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The authors have declared that no competing interests exist.

Coordinated collective behaviors often emerge from simple rules governing the interactions of individuals in groups. We model mechanisms of coordination among ants during cooperative transport, a challenging task that requires a consensus on travel direction. Our goal is to determine whether groups following simple behavioral rules can reach a consensus using minimal information. Using deterministic and stochastic models, we investigate behavioral factors that affect coordination. We define and investigate three types of behavioral rules governing individual behavior that differ in the information available: individuals either 1) have no information, 2) can measure transport success, or 3) measure success while also knowing whether they are aligned with the majority. We find that groups break deadlocks only if individuals more readily give up when they are going against the majority, corresponding to rule type 3 –such groups are “informed.” These behavioral rules succeed through positive and negative feedbacks that are implemented in our model via a single mechanism: individuals only need to measure the relative group sizes to make effective decisions. We also find that groups reach consensus more quickly if they have either a shared bias, high sensitivity to group behavior, or finely tuned persistence. Each of these is a potential adaptation for efficient cooperative transport. This flexibility makes the behavioral rules in the informed case relatively robust to deficiencies in the individuals’ capabilities. While inspired by ants, our results are generalizable to other collective decisions with deadlocks, and demonstrate that groups of behaviorally simple individuals with no memory and extremely limited information can break symmetry and reach a consensus in a decision between two equal options.

Across organizational scales, the patterns and complexity of many biological systems emerge from groups of individuals obeying relatively simple rules, often without a leader [

In collective decisions, groups that deadlock–having approximately equal numbers of individuals aligned with each choice–fail to form consensus. This can result in a split decision, or no decision at all. For some types of decisions this can be catastrophic, and there are behavioral mechanisms to prevent deadlocks in these cases. For example, split decisions during nest-site selection in honey bees can result in colony death [

A collective decision that is particularly prone to deadlocks occurs during cooperative transport in ants. Cooperative transport is the movement of large objects such as food items, intact, by multiple individuals [

Prior research has revealed aspects of cooperative transport, including selection pressures, ecology, recruitment, and more (reviewed in [

We use a proof-of-concept model [

We use this approach to answer two primary questions. First, can realistic, simple behavioral rules reliably overcome deadlocks? As part of this question, we look at what information individuals must minimally receive. Second, what effects do persistence (maximum engagement time with the object) and sensitivity to information have on coordination? In answering these questions we generate hypotheses for cooperative transport adaptations and provide insight into the factors that affect deadlocks during cooperative transport, and during other collective decisions.

We are interested in the minimum information and complexity requirements for deadlock breaking. We therefore assume individuals have minimal capabilities. As described below, we allow them little information. Our simulated ants also have no memory, in that they do not use information from past experiences to shape future behaviors. Real ants have more capabilities and information than the simulated ants in our models, but to find minimum requirements, we exclude several sources of information that have been demonstrated in one or more ant species. We further simplify real cooperative transport efforts by assuming that all ants are identical.

Ants sense a wide range of stimuli (e.g. [

We developed a deterministic, ordinary differential equations (ODE) model that simulates the average behavior of individuals. The model is Markovian–individuals have no memory–but non-linear. We model movement in one spatial dimension implicitly and we use continuous time and continuous abundances of individuals (but see individual-based model below). Individuals are identical, and the total number is fixed at 20; for some analyses we explored the effect of changing group size analytically and by evaluating groups with a total of 6 or 200 individuals. Having a fixed number of individuals is appropriate because the number of workers that can participate in cooperative transport will be limited by the number of grasping points on the object. Furthermore, the behavioral states we model allow varying numbers of individuals to be engaged with the object at any one time. Specifically, our model assumes that each individual occupies one of three mutually exclusive behavioral states: 1) trying to move the object to the left, 2) trying to move the object to the right, or 3) disengaged from the object (

Individuals belong to one of three behavioral states: moving left, moving right, or disengaged. Individuals move between these states at rate constants _{L}, _{R}, _{L}, and _{R}.

