Collective behavior in living aggregations often has a strong anti-predatory function. An efficient group response is crucial to maintain global coherence and cohesion in spite of attacks and disturbances [1–8]. To this end, many groups exhibit collective directional changes in very short times, often triggered by local perturbation events. A beautiful example is the one of bird flocks [9], where collective turns start from a single individual, and the directional information is passed through the group so quickly that the whole flock performs the turn retaining its structure and without attenuation. From the perspective of statistical physics, this behavior is somewhat unusual. Indeed, when considering physical systems with long-range directional order—either at equilibrium or active—perturbing locally the system does not in general change its ordered state, which is another way of saying that the order is stable (i.e. ergodicity is broken in the thermodynamic limit). For instance, if we consider a ferromagnet and apply a magnetic field on a single site of the lattice, the magnetization will remain unaltered. What marks the difference with respect to animal groups is, of course, their size. Living groups might be large, comprising hundreds or thousands of individuals, but they are very far from the order Avogadro numbers that characterize physical systems. Perturbations which would be irrelevant in the thermodynamic limit might in fact change the state of a finite group on observational timescales, if its size is small enough. Response to local perturbations is thus, intrinsically, a finite-size effect. The kind of response exhibited by the system, and the effective timescale for collective adaptation, also depend on the mechanism of information propagation. In flocks, a linear dispersion law has been observed [9], suggesting that second order dynamics for the birds flight directions should be at play [10, 11]. Other animal groups display instead less efficient information transfer. For instance, experiments on fish schools [3, 4, 7] show that groups exhibit evasion manoeuvres arising from induced alarming stimuli. However—contrary to bird flocks—the speed of information propagation decelerates over time, and thus only part of the group follows the initiator of the evasion event, resulting in a distribution of behavioral cascades. In this case, a dissipative dynamics rules individual directional motion, and the local connectivity of the interaction network might dramatically affect the extension of the collective response [4].

These examples indicate that several factors might contribute to the response behavior of finite groups, and it is not always clear how to disentangle one factor from the other. In this work we perform a systematic study to explore the interplay between size, dynamical rules, motility and boundary conditions in determining the response of the system to local directional perturbations. The framework of our study is the one of self-propelled models of collective motion, where the minimal number of parameters allows to exhaustively explore the model's space.

This paper is organized as follows. In section 2 we introduce the model discussed in our theoretical and numerical analysis: the inertial spin model (ISM [10]). This model is a generalization of the Vicsek model (VM) [12] comprising both inertial and dissipative terms in the dynamical equation for the velocities. In the underdamped limit it reproduces the linear dispersion law observed in flocks of birds (for which it was originally proposed), while in the overdamped limit it reduces to the VM. It therefore represents an ideal playground to investigate different dynamical regimes potentially relevant for different behaviors. In the following sections we consider progressively all the factors that might affect the response behavior. In section 3 we start by looking at the ISM with a fixed interaction network. In this case it is possible to derive analytically the equations describing the evolution of the system, when a local perturbation is applied to a single individual. This allows to clearly understand what is the role of the system's size, and how a collective turn manifests itself in the different dynamical regimes of the model. In section 4 we perform numerical simulations that quantitatively confirm our analytical results. Next, in section 5 we consider the off-lattice, active version of the model, and elucidate how motility affects the fixed-network scenario. Finally, in section 6 we discuss what happens when open boundary conditions (OBCs) are considered, and we show how and when the mechanisms studied in the previous sections can lead to a fragmentation of the group.

Collective behavior in animal groups has been described with a variety of approaches, either considering evolution rules for the individuals in the group [13, 14], or by using coarse-grained equations for mesoscopic fields [15–17]. In this study, we work in the context of self-propelled particle (SPP) models, where the aggregation is modelled in a minimal way as a collection of particles with fixed activity interacting with each other via alignment/imitation rules.

2.1. The VM

The most famous among SPP models of collective motion is the VM ([12]), which we now present in its continuous-time version. Consider a system made of N point-like active particles, each labeled by an index i, which move in a d-dimensional space following the equations of motion

Each particle position evolves deterministically according to its corresponding velocity vector of fixed modulus , while the latter undergoes a Langevin overdamped dynamics with a 'social' alignment force given by

We recognize in equation (3) the Hamiltonian of the Heisenberg model, with the coefficient J > 0 setting the strength of the alignment interactions, and where the connectivity matrix nij specifies the interacting neighbors. However, in contrast to the standard Heisenberg model, here the connectivity matrix itself evolves in time through the particle positions, i.e. . Among the possible evolution rules, we will adopt in the following the paradigm of metric interactions: each particle interacts with those lying within a fixed interaction radius rc , i.e. .

