Experimental design and trajectory characteristics

Ring-shaped food items were placed several metres away from a P. longicornis nest in the field. Following discovery of the food and a recruitment phase, the ants commenced cooperative transport (Fig. 1a; Methods; Supplementary Note 1). We repeatedly filmed this behaviour over a distance of ∼1 m (ca. 500 ant lengths) and extracted the positions (Fig. 1b) and angular orientations of the load from a total trajectory length of over 70 m. We also measured the positions of all ants around the object as well as the angles between the carrying ants’ body axis and the object radius (Fig. 1c; Methods; Supplementary Movie 1). We further logged all events of ant attachment to and detachment from the load.

Figure 1: Cooperative transport. (a) A team of P. longicornis ants retrieving a food item ∼350 times the mass of a single ant. (b) Sample nest-bound load trajectories of ∼1-m length each. Scale bar, 10 cm. (c) Snapshot from a tracked movie. Carriers are enumerated in yellow and non-carriers in white. The recent trajectories of different ants are marked by different colours. The object’s direction of movement is depicted by the blue arrow. Scale bar, 2 cm. (d) The natural variability in the number of carriers and their distribution around the load were used to extract median load speed as a function of either total ant number (orange line) or difference in ant occupancy between the leading and the trailing edge (pink line). The green line depicts the efficiency of the carrying in terms of the fraction of energy devoted to rotation. Error bars are the s.d. of a distribution of medians calculated for 1,000 samples bootstrapped from the data. N=637,696 Frames. Full size image

Focusing on global features of the load trajectory, we find that while median speed increases linearly with number of carrying ants (up to 15 attached ants), it does not strongly depend on how these ants are distributed around the object (Fig. 1d, orange and pink correspondingly; see Supplementary Note 2). Furthermore, the relative amount of energy wasted on rotating the object rather than translating it decreases with number of carrying ants (Fig. 1d, green). These properties indicate a high level of conformism that enables all attached ants to efficiently align their efforts. This result stands in agreement with cooperative carrying in other ant species1,2. In contrast, while the trajectories generally head in the correct nest-bound direction they, nevertheless, exhibit substantial sinuosity, looping and detours (Fig. 1b).

Load steering

Understanding load steering is facilitated by the simple mechanical nature of the system. First, whatever the sources of information may be24,25, the movement of the rigid load is determined by the sum of forces and torques applied by the carrying ants. Overall forces dictate linear load velocity and angular speed, both of which are experimentally measureable. Second, the forces that ants can exert on the load are constrained by their anatomy: ants can either lift objects or apply pulling forces that are, in general, aligned with their body axis26, while pushing is rare to absent27 (Supplementary Note 3; Supplementary Fig. 1). Finally, the pulling forces applied by a single ant are of the order of 0.1 mN (Supplementary Note 4).

There are many ways in which a given net force may be distributed between the individual ants. In one extreme case, all ants pull with equal forces and load directionality is set by the angular distribution of attached ants along its edges. However, we find minimal correlation between ant angular distribution and direction of load movement (Pearson Correlation: r=0.104, t-test: P<10−13, N=5,591 frames from 47 experiments; see Supplementary Note 5 and Supplementary Fig. 2a). This tug-of-war assumption is further incompatible with the linear, rather than square root, rise in the median load velocity with the number of attached ants (Fig. 1d).

At the other end of the spectrum is the case where, although all ants can apply a force, they are actually enslaved by a single-informed ant that guides the entire motion. Indeed, in most of our experiments, we could identify at least one ant that acted as a carrier for the full duration of the retrieval. However, the distribution of correlations between the orientations of these persistently attached ants with the direction of motion of the cargo cannot be distinguished from that of non-persistent ants (t-test: P=0.6691, N persistent =14 ants, N non-persistent =186 ants from five experiments; see Supplementary Note 5 and Supplementary Fig. 2b).

We therefore conclude that the collective movement does not arise neither from a wisdom-of-the-crowds type averaging over all opinions nor by the continuous leadership of any single individual. Next, we aim at examining how the total force applied to the load is distributed among the carrying ants.