Disengaged individuals join the transport efforts to the left and right with rate constants _{L} and _{R}, respectively. The realized joining rates depend on the number of disengaged individuals: the instantaneous joining rate for the left state is _{L} multiplied by the number of disengaged ants. We assume the joining rate constants do not change in time but may differ from each other, i.e. individuals may join the “move left” behavioral state at a higher rate than the “move right” state. If _{L} and _{R} are not equal, this ensures a directional bias, which is how we represent individuals having information about the direction of the goal.

In real ants, directional cues about the location of the nest come from one or more sources, such as pheromone trails, visual navigation, or path integration [_{L}, _{R} > 0). Joining rate constants do not vary during the transport effort; for example, we assume groups are not capable of altering their bias in favor of the “winning” direction (here we use the “winning” direction to indicate simply the direction that has more individuals). This makes sense given our conservative assumptions about individuals’ memory and sensory capabilities: individuals that are disengaged, and therefore not in contact with the object to sense information transmitted through it, cannot perceive which direction is winning and have no memory about which direction was winning when they were last engaged.

Individuals in the active behavioral states (left and right) give up at rate constants _{L} and _{R}, respectively. We model three sets of behavioral rules for giving up rates. These sets of rules also differ in the kinds of information individuals act upon (

Rule | Description | Information used |
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If in one of the active states, give up (become disengaged) at a constant rate | None | |

If in one of the active states, give up more readily when transport is unsuccessful, and less readily when it is “successful” (see text) | Must be capable of measuring the “success” (i.e. extent of coordination) but not the direction of the majority relative to one’s own behavioral state. | |

If in one of the active states, and if transport is successful, give up readily if going the opposite way as the majority and less readily if going the same way as the majority. | Must be capable of measuring (i) extent of coordination and (ii) preferred direction of the majority and must compare the latter to one’s own behavioral state. |

Behavioral rules differ among different model runs, but within one run of the model all individuals are identical and have the same rules and parameter values. In “uninformed” groups, giving up rate constants, _{L} and _{R}, are equal and do not change over the course of the transport effort. In “oblivious” and “informed” groups (defined in _{R} and _{L}). Giving up depends on the “success” of transport. “Success” is operationally defined here as a high extent of coordination, measured as the absolute value of _{R}—_{L} divided by the total number of individuals in the system. In other words, the extent of coordination is the degree to which individuals are

In oblivious groups individuals can measure success but they cannot detect if they are contributing to or detracting from that success. Individuals give up less frequently when | _{L}−_{R} | is high, i.e. when there are many more individuals in one active state than the other, regardless of whether they are currently in the “right” or “left” state. Individuals are oblivious to their own contribution. If the transport is successful because many more individuals are trying to move the object to the left rather than the right, individuals moving right, who are going against the majority, still rarely give up. In ants, this would happen if they were capable of determining when the group sizes are uneven (or a proxy, such as the magnitude of the force on the object), but not in which direction. For example, this might occur if individuals are less likely to give up when they are moving, regardless of the direction.

In informed groups individuals are capable of detecting the same information as in the “oblivious” case, but additionally they can determine if their contribution is with or against the majority. Individuals give up less frequently when there is a higher extent of coordination only if their behavioral state matches the majority. For example, when _{L}−_{R} is strongly positive, individuals in the “move left” state give up infrequently while individuals in the “move right” state give up quickly. As discussed above, _{L}−_{R} is a measure of success that could be estimated by ants in multiple ways. For example, large values of _{L}−_{R} (or highly negative values) will correspond to higher speeds over ground, which an ant might measure by estimating optical flow or her own leg movements.

Equations governing the giving-up rate constants, _{L} or _{R}, under each set of rules are listed in _{R} and _{L}, functions for determining _{L} and _{R} depend on one or more parameters (_{1} must be non-zero.

The x-axis indicates a measure of success: the size difference between the two groups. (A) Uninformed rules, _{2}_{1} = 2, _{1} = 4 (solid line) or 0.5 (dashed line). (C) Informed rules, _{1} = 2, _{2} = 0.5 (solid line) or 3 (dashed line). In (B) and (C), dashed lines indicate sharper shape parameters.