Featuring in equation (2) are a viscosity coefficient η and a white Gaussian noise , which are linked by the Einstein-like relation

where the temperature T sets the strength of the fluctuations. Finally, the parameter λi is a Lagrange multiplier which can be used to enforce the fixed speed condition [18].

Due to the time dependence in , it is known that the VM admits, even in , a T-driven transition from an ordered phase (flock), where the individual velocity vectors align and add up to a non-zero total velocity

to a disordered phase (swarm), where the mean group velocity is null. A convenient order parameter to describe the transition is the so-called polarization vector

and its corresponding (scalar) polarization .

2.2. The ISM

The VM successfully describes the large-scale behavior of several living and non-living active systems [13, 16]. However, the VM is not appropriate to explain collective turns in flocks of birds. Experiments indeed show that, in turning flocks, the directional information travels obeying a linear dispersion law, with a propagation speed only depending on the degree of order in the system [9]. The Vicsek dynamics does not reproduce this behavior. An intuition of why this occurs can be grasped by looking at equation (2): its structure is that of an overdamped Langevin equation for the velocity, and this kind of equations usually lead to a diffusive dispersion law [19]. The simplest way to obtain linearly dispersive solutions is to reinstate a second order derivative in equation (2), and this is precisely what the ISM does [10]. As in standard second order equations, it is convenient to write the model as a system of first order equations, by introducing appropriate conjugate variables. The ISM then reads

with the noise correlations given in equation (5). In contrast to the VM, in equation (10) the force term does not act directly on the velocity, but rather on its derivative, which is expressed in terms of the new variable . The vectorial products enforce the constraint on the individual speeds. The particles can thus only change their directions: therefore plays the role of an internal angular momentum, regulating the rotations of the individual velocity vectors, and this is why it is called a 'spin'. The new parameter χ plays the role of a rotational inertia.

2.3. The planar ISM

Let us now specialize the ISM to the planar two-dimensional case, where velocities lie on a plane (i.e. ). This case is simpler to handle algebraically, and it gives a direct intuition of the physical and biological meaning of the spin and of inertial dynamics. Besides, it is also the relevant one for collective turns in flocks (as we will discuss later on), and for a variety of other biological groups. Generalization to the three-dimensional case is straightforward.

In the planar case, we can write

where the phase ϕi specifies the angle of the velocity vector with respect to a reference direction. In terms of the phases, equations (9) and (10) assume a much simpler form, namely

In these equations the spin vector reduces to one single component, whose value si identifies the angular velocity of each particle. In the presence of a force, the spin acquires a non-zero value and, as a consequence, the particle turns. It can be shown that the instantaneous radius of curvature of the trajectory is proportional to the inverse value of the spin [10].

Equations (12) and (13) can be rewritten in terms of the phase only, giving

In the overdamped limit this second order equation reduces to a first-order one, which coincides with the planar Vicsek continuous model derived from equation (2). The ISM is therefore an inertial generalization of the VM, which reduces to the latter in the dissipative limit. The opposite limit in which η → 0 renders a deterministic equation with a Hamiltonian reversible structure (see equations (12) and (13)). In this case, since si represents the generator of the rotational symmetry of the velocity, its total value is a conserved quantity.

2.4. Collective turns and dynamical regimes

Let us now discuss how the ISM can appropriately describe the dispersion law observed in flocks of birds [9]. To do so, we recall first what is known from experiments [9, 20]:

Bird flocks are highly ordered systems, with measured polarization values ;Turns are planar, i.e. each trajectory lies approximately on a 2d plane;Mutual distances do not change during turns, and individuals sweep equal radius paths;Turns have a localized spatial origin, and the signal propagates linearly in time through the group with speed and with negligible attenuation.

Given these premises, we can now focus on the planar version of the model, as in equation (14). This expression becomes even simpler if the system is in the deeply ordered phase, as flocks are. In this case, if we choose as a reference direction the one of the global velocity , the individual phases are very small, i.e. . One can then expand the potential in equation (3) up to quadratic order—which is called the spin-wave approximation (SWA)—to find

where we have introduced the discrete Laplacian Λ with

Since individuals do not change their mutual positions during turns, we can assume that the interaction network nij remains approximately the same. If the phases vary slowly from one individual to the next, we can further treat phases as a continuous field . The discrete Laplacian then becomes a continuous one, (a being the mean nearest-neighbor distance), and equation (14) becomes

with (assumed to be constant on a regular network). While computations can be easily performed also in terms of the discrete variables, the continuous form is more convenient for visualizing the dispersion relation. Indeed, let us now set ξ = 0 into equation (17) (or equivalently focus on ), and Fourier transform in both space and time to get

From this relation we can immediately see that in the Vicsek case (χ = 0) the dispersion law becomes purely diffusive, i.e. , leading to strong attenuation and a non-linear relation between space and time. Conversely, when χ is different from zero a more complex dispersion law is obtained,

where we introduced the propagation speed and the effective dissipation γ as

and where .

