Animals moving together in flocks, schools, and herds are ubiquitous in nature. The standard explanation for how individuals in these groups coordinate to generate the group level behavior we observe is that they interact locally with nearby individuals via some combination of attraction, repulsion, and alignment interactions [1, 2]. More specifically, attraction allows individuals to aggregate, repulsion prevents collisions, and the alignment interaction where individuals align with the average heading of their neighbors is proposed to drive the collective motion [3].

Much of our understanding of how animals in moving groups coordinate their motion has come out of the study of self-propelled particle (spp) models [3]. In spp models a number of particles move and interact with nearby particles via a set of local interaction rules, for example, attraction, repulsion, and alignment [2, 4]. Models including these three interactions have been shown to generate the standard groups: mills, swarms, and aligned (or polarized or dynamic) groups [4, 5], and have been widely adopted in modeling collective motion in specific real animal groups (e.g., [6, 7]) and as base models for general theoretical investigations (e.g., [8, 9]). The capacity of these models to produce group level alignment via the explicit alignment interaction has been critical in modeling groups of animals that move collectively through the environment. From now on we refer to group level alignment as “polarization,” and aligned groups as “polarized groups,” to clearly distinguish these from the explicit alignment interaction.

Over the past decade experimental studies has failed to detect explicit alignment interactions between individuals despite observing polarized schooling [10, 11]. It has also been established that explicit alignment interactions between particles are not required to produce polarized groups in spp models [12–17]. In addition, calculating explicit alignment has been described as a computationally intensive process [15], and [11] argue that it is unlikely that real animals will measure and store the speed and heading of neighbors that are required to compute the explicit alignment. Furthermore, [17, 18] speculate that the inclusion of explicit alignment interactions in spp models might explain why these models tend to fail to produce disruptive phenomena that are ubiquitous in nature, for example, bistability and switching between group types in fish [19]. Combined this suggests that alternatives to explicit alignment for generating polarized groups is required to explain collective motion in some animal groups and to address a number of issues related to a theory of collective motion based on explicit alignment interactions.

A number of specific mechanisms that can induce polarized collective motion from non-alignment interactions are known [12–17]. In particular, it is known that asymmetric interactions [13], anticipation [17], and asynchrony [16] induce polarization in combination with attraction. Asymmetric interactions, via blind zones, have been extensively studied in spp models [5, 15, 20–22]. In these models particles that are in a blind zone behind a particle relative to its direction of travel do not contribute to some, or all, of the interactions between the particles, resulting in asymmetric interactions. Given that many animals have restricted fields of vision [23, 24] and/or interact more strongly with nearby individuals in certain directions than others [25] asymmetric interactions represent a biologically plausible alternative to explicit alignment for explaining collective motion in some groups. Anticipation, where individuals use the future anticipated positions and headings of other individuals, rather than their current positions and headings to update their own headings is used by a number of animals [26–28], including humans moving in crowds [29–31]. This type of anticipation has been included in spp models that contain explicit alignment interactions [32, 33] and models that do not include them [17, 34]. In alignment based models anticipation has been reported to inhibit polarization and promote milling and swarming [32, 33], whereas in attraction based models anticipation has been shown to induce polarized collective motion [17]. In most spp models particles update their headings and positions synchronously, i.e., all particles update at exactly the same time, however, given stochasticity and other factors it is likely that individual animals in a moving group update at different times, in an asynchronous fashion [16]. A number of studies have investigated asynchronous updating in spp models [16, 35–37], in particular, [16] has established that sequential random asynchrony in update in combination with attraction induces polarization. However, sequential random asynchrony, where individuals update in a random sequential order on each time step, is unlikely to occur in real animal groups, but other types of asynchronous intermittent locomotion has been observed in animals across taxa [38]. In particular, the burst-and-glide and burst-and-stop type dynamics observed in fish [39–41], mammals [39, 42], birds [43], and insects [44] would be a more biologically plausible type of asynchrony. However, at present no study is available that has established that this type of asynchrony has similar polarization inducing capabilities, in combination with attraction, or other interactions.

Attraction is a fundamental biologically plausible interaction operating in animal groups [1, 3], and a component of many spp models [2, 13], but at present its perceived role is often limited to explain aggregation [3]. However, given that alternatives to explicit alignment are needed to explain collective motion in some animal groups, and to address issues related to the current alignment based theory of collective motion, and the recent discoveries of several mechanisms that induce polarization in combination with attraction, its role in the context of collective motion might require revision. Unfortunately, the information relating to the polarization inducing capacity of attraction is scattered throughout the literature and no direct comparison of the discovered polarization inducing mechanisms among themselves, or with explicit alignment is available, so how they differ with respect to the dynamics induced is largely unknown. Therefore, beyond biological plausibility arguments it remains unclear how the available polarization inducing mechanisms can help advance our understanding of collective motion in moving animal groups.

