Several animal species exhibit forms of collective motion in which two or more individuals move together coherently. Examples include flocks of migrating birds, schools of fish, murmurations of starlings, swarms of locusts, and many others. In general, the same group of animals can produce various types of collective patterns, including disordered aggregations, milling, or schooling depending on both internal states (e.g. hunger level) and external conditions (e.g. in response to a predator).
Much of our current understanding of collective motion of animal groups comes to us from the study of theoretical models, and in particular of a class of models known as ‘self-propelled particle models’. These models indicate that a small set of ‘rules’ of interaction is sufficient to generate group level patterns that resemble, at least visually, those formed by real animal groups. For instance, Reynolds [1] proposed a model that implements only three different rules. The first rule consists in a repulsion behaviour, through which each individual turns away from its local neighbours and avoids local crowding and collisions. The second rule is an alignment behaviour, or a turning response towards the average heading of local neighbours. The third rule is a turning response towards the position of more distant neighbours; this is an attraction rule, that contributes to maintain the members of the group together. Several alternative models of collective motion have been proposed (see [2] for a review), each implementing a slightly different set of interaction rules. In spite of their differences, almost all the models existing in the literature are able to produce realistic looking patterns of collective behaviour, at least within a certain range of parameters.
In order to make meaningful predictions about the collective movement patterns of a given animal species, it is important that the interaction rules implemented in the models match those actually used by animals of that particular species. In order to determine how real animals of different species interact together, several research groups have started to collect empirical data on the movement patterns of real animal groups. Traditionally, this has been done either focusing on the collective level, or on the individual level. The collective-level approach consists in collecting data on the spatio-temporal organization of the group, such as e.g. the mutual positions of close neighbours, and testing which theoretical models are compatible with the data; the individual-level approach operates instead by selecting a ‘focal individual’ within the group, and recording all the changes of speed and direction of movement of that individual in response to the position and movement of its neighbours [3]. Here, we provide a brief review of this literature, with particular emphasis on articles that either measure or predict the mutual positions of close neighbours.
As an example of the collective-level approach, Ballerini et al. [4] tracked the 3D positions of starlings flocking together in natural flocks, with the aim of characterising the spatial organization of the group. These authors observed that nearest neighbours consistently occupy the same positions with respect to each other, determining an anisotropic arrangement at the local scale. The anisotropy did not spread to the scale of the entire flock, but dropped quickly to a completely isotropic distribution between the sixth and the seventh nearest neighbour. The fact that the anisotropy cut-off depended on the number of neighbours, but not on the density of the group, was interpreted as evidence that starlings ‘pay attention’ to a fixed ‘topological’ number of six - seven neighbours, instead of responding to all neighbours within a fixed ‘metric’ distance. The topological nature of interactions in starlings was later confirmed also by an alternative maximum entropy approach, based on the relative alignments of nearest neighbours, instead of their positions [5]. A similar collective level approach was adopted by Lukeman et al. [6]. These authors recorded the positions and orientations of surf scoters sitting on the water surface. The observed arrangements of neighbours around a focal individual were consistent with models implementing repulsion, alignment, and attraction, but also required the existence of a more direct interaction with one single neighbour situated in front. Buhl et al. [7] measured the relative positions of swarming locusts, and observed isotropy in the radial distribution of neighbours around a focal individual. This distribution was compatible with both metric and topological models of interactions, but not with a third class of ‘pursuit/escape’ models [8] in which individuals try to reach neighbours ahead of and moving away from them, while they escape from other individuals that approach them from behind. Hemelrijk et al. [9] measured how the overall shape (length vs. width) of schools of mullets scales with group size. Their empirical data were consistent with a model in which the oblong shape of some schools, results from individuals slowing down to avoid collisions.
As examples of studies that have adopted the individual-level approach, we can mention Katz et al. [10], who reconstructed the ‘force maps’ that describe the acceleration and turning of schooling golden shiners, and Herbert-Read et al. [11], who reconstructed the force maps of mosquitofish. These studies indicated that a fundamental component of how fish of both species interact are changes of speed: the fish consistently increased or decreased their speed to catch neighbours that they had respectively in front or behind; but when a neighbour was too close by, the speed responses were reversed, so speed changes also mediated collision avoidance. Both studies found only weak alignment responses, in comparison to attraction and repulsion forces. While both mosquitofish and golden shiners formed aligned groups, this was more a consequence of the fish following each other (and eventually becoming aligned) than an explicit alignment response. More recently, Pettit et al. [12] applied a similar approach to the study of flight interactions in pigeons. The observed flocking responses of pigeons where different from those found in fish: alignment responses were explicit and strong, and collision avoidance was mainly mediated by turning, while speed remained relatively constant. These observations could be interpreted in terms of the different needs and constraints associated with flocking, which are different from those experienced by fish during schooling. Explicit alignment responses, for instance, might be necessary to achieve the high cohesion of pigeon flocks, that can fly without splitting for several kilometers. Avoiding collisions by turning away from the neighbour, instead of slowing down, might respond to a necessity to maintain a relatively constant speed, required to produce a sufficient lift force.
Gautrais et al. [13] used an intermediate approach to build a model of the shoaling behaviour of fish: they first characterized the motion of isolated fish, and progressively added interaction terms to the model through visual observations of how fish interact with obstacles and other fish, using quantitative methods to fit the parameters of these interaction rules to the tracked movements of the fish. The model was then tested at the collective level, by collecting statistics of the alignment and distance of real fish. In spite of its nice data driven formulation and good fit to experimental data, the model introduced by these authors does not formulate predictions about the mutual positions of nearest neighbours, and does not quantify these mutual positions in the empirical data; for these reasons it will not be discussed further in the context of our simulations which insist precisely on these aspects.
While some work has characterised directly the interaction responses of individuals, and other work has derived interaction responses indirectly, by selecting the interaction rules that reproduced better the observed configuration of a group, it is clear that different interaction rules lead naturally to different local configurations of the group. Consider for instance the case of an animal that avoids collisions by changing speed (like mosquitofish or golden shiners). Its acceleration response will be positive when the neighbour is in front and negative when the neighbour is behind, but will invert sign in the repulsion zone. The only region where there is no acceleration response is on the border between attraction and repulsion zone. Similarly, if turning does not mediate collision avoidance, the turning response will be simply directed towards the neighbour, that is, to the left if the neighbour is on the left and to the right if the neighbour is on the right. There are only two ‘fixed points’, for which both the turning and the acceleration response are zero: one directly in front and the other directly behind the neighbour. Not surprisingly, these positions are those at which both mosquitofish and golden shiners are most likely to have their neighbours [10,11]. Similar arguments can be used to explain that when collision avoidance is mediated through turning away from the neighbour (as in pigeons), a side by side configuration is the one which is stable (this is the configuration that was most frequently observed in pigeons [12]). In other words, different interaction rules lead naturally to different local arrangements of neighbours within the group.
In the present paper, we examine the different implications of this duality between interaction responses and mutual positions in flocks and schools. Unlike in previous studies, where mutual positions result from the interactions, here we consider the theoretical situation of animals maintaining stable mutual positions, and we address the question of what ‘apparent’ interaction responses would be observed as a mere consequence of the imposed mutual positions and noise.