Ranges of parameter values explored are in parentheses.

_{L} |
_{R} |
Max ^{-1}) |
Shape parameter | |
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Uninformed ( |
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Oblivious ( |
_{1} (0.1–100) |
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Informed ( |
_{1} (0.2–20) |
_{2} (0.1–100) |

The giving up rates described above are tunable based on individuals’ persistence and sensitivity to information. These parameters govern the shape and maximum values of the giving-up functions (_{L} = _{R} in the oblivious case and when the difference between _{L} and _{R} is largest and opposed to the individual’s state in the informed case. We refer to this engagement time as persistence [

Persistence is individuals’ resistance to changing their behavior based on information [

For the oblivious and informed cases we can also tune the parameters to change the sensitivity of individuals to the success of transport, that is, the magnitude of |_{L}−_{R}|. We do this by changing the shape of the giving-up functions through manipulations of the shape parameters (_{1} and _{2}; _{L} = _{R}), small changes in success lead to only small changes in the frequency of giving-up; individuals with a gradual shape therefore have low sensitivity to transport success. On the other hand, for sharp shapes, a small change in success when _{L} ≈ _{R} leads to a dramatic change in this frequency; this means individuals are highly sensitive. Differences in sensitivity could be caused by a number of factors, including error in sensing the group sizes. This shape parameter can be quantified for real organisms by fitting functions to data on individuals, for whom cooperative transport efficiencies are experimentally manipulated.

The model consists of the following set of differential equations giving the rates of change in the numbers of individuals in each behavioral state (moving left, moving right, or disengaged, respectively):
_{D} is the number of individuals in the disengaged state (_{R} and _{L} on _{R} and _{L}. There is a constant number of total individuals (i.e., _{D} _{R} + _{L} = _{L} = _{R}, so _{R} and _{L} are nonlinear functions of _{R} and _{L} in the oblivious and informed cases, it is difficult to solve this system of differential equations analytically. We numerically solved the ODE for each of nearly fifteen thousand sets of parameters, running the model under different sets of behavioral rules, global directional biases, and persistence and sensitivity. The range of parameter space explored for giving-up parameters is shown in

Our ODE model makes certain assumptions required for any ODE, including instantaneous updating of information and continuous, rather than discrete, individuals. To test whether our conclusions are robust to these assumptions, and to look at the potentially important influence of stochasticity, we extended the model to a stochastic framework. The stochastic extension is an individual-based model operating in discrete time. We converted the instantaneous joining and giving-up rate constants (_{L}, _{R}, _{L}, and _{R}) to probabilities of joining or giving up in a given time step with the equation
_{t} is the probability of a behavioral shift in one time step, δ_{t} is the length of a time step (here, time steps were always unit length), and _{L}, _{R}, _{L}, or _{R}. We ran the stochastic simulation for 60 time steps; this duration was more than sufficient to capture transient dynamics. All other model assumptions and parameters were the same as in the deterministic model, including the three sets of rules.

In each time step we allow individuals to change their behavioral state. An active individual changes its status by giving-up with a probability equal to the giving-up probability for that individual’s current state (left or right), and disengaged individuals can change their status by joining. Because disengaged individuals can change their status in one of two ways (joining the left group or the right group), we first calculated the joint probability of an individual joining at all. For individuals that were to join, we then stochastically determined whether they joined left or right using the relative probabilities of each. We ran the stochastic model under the same parameter sets as the deterministic model, querying 1,000 simulations for each set of parameters. As with the deterministic model, we examined the extent of coordination. We performed and analyzed stochastic simulations in R (RStudio version 0.98.977). Code is included in

Our primary measurement of success is the extent of coordination, which is the difference in the number of individuals in the active behavioral states (left and right) divided by the total number of individuals in the system. If the transport is uncoordinated, there are roughly equal numbers of individuals pulling each direction, and/or most individuals are disengaged. Streamplot representations of the vector fields portray the dynamical behavior of the system in _{L} and _{R}). The number of disengaged individuals, _{D}, is not shown explicitly because the total number of individuals is fixed at 20 (i.e., _{D} = 20 –(_{L} + _{R})).