Here, we present a comparison of a number of known polarization inducing mechanisms in combination with attraction: explicit alignment [45, 46], asynchrony [16], anticipation [17], and asymmetric interactions (via a blind zone) [13]. We also introduce two previously unreported polarization inducing mechanisms in combination with attraction, burst-and-glide and burst-and-stop update, and add these, as well as explicit alignment alone, to our comparison.

We use the synchronous local attraction model (LAM) [16], known to not produce polarized groups, as our base model and add the polarization inducing mechanisms to it to allow for a direct comparison of their polarization inducing capabilities. We start by summarizing this model and then describe how it was adapted for each of the polarization inducing mechanisms. We consider eight models in total and label them (I)–(VIII). We note that all of these, except (V) and (VI), have been previously described in the literature and we provide the main references in the description of each. Models (I)–(IV) are identical to those presented in the references, but for (VII)–(VIII) we have only included the main interactions from the listed sources into the common framework of (I)–(IV) to facilitate focused comparison of the effects of the mechanisms themselves. Throughout this manuscript we use “hat” notation for normalized vectors (e.g., D^), and “bar” notation for non-normalized vectors (e.g., D̄).

I. Synchronous LAM [16] This is a self-propelled particle model in which N particles move at constant speed δ in two dimensions and interact via local attraction only (see Figure 1A). On every time step, each particle calculates the position of the local center of mass (LCM) of all particles within a distance of R from it (its neighbors). The new heading of particle i (D̄t+1i) is a linear combination of the normalized direction toward the local center of mass (Ĉti) and its normalized current heading (D^ti)

The parameter c specifies the relative strength of attraction to the LCM when the relative tendency to proceed with the current heading is 1. Once all particles have calculated their new headings based on the current positions of their neighbors, all particles are simultaneously moved a distance of δ in the direction specified by D̄t+1i.

II. Synchronous LAM with a blind zone ([16] with blind zone from [13])

Identical to (I) except that a particle does not interact with other particles in a blind zone defined by the angle β behind it relative to its direction of travel (see Figure 1B).

III. Sequential random asynchronous LAM ([16])

Identical to (I) except that particles update their headings and move in random sequential order on every timestep.

IV. Positional anticipation in the synchronous LAM ([17])Identical to (I) except that instead of using the actual local center of mass of the neighbors for the heading update each particle uses the anticipated local center of mass of the neighbors instead (see Figure 1C).

V. Burst-and-glide asynchrony LAM

Identical to (I) except that instead of interacting on every timestep the particles only update their headings at random “burst” times T defined by T_j+1 = T_j+γ where γ is a glide time (time between bursts) drawn from a Weibull distribution Γ = Γ(κ, λ), where κ is shape parameter and λ is the scale parameter. The choice of the Weibull distribution is motivated by empiricial findings [41]. If a particle is scheduled to update its heading on timestep t it will update it according to Equation (1) and move a distance of δ in this direction (see Figure 1A), and for the subsequent timesteps until its next burst time T_j+1 the particle proceed with unchanged heading but exponentially decreasing speed δ~(t)=δek(Tj-t) (See Appendix A for more details).

VI. Burst-and-stop asynchrony LAM

Identical to (V) except that the particles only move when they are updating their headings, and on these timesteps they move a distance of δ. For timesteps between updates the particles remain stationary.

VII. LAM with explicit alignment (Interactions similar to [46, 47])

Identical to (I) except that an explicit alignment term aÂti has been added to Equation (1) resulting in

where

with N_n representing the number of neighbors of particle i and D^j the normalized current heading of neighbor j (see Figure 1D).

VIII. Explicit alignment only. (Interaction from [45])

Identical to (VII) but without the attraction term cĈ. Here, particles only align with the average heading of their neighbors without any attraction (see Figure 1E).

We establish that both burst-and-glide (V) and burst-and-stop (VI) update induces polarized group formation in combination with attraction, in addition to producing mills and swarms. Figure 2A shows the group types produced by burst-and-glide, and Figure 2B the groups produced by burst-and-stop. Figure 2C shows the common modified Weibull distribution Γ(4, 3) as measured in simulations, and Figures 2D,E show the speed over time for two particles in the simulations with asynchronous burst-and-glide update (D) and asynchronous burst-and-stop update (E). Finally, Figures 2F,G show examples of the polarization process through a simulation with burst-and-glide (F) and burst-and-stop (G). We note that in both models the polarization increases up to and flattens out around α = 0.85–0.9. A detailed investigation of the polarization inducing capacity of these two mechanisms in combination with attraction, and their polarization and repolarization characteristics are described in the comparisons below.