Transient influence of newly attached ants

Freely moving ants are well informed of the correct nest-bound direction (Supplementary Note 6; Supplementary Fig. 3a). When such ants attach to the load they steer it so that it moves more accurately towards the nest. Quantitatively, we found that within the several seconds that follow their attachment, ants inject about 0.5 bits of directional information into the system (see Fig. 2a, Supplementary Note 6 and Supplementary Fig. 3b). This causal effect implies that newly attached ants adopt an influential role.

Figure 2: Transient guidance. (a) The information in the angular spread of the load’s direction of motion immediately following the attachment of a new ant at t=0 (N=134 attachments). Errors were calculated from the entropy of artificially generated histograms with added binomial noise. (b) A half-polar histogram of the angles between the attachment/detachment point of an ant and the change in velocity (relative impact direction) that follows different events (N=252). (c) Relative impact direction as a function of time (blue line N=134 attachments) and difference between distributions of time since attachment of ants in the leading and trailing edge of the load (turquoise line). The insets illustrate this process for a sample newly attached ant (marked by a yellow circle). Load velocity (green arrow) and the acceleration caused by this ant (dashed arrow) are overlaid. (d) An example of a series of switches between steering ants along a trajectory. Overlaid colours mark trajectory segments where different ants steered the load. Scale bar, 10 cm. e) Mean magnitude of velocity change caused by newly attached ants (denoted by on y axis) as a function of number of ants already attached (N=134 attachments). Error bars are standard error of the mean. Full size image

The peak in directional accuracy, evident in Fig. 2a, implies that carrying ants are less informed than newly attached ants. The quick deterioration of accuracy after its initial rise shows that while newly attached ants have a transient positive influence, their directional knowledge rapidly decreases following the time of attachment (Fig. 2a). A likely explanation to this effect is obstruction of ants’ antennae by the large load (these ants do not strongly rely on vision for their navigation; see Supplementary Note 7 and Supplementary Fig. 4). The timescale of a newly attached ant’s influence is much shorter than the entire duration of the transport, which is qualitatively different from more stable forms of influence observed in systems with distinct leadership9,28,29. Finally, note that the high variation of directional accuracy in time (Fig. 2a) cannot be explained by a model in which carrying ants continuously reorient the load similar to the way a single ant follows a pheromone trail30.

Microscopically, we estimate the effect of an individual carrier by the change of load speed immediately following her attachment or detachment. In general, we find that ants at the leading edge of the object (Fig. 1c) tend to pull (evident by the recoil of the load on detachment, Fig. 2b, red), while those at the back assist the motion by lifting (see the non-specific direction change on detachment; Fig. 2b, green). Considering forces together with the ants’ angular locations on the load and their carrying durations allows us to illustrate the typical time course of a carrier: on attachment, a new carrier tends to change the load’s velocity towards her own pulling direction (Fig. 2b, blue). This influence lasts for several seconds (blue line, Fig. 2c) in agreement with the timescale of the initial improvement in load directional accuracy (Fig. 2a). The initial steering works to bring the carrier to the leading edge of the load (Fig. 2c, turquoise line), where she remains for about 20 s and continues pulling. At longer times that carrier’s angular position becomes indistinguishable from that of any other ant (Fig. 2c, turquoise line) and the forces she applies decrease (Supplementary Note 8; Supplementary Fig. 5). In particular, the ant often ends up at the trailing edge of the load from where she cannot guide the motion. This provides strong evidence for the fact that all useful steering is provided by newly attached ants.

Multiplying the steering timescale of 5–20 s (Fig. 2a,c) by the mean attachment rate (Supplementary Note 9), we estimate the number of mean concurrent steering ants at 0.35–1.4. Figure 2d illustrates changes between these steering ants along one trajectory and demonstrates how a high turnover can bridge the scale gap between that of seconds, relevant to the effect of a single ant, and that of minutes, which characterizes the duration of the collective transport process (see Supplementary Note 10). The ants are able to compensate for the scarcity of information imposed by the small number of concurrent steering ants by maintaining high responsiveness to these ants, even as the number of carriers increases (Fig. 2e).