These show the direction the system tends towards for various abundances in each behavioral state. A-C: Uninformed rules, _{1} = 4, _{1}/_{2} = 1; G-I: Informed rules, _{1} = 1, _{2} = 0.5. A, D, and G: Strong directional bias, _{L} = 0.01, _{R} = 0.7; B, E, and H: Weak directional bias, _{L} = 0.3, _{R} = 0.7; C, F, and I: No directional bias, _{L} = _{R} = 0.3.

In the absence of a directional bias (_{L} = _{R}), both the uninformed and oblivious rules have stable equilibria (_{L} = _{R} (_{2}, the equilibrium is stable and deadlocks are maintained. Thus there is a critical value of _{2} at which a phase transition occurs, from stable to unstable equilibrium. Using fixed-point analysis [_{2} has the following value:
_{L} = _{R}) and _{2}) to break deadlocks even in the informed case, and sensitivity is less important for large groups. Details of the fixed-point analysis are included in

When a directional bias is present (_{L} ≠ _{R}) more individuals attempt to move the object in the direction favored by the bias, regardless of the set of rules (

The stochastic results are very similar to results from the deterministic model.

Includes two example simulations with each set of parameters. Blue: number moving right, Red: number moving left, Black: number disengaged. Dashed lines show deterministic model behavior. Columns are different directional biases and rows are different sets of behavioral rules. The parameter values are the same as in the analogous panels in _{1} = 4, _{1}/_{2} = 1; informed rules: _{1} = 1, _{2} = 0.5. Strong directional bias: _{L} = 0.01, _{R} = 0.7; weak directional bias: _{L} = 0.3, _{R} = 0.7; no directional bias: _{L} = _{R} = 0.3.

The x-axis shows the number of ants, and the y-axis shows the number of simulations for which the given behavioral state had that many ants at that time. Blue bars are for ants moving right, red bars are for ants moving left, and black bars are for disengaged ants. Bars appear purple when red and blue overlap. Dashed lines show the abundance of each behavioral state in the deterministic model. The parameter values are the same as in the analogous panels in Figs _{1} = 4, _{1}/_{2} = 1; informed rules: _{1} = 1, _{2} = 0.5. Strong directional bias: _{L} = 0.01, _{R} = 0.7; no directional bias: _{L} = _{R} = 0.3.

In all other respects, deterministic and stochastic results were very similar despite differences in the formulations of these models. When a directional bias is present more individuals try to move the object in that direction than in the other direction under our initial conditions of all individuals beginning as disengaged. In the absence of a directional bias, roughly equal numbers of individuals are in each active state in uninformed and oblivious groups, while individuals converge on either direction in informed groups. In each of 1,000 simulations, the informed case allowed for convergence to a pure state (every individual or nearly every individual in the system transporting in the same direction) even with no directional bias (

When a directional bias is present, the informed case still leads to strikingly different performance than either of the other sets of rules. Individuals converge rapidly in informed groups, while in oblivious or uninformed groups, convergence, which we define as an increasing coordination through time until all individuals are pulling the same direction, does not occur. There are more individuals pulling in the direction of bias but coordination does not increase over time (Figs

Maximum giving-up rate constant is the maximum possible as defined by the function (

The effect of persistence depends on the behavioral rules (

For oblivious and informed groups, the sensitivity changes the effect of persistence (Figs _{1}) increase coordination for a given persistence value. In the informed case there is a critical sensitivity below which deadlocks cannot be broken, as discussed above and in _{2}) further increase coordination, which has the effect of widening the range of persistence values that lead to coordination. For a moderate group size of 20 individuals, with a gradual shape and a small directional bias, there is a narrow range of persistence values that allow for high coordination. At small group sizes only groups with higher sensitivity or relatively strong directional bias coordinate successfully regardless of persistence, while large groups successfully coordinate across a wide range of persistence values regardless of sensitivity and bias (

Maximum giving-up rate constant is the maximum possible as defined by the function, actual values will depend on the number of individuals in each group. Extent of coordination is defined as the difference in the number of individuals pulling right and left, divided by the total number in the system. Maximum coordination is the maximum observed over a given time period, rather than an absolute maximum; higher values on the y-axis indicate faster convergence. Left column: shape parameter, _{2} = 0.5, which corresponds to the solid line in _{2} = 1, which is less gradual.