From equation (19) we can clearly evince that the two conditions observed in flocks of birds, i.e. linear dispersion law and no attenuation, are reproduced when and . In this deeply underdamped regime, the ISM predicts a linear dispersion law with propagation speed given by equation (20). This is a highly non-trivial relation linking together the way directional information propagates through the system, and the degree of order (set by J). Remarkably, this prediction is very nicely verified in experimental data [9], thus supporting the idea that the ISM in the strongly underdamped regime provides a good description for collective turning in flocks.

More generally, according to the ratio of inertia and dissipation, and depending on the size of the system, the ISM interpolates between the dissipative Vicsek limit, and the efficient limit of flocks. For all the modes are overdamped, while for the system sustains linear propagation with speed and constant attenuation with damping time . Since k0 is the inverse of a length scale, it sets a limit on the size L of a flock through which a signal can linearly propagate: indeed, imposing implies . Another equivalent way to understand this constraint comes by thinking in terms of timescales. Indeed, the time needed for a signal of speed to cross the entire flock of size L is

but the same signal gets attenuated over a time scale

We can then identify two dynamical propagation regimes:

Underdamped regime (inertial propagation): . In this case directional changes travel linearly through the whole system before attenuation can deteriorate the signal. Flocks of birds observed in experiments [9] belong to the deep extreme of this regime, where attenuation is negligible for all the individuals. In terms of the parameters of the model, the condition defining this regime is . Overdamped regime (dissipative propagation): . Propagation of information is inefficient because the signal gets dissipated before reaching the other end of the group. In this case, the signal travels unaffected up to certain scales , but it is damped on larger scales. This might result in a strong deformation of the group, or even in its fragmentation. It is likely that fish schools investigated in [4] belong to this regime.

The dispersion law and the way information is propagated through the system determine the dynamical behavior of the scalar polarization and of the correlation functions [18]. In this work, we are interested in understanding how collective turns might arise in finite groups, and we shall find that the dispersion law affects in a crucial way the occurrence of these events. To do so, we will study the response of finite systems to a local perturbation under different boundary and motility conditions, and we will explore the dynamical regimes numerically by tuning the various parameters of the model.

In the previous section we introduced the ISM and we discussed the predictions of the model, under specific approximations, concerning the dispersion law in the system. In this section, we want to generalize our discussion to the study of perturbation events. When a group changes its global direction of motion, this often happens because some external perturbation or disturbance acts upon one or more individuals in the group. This is not the only possibility (see e.g. [21]), but it is certainly one of the most interesting. A reasonable choice is to model such a perturbation as a field linearly coupled to the individual velocities, in which case

We note that, in principle, in the ISM it would also be possible to couple a field to the spins . This would however change the angular velocity of the perturbed individuals, while it would not bias them directionally, which is here our main purpose (see however [22]). In the planar case, the equations of motion then read

These are the equations that we will consider in the numerical simulations performed in the remaining of this work. To make analytical progress, however, we still need to perform some simplifying approximations.

3.1. High order and slow network

First of all, we consider a system in its polarized phase. Indeed, our aim is to investigate how a state of collective motion can change under external perturbations. We then perform the following approximations:

Spin-wave approximation: the system is highly polarized, and the individual velocity vectors deviate weakly from the orientation of the global polarization . If the latter is chosen to be initially aligned to the x-axis when the external field is applied, then in the initial phases of the collective turn. Fixed network: we assume that the adjacency matrix nij appearing in the equations of motion is no longer a function of time. This assumption is reasonable when the timescale over which the collective turn develops is much shorter than the typical reshuffling time of the interaction network nij . This condition of local equilibrium [23] is precisely what happens in flocks of starlings. A deviation from this condition is however expected to arise for sufficiently large values of v0. This analysis will be the subject of section 5.

Under these approximations, the equations of motion for si and ϕi take the form

where αi denotes the direction of , while is its magnitude. We choose as the relevant observable the average polarization angle

This coincides, for small angles ϕi , with the angle formed by the polarization measured at time t (after the perturbation) with the initial polarization (i.e. the one maintained in the absence of perturbations). It therefore characterizes the way the system changes in time its collective flight direction.

3.2. Analytical results

The derivation of