Although our results demonstrate that, over time, a large number of ants steer the collective motion, we do not claim that all ants are equal in their susceptibility to guide the motion or the degree of influence that they exert. Figure 2d shows that following attachment, some ants guide the motion for a longer duration than others. It is also possible that by detaching from the load and then reattaching to it an ant may extend her influence even further (examples for this are evident in Fig. 2d). It has been previously shown that the first Formica schaufussi ants to locate food are crucial for the transport process29. Differently, cooperative transport in P. longicornis relies on a larger number of transient steering ants and in Supplementary Note 11, we follow the recruitment process in this species and discuss its relevance to the subsequent transport (also see Supplementary Fig. 6).

Finally, the carrying ants’ susceptibility to influence (Fig. 2a,e) seems to contradict the conformism7 that allows the ants to coordinate their forces and achieve high speeds (Fig. 1d). In the next section, we present a theoretical model that describes the cooperative transport and use it to reconcile this seeming contradiction.

Theoretical model

On the basis of the experimental properties outlined above, we constructed a theoretical model as specified in Fig. 3a, Supplementary Notes 12–14, Supplementary Fig. 7 and the Methods section. In short, the model is based on the minimal assumption that carriers interact uniquely through local forces transmitted to them by the load31. Informed ants are assumed to ignore these forces and attempt to pull the load in the correct nest-bound direction. Our experimental data suggests that the information held by ants deteriorates after they had been attached for a certain period (Fig. 2a,c). For simplicity, the model assumes that these ants become completely uninformed.

Figure 3: Microscopic model. (a) Model sketch including the possible transitions for non-informed individual ants. (b–e) The four model parameters were set by fitting experimental data of b. The distribution of the object’s velocity (projected on an arbitrary direction) in periods of continuous motion (N=56,030 frames). (c) Correlation distance functions (N=17 trajectories). (d) Median speed (N=56,030 frames). (e) Median angular speed (N=56,030 frames). In each of these panels, the coloured lines represent the experimental data for ants transporting a load in the absence of informed ants. The solid black lines denote the results of our model. Error bars in c are the maxima and minima of correlation functions produced by partitioning the data into four parts. Error bars in d and e are the s.d. of a distribution of medians calculated for 1,000 samples bootstrapped from the data. Full size image

Non-informed ants either try to align their pull with the force vector on the load (but align against the local torque) or simply lift the object to decrease friction. An uninformed ant’s decision of which role to perform is random and is biased by her alignment with the direction of the centre-of-mass force vector such that ants in the leading edge tend to pull (and ants in the rear tend to lift). In this way, the roles of the ants depend on their location around the load with respect to the direction of motion. An individuality parameter, F ind , determines the tendency of non-informed ants to randomly switch between pulling and lifting. A low individuality value indicates that an ant's decision is governed by her alignment with the group while a high value indicates random role switching, irrespective of the pulling force of the group. We denote the basal rate of this switching by K c . In addition, ants can attach or detach from the load. Finally, the object’s speed and angular velocity were taken to be linearly correlated with the total force and torque as applied by all ants (Supplementary Note 12). The mathematical form of our model is motivated by models of collective transport on the subcellular32 and single-cell33 levels. While the population of ants may have a distribution of internal properties34, here we consider, for simplicity, a population of identical ants (for the behaviour of populations with non-uniform F ind ; see Supplementary Note 15 and Supplementary Figs 8–9); in particular, the forces applied by informed and uninformed ants are taken to be equal.