Can relatively simple individuals with minimal information break deadlocks? Our results show that, indeed, individuals with simple behavioral rules and no memory can break deadlocks. However, only individuals in our informed case convincingly succeeded. These individuals followed simple rules: 1) give up more readily if one is moving against the majority and 2) do this to a greater extent for extreme majorities than slight majorities. Using these simple rules, with minimal information available, groups rapidly converge on a single travel direction, even when this required symmetry-breaking. Our deterministic and stochastic models agree, despite being formulated differently and having contrasting assumptions about individuals and time. This suggests that our conclusions are robust to specifics of model formulation.

In terms of information, it is sufficient for coordination for individuals to only be capable of measuring the direction that the majority of the group is trying to move the object and the relative sizes of the groups moving each direction (or a proxy). This information is crucial; with insufficient sensitivity to these group sizes (low _{2}) groups do not form a consensus. While sensitivity must be sufficient, it does not need to be high. As further discussed below, groups with only modest sensitivity were still coordinated across a wide range of other parameters. Thus, various proxies for relative group sizes may be accurate enough to break deadlocks. For example, individual ants could gain this information through stigmergy on the object being carried. If this is the case, a single sensory mode may provide all necessary information in informed groups. In nature, ants may have other information available, or may use different behavioral rules, but we show that by using these simple rules, groups are successful.

If individuals have global directional cues that correspond to a shared directional bias, this helps promote coordination regardless of the other information available. Additionally, if there is only one correct direction, for instance if there is a single nest entrance, a shared bias towards the nest would help ensure the group converges on the appropriate direction. But directional bias is neither necessary, nor sufficient, for convergence on a decision.

This makes sense considering the high variation in cooperative transport ability among ant species. We expect workers of all species to be good at knowing the direction of the nest. So we expect directional biases to be common among species, at least for situations with only one correct direction. Considering that efficient cooperative transport is comparatively rare among ants [

We also investigated the effects that persistence and sensitivity (the sharpness of the giving-up function) have on coordination. These effects are complex and depend on the total group size and the behavioral rules. In the uninformed and oblivious cases, groups are most coordinated if individuals are highly persistent. While somewhat surprising, this makes sense in light of a tradeoff in persistence. Groups of highly persistent individuals may pull in opposing directions for a long time, but if movement does occur, either because of a directional bias or due to random fluctuations, the progress continues; they are unlikely to change their direction.

This suggests that high persistence allows species without other adaptations for cooperative transport, for instance those with behavioral rules similar to our uninformed or oblivious rules, to at least sometimes succeed at bringing a large object home to the nest. In such species, individuals are equally likely to give up whether they are helping or hurting the effort; even when successful movement occurs, individuals pulling with the motion may give up. High persistence makes it less likely that anyone will give up, allowing existing movement to continue. If, as in our model, individuals are identical, the individuals going the wrong way will also be unlikely to give up, so to minimize the length of deadlocks there should only be a small number of these individuals. A sufficient directional bias would accomplish this, and directional biases should be common in many circumstances (such as if the object is relatively far from the nest). So if high persistence is paired with a directional bias, it may allow ant species with rudimentary behavioral rules to conduct cooperative transport. Analogously, agents involved in any decision between two options, when they are unable to determine which option is winning, should be persistent to maximize the chance that a single option will be chosen.

