the equal-PDR result suggest that colonies with different numbers of individuals (or other similar constraints, such as different macronutrient regulation requirements) will converge on PDR values within a colony that will differ from equilibrium PDR values for other colonies. In other words, if fitness functions are evaluated at the level of a colony, colonies with more foragers (or different overall energetic demand constraints) should allocate to lower PDR levels in the same way as a single forager experiencing a greater time between patches in OFT will tend to spend longer in each patch at the MVT equilibrium. Testing the agreement in PDR across different colonies is thus an important area of future research to validate these implied connections among foraging theories and collective transport.

### Constituent measures of group performance

We found that the challenge posed by steeper inclines qualitatively differed from that of increasing load mass. We also found no evidence of their interaction effects between mass and incline. As with heavier loads, teams transporting over steeper slopes took more sinuous trajectories and had more ants engaged with the loads. But even though vertical surfaces were also associated with more backtracking (slipping), loads were transported up these slopes in the same amount of time. In agreement with this result, similar dynamics have been incidentally observed in the African species of weaver ant, *Oecophylla longinoda*; an unpublished study noted no difference between collective transport displacement speeds over horizontal and 35 deg inclines (Lioni, 2000). Additionally, higher variation in average displacement speed among light loads likely results from the regulation of a constant PDR. When mass per ant is low (as with lighter loads) small variations in the mass per ant should produce large variations in speed (as per the hyperbolic dashed line in Fig. 2B). However, unmeasured individual variability among transporters would also exert an outsized influence when groups are smaller.

### Faster average instantaneous speeds on vertical surfaces

Contrary to the conceptual model that incline should have a similar effect to mass, we found that loads were dragged significantly faster across vertical surfaces than over horizontal ones. We investigated two possible explanations by collecting further data from our recordings. Firstly, we hypothesized that the effort individual ants exert might be greater on vertical surfaces. McCreery (2017) suggested that longer engagement and persistence with loads correlates with more successful collective transport. Thus, if individual weaver ants were more engaged on steeper inclines, then the resulting increase in time engaged with the load could lead to more successful collective transport on those vertical surfaces. Secondly, we hypothesized that the increased group sizes on vertical surfaces might lead to faster transport. Steeper slopes had larger team sizes, and larger team sizes might be the causal reason why loads moved more quickly.

Our results suggest that the quantity of transporters (and not the persistence of individuals) is likely the direct reason for faster movement on vertical surfaces. A possible alternative explanation is that prey loads have less frictional interaction with the substrate on vertical surfaces; however, the biomechanics of collective transport are more complicated than they might first appear. Ants climb by pulling into the walls (Endlein and Federle, 2015), and so any load being lifted by a tensile force will also experience a concomitant force pressing it towards the wall (unless ants underneath the load are actively pushing it off the substrate). Regardless, it is not clear how this vertical condition leads to a larger number of ants in the transport team and why each team reaches the size that it does.

Our study raises many questions about the coordination of collective prey transport, both vertical and horizontal. Although the ultimate explanation for the weaver ants' constant PDR might involve an optimized allocation of transport effort, the mechanisms behind this dynamic allocation are unclear. How are individual ants able to assess the marginal utility increase from engaging with a given load? We speculate that potential transporters may be able to measure the resistance and/or change in speed that results from an initial tug on a load. Previous studies have shown that some ant species will frequently over-recruit nestmates to pinned prey items, presumably because scouts are modulating recruitment based on the tractive resistance of the potential load (Hölldobler, 1983; Detrain and Deneubourg, 1997). We propose that a similar process may be used to assess loads during transport; if a moving load offers little resistance when pulled upon by a new ant, that potential transporter may be superfluous and not contribute to increases in speed (thus leading to an overall decrease in PDR if she attaches). Alternatively, the speed (or change in speed) of a load may also be important for this behavior. Swiftly moving, light loads may be less attractive to unengaged recruits, while slower, heavier loads may be more attractive (even if workers disengage at higher rates from heavier loads).

### Future directions

This is the first study to investigate collective transport up vertical surfaces, and our findings suggest many potential avenues of further research, both for our system specifically and more generally across ant species. For example, the mere fact that transport efficiency did not decrease across any of our treatments suggests that the challenges given to these weaver ants were surmounted without detriment. The extent to which PDR can be regulated over a wider range of parameters remains an open question.

Following the example of a previous study of horizontal collective transport in weaver ants (Lioni, 2000), we weighted our prey crickets with lead sinkers, placed the loads in high-traffic areas and observed transport over relatively short distances. Our heaviest loads were much more dense than realistic prey items, and recent work has shown that load size and mass have different effects on transport in other ant species (McCreery et al., 2019). Our small but heavy crickets may have artificially limited the number of transporters that could easily engage with the load or, alternatively, decreased the usual friction for such a weight.