We begin by comparing the model with the more elementary experimental condition from which informed ants are absent. This condition was achieved by carefully transferring the cargo along with the attached ants about 75 cm away from its initial location to a nearby clean board. Note that the relocation itself does not significantly affect the transport behaviour (Supplementary Movie 2). Under these circumstances, the ants-cargo system exhibits a persistent random walk (Fig. 3b,c; Supplementary Note 16; Supplementary Fig. 10). The model has four free parameters (see Methods): two (F ind , K c ) are associated with the ants’ decision-making process, while the other two are related to the mechanics of the specific ant-load system. We adjusted these parameters (Methods; Supplementary Notes 13,17 and 18; Supplementary Figs 7 and 11) to fit a large number of experimentally measured features (Fig. 3b–e). The good agreement demonstrates that the observed behaviours can be reconciled with a model that is based solely on mechanical information transfer; that is, the newly attached ant exerts its influence by causing a transient disturbance in the total force vector to which the rest of the ants react, all this without an active signalling mechanism14. Particularly, any form of influence must be implicit and is similar, in this sense, to the implicit influence that characterizes an effective leadership13. The sensitivity of the calculation to the chosen values of the free parameters is shown in Supplementary Note 19 and Supplementary Fig. 12.

Balancing individuality and conformism

In the simulation, we define the response of the system to an informed, newly attached ant (Methods; Supplementary Notes 15 and 20) as the distance that the load travels towards the nest in the characteristic time between two consecutive attachments. Following Fig. 2a,c, we assume that an informed ant disorients, turning into an uniformed ant on a timescale of 10 s. This simple model assumes a discrete (but stochastic) process of forgetting. In Supplementary Note 15, we show that a more gradual forgetting process gives essentially the same behaviour, even when combined with a value of F ind that is not constant over all ants (Supplementary Figs 9 and 13).

We fix three of the four free model parameters and check for possible optimality in terms of system response as a function of F ind . We find that both complete conformity (small values of F ind ) and strong individualism (large values of F ind ) reduce the effectiveness of a newly attached ant in steering the load (Fig. 4a). The fitted value of F ind lies between these two extremes and this suggests that the ants operate in the transition region between strong and weak conformity, possibly to optimize their responsiveness to a limited influx of information (Fig. 4a, upper left inset). In addition, we find that the working regime of the ants is such that the velocity distribution of the load lies in the transition region between unimodal (tug of war or random walk) and bimodal (persistent motion) behaviours (insets of Fig. 4a).

Figure 4: Group optimality. (a,b) Simulation data of the response of the object to a single attachment of a single-knowledgeable ant as a function of the individuality parameter F ind (a) or of the object’s radius (b). Insets depict velocity component distributions for small (orange) and large (blue) values of F ind as well as its fitted value (pink). Upper left inset depicts trajectories (all starting at the yellow dot, colour coded as before) that take into account the continual arrival of informed ants (scale bars represent 10 cm). The pink dot in b marks the radius of the experimental load. (c,d) Exact solution of the Ising spin model. (c) Normalized (dimensionless) system response as a function of the individuality parameter F ind . The blue curve marks the short-term response to a newly attached ant and the red curve the mean-field susceptibility, which diverges at the critical point. (d) Normalized (dimensionless) short-term response to a newly attached ant as a function of the mean number of ants attached to the load (a proxy for load size). Dotted lines mark the critical transition points. (e–g) Experimental verification. (e) Mean absolute curvature of trajectories of objects of different size (total N=90). Synthetic and non-synthetic materials that were used for the small item exhibited similar curvatures (medians: synthetic: 8.84, non-synthetic: 6.53, unpaired two-sample t-test: P=0.6285, N=9) and were therefore pooled together. Thus, the effect presented in the figure is a size effect that cannot be attributed to load substance composition. Inset: mean curvature of simulated tracks of objects of different sizes (calculated on clean board conditions). (f) Time spent at ν>75% transport speed for one (green) versus 2–4 (blue) ants carrying a small load (total N=20). (g) Top: time to negotiate an obstacle (t-test: P<0.01) and (bottom) the maximal backwards displacement (t-test: P<0.0001) towards a successful crossing of a U-shaped block (which required 5-cm backtracking) for two load sizes (total N=11). Scale bar, 10 cm. Full size image

Our model suggests a correspondence between load size and the ants’ individuality (Supplementary Note 21). Namely, larger loads correspond to more conformist ants and vice versa. This behaviour naturally arises from the mean-field nature of this system (see below). We fixed F ind to its fitted value and simulated ring-shaped objects of different radii (such that the mass per ant remains constant) to calculate the normalized response function to the attachment of a new ant. We find an optimal load size regime which is on the order of 1 cm (Fig. 4b). This scale is compatible with natural prey and nest entrance dimensions.