In contrast to these results, in the informed case there is an optimum persistence value; groups with individuals more or less persistent than this value will be less coordinated. But the importance of persistence depends on directional bias, on the sharpness of the giving-up function, and on the total group size. In most of the parameter space of our model, the range of persistence values that lead to high coordination is wide. Only when the directional bias is low and the sensitivity is above the critical threshold but still gradual does one find a narrow peak in coordination around the optimum persistence. This was especially true for smaller group sizes. Large groups had a wide range of persistence values that would lead to coordination regardless of sensitivity, indicating that it may be easier to coordinate in a large group rather than a small group. This makes sense given that small groups will be more affected by the behavior of single individuals. In order for informed individuals in groups of small to moderate size to be highly coordinated, they must have one, but do not need more than one, of the following: high directional bias, high sensitivity to the sizes of the two groups, or finely-tuned persistence. Each of these is a potential adaptation for efficient cooperative transport in informed groups. This flexibility makes the behavioral rules in the informed case relatively robust to deficiencies in the individuals’ capabilities as long as they have at least minimal accuracy in sensing group sizes.

Because we did not constrain our model by tuning it to a particular species, our results are applicable to other collective decisions. A system in which groups must decide among multiple options is vulnerable to deadlocks, especially when the options are relatively equal (analogous to having no directional bias); small group size may also make deadlocks more likely. One of the best studied examples of collective decisions is nest-site selection in social insects (reviwed in [

The outcome of our deterministic model with respect to the effect of behavioral rules looks similar to the results of Seeley et al. [

The Seeley et al. [

Our model demonstrates that simple behavioral rules can lead to a consensus about travel direction during cooperative transport, even without a directional bias. Our simulated ants had no memory, limited sensory ability, and followed only simple rules, yet made decisions rapidly in informed groups. We identify a potential adaptation–giving up more readily when going against the majority–that allows for deadlock-breaking, and may explain why we see such large variation in cooperative transport ability among ant species. While it is currently not possible to directly measure this adaptation in ants, the consequences we have modeled here can, and should, be measured to see if real ants use this behavioral rule. Our model reproduces dynamics similar to those of other decision-making processes [

All code is included here. S1 Code also includes example parameter sets necessary to reproduce Figs

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Effect of persistence (inverse of maximum giving-up rate constant) on maximum coordination for small, moderate, and large groups. Maximum giving-up rate constant is the maximum possible as defined by the function, actual values will depend on the number of individuals in each group. Extent of coordination is defined as the difference in the number of individuals pulling right and left, divided by the total number in the system. Maximum coordination is the maximum observed over a given time period, rather than an absolute maximum; higher values on the y-axis indicate faster convergence. Top row: uninformed rules, middle row: oblivious rules, bottom row: informed rules. Left column: total group size = 6, middle column: total group size = 20, right column: total group size = 200. Lines with smaller dashes indicate lower directional bias.

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Effect of persistence (inverse of maximum giving-up rate constant) on maximum coordination in small, moderate, and large informed groups at low (gradual) shape values. Maximum giving-up rate constant is the maximum possible as defined by the function, actual values will depend on the number of individuals in each group. Extent of coordination is defined as the difference in the number of individuals pulling right and left, divided by the total number in the system. Maximum coordination is the maximum observed over a given time period, rather than an absolute maximum; higher values on the y-axis indicate faster convergence. Top row: shape parameter, _{2} = 0.5, which corresponds to the solid line in fic. 2C. Bottom row: _{2} = 1, which is less gradual. Left column: total group size = 6, middle column: total group size = 20, right column: total group size = 200.

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Movie of histograms showing the state of 1,000 simulations of the stochastic model for each of nine parameter sets. Each frame is a time point. Each panel is a set of parameter values, corresponding to the analogous panels in Figs

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We thank James Marshall and Andrew Koller for advice on the model and implications. Ted Pavlic, Tomer Czaczkes, and anonymous reviewers provided valuable comments on the model and manuscript. We also thank Tim Szewczyk, the Modeling Group, QDT, the Writing Cooperative, and the Breed Lab at the University of Colorado for comments on the model, manuscript, and code.