The models that include attraction induce polarization over different attraction ranges (see Figure 3). We note that three of the mechanisms induce polarization only when attraction is weak, random sequential asynchrony (III) (c = 0.1and0.2) [16], burst-and-glide (V) (c = 0.2), burst-and-stop (VI) (c = 0.1and0.2). Asymmetric interactions (II), on the other hand, only induce polarization when attraction is stronger c ≥ 0.3. Explicit alignment (VII) induces polarization for c ≤ 0.4 when a = 0.01, and this upper c limit will increase with a (see Supplementary Figure 3). In contrast to all other mechanisms anticipation (IV) reliably induces polarization in different parts of the attraction parameter space, in particular, for weak attraction c ≤ 0.2 and strong attraction c > 1 [17].

The route to polarization from uniformly random initial conditions is markedly different for the different mechanisms. Some are stronger polarization inducing mechanisms and polarize the groups fast, and some are weaker and polarize the group more slowly, on average. Figure 4A shows that explicit alignment alone (VIII) and anticipation with larger c (IV with c = 2) polarize groups very fast on average (t < 100). Explicit alignment in combination with attraction (VI) is initially on par with (VIII) and (IV with c = 2) but is slower in the final approach to max polarization. Anticipation when attraction is weak (IV with c = 0.2) is not far behind (t < 600) and asymmetric interactions (III) generate polarized groups at an intermediate rate (t < 2, 000), on average. Figure 4B shows that the three asynchronous update models (III,V,VI) tend to take longer to polarize the groups (t > 2, 000). (See Supplementary Figure 1) for the median curves used to create Figures 4A,B with mad error bars. Figure 4C shows the relative frequency distributions of the time to polarized group formation, and the median ± mad for the eight models are (II) 1051 ± 848, (III) 3729 ± 1414, (IV with c=0.2) 471 ± 117, (IV with c=2) 72 ± 72, (V) 12959 ± 4953, (VI) 7319 ± 1765, (VII) 365 ± 129, and (VIII) 74 ± 20. (See Supplementary Videos 1, 2 for simulations showing the polarization process for each of these models).

The polarized groups generated by the different mechanisms exhibit different repolarization behavior following strong perturbations. Figure 5A shows the standardized median repolarization curve for each model. We see that the alignment based models (VII,VIII) accumulate most of the polarization very early in the process, on average, and that some models accumulate polarization at an almost linear rate (II,V). (See Supplementary Figure 2) for the median curves used to create Figure 5A with mad error bars. Figure 5B shows the relative frequency distributions of the time to polarization, and the median ± mad for the eight models are (II) 714 ± 848, (III) 799 ± 1203, (IV with c=0.2) 280 ± 140, (IV with c=2) 8 ± 18¹, (V) 9853 ± 4904, (VI) 5353 ± 1560, (VII) 466 ± 134, and (VIII) 460 ± 138.

The route to polarization (Figure 4) and repolarization behavior (Figure 5) are significantly affected by particle number (N) for some mechanisms, but left effectively unchanged for others (see Figure 6). The top panels in Figure 6 shows that both burst-and-glide (V) and burst-and-stop (VI) generates polarized groups faster as the number of particles increase, and that the polarization curves for the other mechanisms are relatively unaffected by change in particle number. The bottom panels in Figure 6 shows that the repolarization behavior of burst-and-glide (V), and to a lesser extent burst-and-stop (VI), are affected by particle number. In particular, for particle numbers less than 100, burst-and-glide exhibits sublinear polarization curves, but for N > 100 the curves become superlinear. The repolarization behavior of burst-and-stop (VI) also becomes more superlinear with increasing particle number, but the basic shape of the repolarization curves for the other mechanisms remain relatively unaffected.

How collective motion in moving animal groups emerges has been a question of debate. Explicit alignment was, and to a large extent still is, the standard explanation. However, motivated by both experimental and theoretical findings over the past decade the role of explicit alignment as the driver of collective motion has come into question. In particular, a number of biologically plausible auxiliary mechanisms have been shown to induce polarized collective motion in combination with attraction, another fundamental interaction operating in moving animal groups. However, a comparison of the effects of these recently discovered mechanisms between themselves or with explicit alignment has not been conducted and it is still unclear exactly how one can distinguish the effects of the different mechanisms. Here, we present such a comparison, including two previously unreported mechanisms that generate collective motion in combination with attraction; burst-glide asynchrony in update, burst-stop asynchrony, in update.