Finite-size criticality

Figure 4a is reminiscent of the divergence of susceptibility near a second-order phase transition, and thereby suggests an analogy between F ind and temperature, and between the response to a newly attached ant and susceptibility. To ascertain this analogy, we constructed a simple Ising model in which the group moves along one dimension (see Methods, Supplementary Notes 22–24 and Supplementary Fig. 14). This model is analytically tractable and its equilibrium state describes the centre-of-mass motion of the load. Note that although the model describes a one-dimensional (1D) motion the spin connectivity pattern has no spatial dimensionality (all spins interact with each other over a complete graph).

The spins in this Ising model denote the ants’ roles: +1 for puller and −1 for lifter, while the external field is analogous to the force applied by an informed ant. Since all carrier ants are attached to the same cargo, each of them senses the total force exerted by all others and this makes the spin model inherently mean field. The mean-field solution (Supplementary Equation (41), which is an approximation for a finite-size system) reveals that, similar to the extended model described above, the response of the spin system to a transient pull by an informed ant (Equations (2) and (3)) peaks at Fc ind (Fc ind =4.3 compared with the peak at F ind =4.25 in the extended model) for fixed N (Fig. 4c), and at N c for fixed F ind (Fig.4d). The critical value Fc ind indicates the transition of this Ising model between ordered (F ind <Fc ind ) and disordered (F ind >Fc ind ) spin states, where the order parameter is related to the mean speed (Supplementary Note 24; Supplementary Fig. 15). The susceptibility of the simplified 1D model diverges at the same critical point (Fig. 4c; Equation (4) in Methods). The good agreement between the critical points of the simple 1D model and the full two-dimensional model arise from the fact that rotations (which are absent in the 1D model) do not contribute to the ordering transition (Supplementary Note 24; Supplementary Fig. 16a), and in addition because reorientations of the ants make the two-dimensional system more 1D like (Supplementary Fig. 14).

Since the model is mean field, and unlike the typical behaviour of systems with short-range interactions, the critical point Fc ind increases linearly with N (see equation (1) in Methods). This unique property allows us to explore the phase transition by varying system size (Fig. 4b,d; Supplementary Fig. 16b) rather than F ind (which is an inherent property of the ants and therefore difficult to manipulate experimentally). For example, large cargo sizes imply large N and therefore map to a large value of Fc ind ; since F ind is fixed to some given value this implies that F ind <Fc ind and that, as a consequence, large loads are expected to be in the ordered (persistent motion) phase (Supplementary Fig. 17). Conversely, small systems have Fc ind <F ind and are in the disordered (random walk motion) phase.

Experimental evidence for a mesoscopic phase transition

We used loads of varying sizes to experimentally test the models’ predictions. First, our model predicts that ants can carry ring-shaped loads with arbitrarily large radii. Indeed, we could experimentally induce the ants to move a load of radius 8 cm, much larger than anything they naturally carry. We find that the curvature of the object’s trajectory decreases with load size (Fig. 4e). The model shows the same dependence of the curvature with load size (Fig. 4e, inset); as the system becomes more ordered the curvature of the path decreases. Note that, in the case of small objects, a single ant involved in individual transport achieves a more direct trajectory than a small group of ants. This lack of coordination in small groups leads to non-optimal transport evident as a tug of war and a decrease in speed when compared with a single carrier (Fig. 4f). To demonstrate the suboptimality of a team of highly conformist ants, we used large (4 cm) radius objects that can occupy over 100 ants. In agreement with the model’s predictions, these objects exhibit highly persistent motion (Fig. 4e; Supplementary Note 25; Supplementary Fig. 17). However, this large ant team was unable to traverse a U-shaped barrier placed in their path to imitate the ragged conditions encountered in the field (Fig. 4g; Methods; Supplementary Note 1). Note that smaller objects (1-cm radius) do pass the obstacle (Fig. 4g) and this is facilitated by the attachment of informed ants. This demonstrates, as predicted by the model, that ants lose their ability to exert positive influence in the case of large loads.