By placing the loads on pre-existing ant trails, we provided a steady stream of potential recruits that would not necessarily be present during normal prey retrieval. Although this choice decreased recruitment latency in our study, transport events further from the nest may exhibit different dynamics. We examined only the first ∼50 cm of collective transport, but the full journey is expected to cover several meters. Our smaller sample is similar to that in other studies and thus allows useful comparison of metrics such as PDR. However, another study of horizontal collective transport showed that load speed may evolve non-linearly over time (Lioni, 2000). Future work may benefit from considering the entire effort from the site of prey capture to the nest.

Lastly, there are other types of inclined surfaces that we did not test. Weaver ants will routinely transport prey along the underside of surfaces, and performance could change whether the teams are transporting loads up or down an incline. We also did not test how teams respond to transitions between horizontal and inclined surfaces; such transitions are common in nature and might pose challenges in and of themselves.

## Acknowledgements

We would like to thank Kelly S. O'Meara for her help in recording the transport events and the support she offered in the field. We also appreciate the helpful comments from three anonymous referees on the original submission.

## Footnotes

**Author contributions**

Conceptualization: A.T.B., T.P.P., S.C.P., C.R.R.; Methodology: A.T.B., T.P.P., S.C.P., C.R.R.; Software: A.T.B., C.R.R.; Validation: A.T.B.; Formal analysis: A.T.B., T.P.P., S.C.P., C.R.R.; Investigation: A.T.B., T.P.P., S.C.P., C.R.R.; Resources: A.T.B., T.P.P., S.C.P., C.R.R.; Data curation: A.T.B.; Writing - original draft: A.T.B., T.P.P., C.R.R.; Writing - review & editing: A.T.B., T.P.P., S.C.P., C.R.R.; Visualization: A.T.B.; Supervision: A.T.B., T.P.P., S.C.P., C.R.R.; Project administration: A.T.B., T.P.P., S.C.P., C.R.R.; Funding acquisition: A.T.B., T.P.P., S.C.P., C.R.R.

**Funding**

This work was supported by an Australian Government Department of Education and Training Endeavour Research Fellowship (to A.T.B.), a National Science Foundation Graduate Research Opportunities Worldwide grant (to A.T.B.) and an Australian Research Council Discovery Early Career Researcher Award DE190101513 (to C.R.R.).

**Data availability**

All relevant data can be found within the article and its supplementary information.

## References

*Optimization and Stability Theory for Economic Analysis*

*Novomessor cockerelli*

*Insectes Soc.*

*Novomessor cockerelli*

*PLoS ONE*

*Integr. Zool.*

*Myrmecol. News*

*Proc. Natl. Acad. Sci. USA*

*Aphaenogaster senilis*

*Behav. Ecol. Sociobiol.*

*Am. Nat.*

*Pheidole oxyops*

*Insectes Soc.*

*Pheidole pallidula*

*Insectes Soc.*

*Pheidole pallidula*: a key for understanding decision-making systems in ants

*Anim. Behav.*

*PLoS ONE*

*Eciton burchelli*, Hymenoptera: Formicidae)

*Behav. Ecol. Sociobiol.*

*Proc. R. Soc. B*

*Anim. Behav.*

*Acta Biotheor.*

*eLife*

*Oecophylla smaragdina*)

*Biotropica*

*Oecophylla longinoda*(Latreille) (Hymenoptera: Formicidae)

*Behav. Ecol. Sociobiol.*

*Behav. Res. Methods*

*Behav. Ecol.*

*Oecophylla longinoda*

*.*

*J. Insect Behav.*

*Camponotus*

*J. Exp. Biol.*

*Insectes Soc.*

*Insectes Soc.*

*PLoS ONE*

*J. Exp. Biol.*

*J. Exp. Biol.*

*Leptogenys cyanicatena*(Formicidae: Ponerinae) in northern Thailand

*Asian Myrmecol.*

*Natl Geogr. Res. Explor.*

*Pheidologeton silenus*(Hymenoptera: Formicidae: Myrmicinae)

*Ann. Entomol. Soc. Am.*

*Leptogenys*ants from Cambodia

*Insectes Soc.*

*Sci. Rep.*

*Methods Ecol. Evol.*

*Nat. Methods*

*Biometry: The Principles and Practice of Statistics in Biological Research*

*Foraging Theory*

*Formica schaufussi*

*Behav. Ecol. Sociobiol.*

*Oecophylla longinoda*(Latreille 1802)

*Trop. Zool.*

**Competing interests**

The authors declare no competing or financial interests.