The finding that burst-glide and burst-stop asynchrony induce polarization in combination with attraction here is important in its own right. While it was previously known that sequential random asynchrony induces polarization [16], this particular type of asynchrony is less biologically plausible than the burst-and-glide/stop asynchrony that has been observed in animals across taxa [38–44]. Our findings suggest that it might be the attraction and burst-and-glide/stop dynamics that leads to the polarization observed in some of these groups, not explicit alignment interactions. In particular, [41, 46] has modelled experiments involving burst-and-glide moving rummy nose tetra fish using a model based on attraction and explicit alignment interactions. Investigating how a model based on attraction and burst-and-glide asynchrony compare with the attraction and alignment model originally proposed for this system could be useful as a benchmark to examine if their respective effects can be distinguished when parameterized by, and compared to, the same data of a specific system. Similarly, while the current explanations for the schooling behavior of fish in [10, 11] do not involve explicit alignment, they do involve attraction and asymmetric interactions (potentially via blind zones). Given that some fish species have a very large field of vision [48], including the Golden shiner used in [11], perhaps blind zones are not the main polarization inducing mechanism at play in these systems. At least, attraction and burst-and-glide dynamics might offer an alternative explanation.

It is well-known that a large number of different spp models can produce the same type of groups, in particular, polarized groups, mills and swarms, and thus we cannot infer which mechanisms led to a particular group from the final result alone [2, 49]. Our work shows that focusing on the polarization process might provide additional ways to distinguish between mechanisms proposed to be operating in groups that tend to polarize. As a complement to currently used methods focusing on macroscopic properties of the groups and microscopic analysis of individual behavior and interaction patterns [2, 25, 49]. In particular, our comparison shows that some of the polarization inducing mechanisms included here induce polarization in different parts of the parameter space (Figure 3). This observation might help narrow down the potential candidate mechanisms, given that attraction may be gauged from trajectory data [25]. For example, for animals that exhibit polarized collective motion and very strong attraction, asymmetric interactions or anticipation may be more plausible explanations than sequential random and burst-and-glide/stop, because the latter do not induce polarization for strong attraction, at least in our framework. The observation that some mechanisms polarize the groups faster than others (Figure 4) may also be used to distinguish between proposed candidate mechanisms for a given situation. For example, if very rapid polarization from uniformly random initial configurations are observed, out of the mechanisms included here, only explicit alignment, anticipation, and asymmetric interactions are likely candidates. However, if the process is slower the other mechanisms might be more plausible drivers. The repolarization curves (Figure 5) could similarly be used to distinguish between mechanisms from trajectory data in experiments where groups are repeatedly perturbed. Given that inferring the interaction rules from data of steady state configurations often is less informative than inferences based on the approach towards the stable configuration [49] a perturbation approach might be particularly fruitful to address certain questions. In particular, it seems unlikely that any statistical method that allows for the detection of an explicit alignment interaction will fail to detect it in data collected mainly from a stable polarized group, regardless of how that group level alignment was induced. Perturbation experiments, similar in setup to experiments designed to study fish escape behavior [50, 51], but with repeated perturbations may help resolve this issue in some cases.

Here, we have focused on the polarization inducing capacity of the mechanisms, and thus on polarized groups, however, it is worth noting that all included models, except explicit alignment (VIII), also generate mills and swarms. While this is not surprising given that mills and swarms are produced by the synchronous local attraction model [16], to which each of the mechanisms were added in (II)–(VII), the fact that all attraction based alternatives here produce all the three standard groups: polarized groups, mills, and swarms, but the polarization is induced by different mechanisms in each case suggests that this is a useful and versatile class of models for collective motion.

In conclusion, our work shows that alignment based and non-alignment based mechanisms may be distinguished from their polarization processes and how they interact with attraction. In particular, models containing explicit alignment exhibit exponential polarization accumulation processes whereas most non-alignment based processes exhibit more moderate polarization accumulation processes, anticipation being the exception. In addition, explicit alignment based models polarize faster from uniformly random initial configurations than they repolarize following strong perturbations, in contrast to all attraction based models that repolarize faster than they polarize from uniformly random configurations. These insights could potentially be used to probe whether explicit alignment is operating in a particular group or not via perturbation experiments. However, as described in the simulation and analysis section the analysis presented here is based on the limited ranges/values of the auxiliary model parameters (c, R, δ, τ) used in [13, 16, 17], and the results may depend on the actual parameters used. Therefore, anyone planning to utilize the approach introduced here for situations beyond these parameter ranges should generate the polarization and repolarization curves corresponding to their situation and parameter values of interest first. To facilitate such analysis for specific experimental systems, or further theoretical study, we provide the full code needed to generate the polarization curves (Figure 4) and repolarization curves (Figure 5) for parameter values beyond those considered here. See the Data availability statement for how to access it.

All code required to verify the results in this manuscript, with information about how to modify/extend the analysis, can be found at: https://github.com/danielstrombom/AttractionVsAlignment.

DS conceived the study and analyzed the simulation results. DS and GT finalized the design, performed simulations, and wrote the manuscript. Both authors contributed to the article and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.