Skip to main content
European Commission logo print header

Statistical and dynamical analysis of collective behaviour in a three-dimensional motion: Empirical studies and modelling

Final Report Summary - PASSAROLA (Statistical and dynamical analysis of collective behaviour in a three-dimensional motion: Empirical studies and modelling)

Project context and objectives

Collective behaviour is widespread among animal groups. Bird flocks, in particular, represent paradigmatic examples of self-organised collective motion, where global coordination emerges as a unique consequence of mutual interactions between individuals in the group. In this respect, there is an intriguing analogy with what happens in physics, where ordering phenomena occur in systems of many interacting units (either particles or spins). For this reason, many of the concepts and techniques developed in statistical and condensed matter physics can be precious to analyse, quantify and understand collective phenomena in biological systems. The research group at ISC-CNR that hosted the present project has pursued in the last years a whole research line devoted to the study of collective animal behaviour, where experimental work is combined with data analysis, theoretical analysis and modelling, and where the physics background is a crucial methodological ingredient.

In particular, our group has realised several experiments on bird flocks in the field and obtained three-dimensional data on individual positions, velocities and trajectories for very large groups (hundreds to thousands individuals). These data allow a statistical characterization of collective behaviour, which was unthinkable before. In general, our main objective is to use this characterization to better understand the microscopic origins of collective behaviour: what are the interactions between individuals and how they concur to produce the collective patterns that we observe. The project PASSAROLA and the work of Dr Duarte Queiros developed within this more general research activity, focusing on the analysis of individual trajectories in time. Its specific objectives are related to better understand how individual birds move one with respect to the other and how this mutual dynamics is connected with the collective nature of flocking. On the hand, this implies analysing empirical data and making sense of them. On the other hand, this also implies to develop more abstract theoretical models to address the many open questions related to dynamical inference, finite time series investigation and the role of a dynamical network of interactions.

Project work

The work carried out during the project and the main results can be summarised as follows:
- Statistical and dynamical analyses: the largest stake of the activities within the project was related to the assessment of the impact of the dynamics of the network of interactions in a flock of birds. The coordination between birds in a flock is the outcome of the interaction network between individuals, particularly nearest neighbours. Recent empirically based results obtained from the hosting group indicate that during flocking each bird aligns its direction of motion with those of closest neighbours. The number of these close interacting neighbours has been found to be of order of seven and - interestingly - is independent of density, a feature that has been showed to enhance cohesion and robustness to perturbations. These interacting neighbours are not always the same though. In other words, since individuals in the flock move and exchange relative positions, the interaction network dynamically evolves as well. Within this context, a leading question is thus to investigate how neighbourhood relationships change in time, and the magnitude of the mutual diffusion of the birds in the flock. To this aim, we considered a data-set of individual trajectories in large flocks of starlings, obtained from field studies, and performed a statistical analysis of diffusion properties.

Project results

Diffusion properties: during flocking, birds move in a very polarised way, following approximately the same direction of motion. Still, when we looked in detail at how they move one with respect to the other and within the group we discovered that they exchange positions and rearrange their location with respect to each other with a rather fast dynamical process. They perform 'relative' trajectories that are erratic, in what constitutes a qualitative indication of diffusion. More quantitatively, when looking at diffusion in the centre of mass reference frame and at relative diffusion, we find a super-diffusive behaviour (i.e. faster than Brownian motion), with exponents that are different than theoretical predictions for living active matter. Spatially, diffusion is anisotropic and concentrates in the direction along the wing-span. This finding agrees with energetic considerations and with some very general theoretical arguments. Interestingly, the results on the diffusion properties can be used to fully and quantitatively explain the dynamical rearrangement of the neighbourhood relationships. In this way we have been able to show that the way the interaction network between individuals dynamically evolves is uniquely due to the relative motion in space, while there are no indications of additional forces, which might tend individual to cluster or correlate.

Border properties: a much debated question in animal grouping is whether individuals at the border exhibit specific properties. Being at the border means being exposed at higher risk towards predation, and one might ask whether this influences their behaviour. We investigated the dynamics at the border and indeed found that birds in the border exhibit a particular behaviour. In particular, permanence in such a border state is lower than in the rest of the flock, which alludes to an overall selfish intention in the state of collective motion.

Analysis of velocity fluctuations: to better understand how it is possible to describe flocks in terms of a stochastic process, we analysed the statistics of the flight directions during motion. The velocities fields are typified as a non-stationary process with a non-Gaussian distribution of the velocities with a slow convergence of the rate function that can be calculated within the framework of large deviation theory. This slow convergence of the velocity fluctuations concurs with the slowly decaying self-correlation of the velocity fluctuations and with the (sub-ballistic) super-diffusive behaviour of the birds.

Modelling: parallel to data analysis, some of the relevant questions at technical and conceptual level have been addressed with simplified models. In particular, theoretical and computational techniques have been developed to investigate finite time series:
-It was investigated the combined effect of dissension and diversity of random forces in a mean field model of collective behaviour. For the same mean field parameters, it was verified that the introduction of dissension can actually help the system achieve a global ordered state, thus bringing forth evidence over the ordering role that the introduction of disorder might have.
-It was introduced a new non-parametric method for the identification of patches of quasi-non-stationary in non-stationary time series. This method has got several applications, from the description of insect moving in a swarm to finance a physiological diagnosis.
- Within the context of out-of-equilibrium statistical mechanics, it was assessed how a coupled system exchanges heat between its elements as a function of the type of noise that acts on it. It was shown that the noise only plays a significant role when the coupling between the elements is non-linear. Bearing in mind the relation between exchange heat, entropy and information this work constitutes a first step to the definition of a quantitative observable describing the information flux between elements of a non-equilibrium dynamical system where the thermodynamic concept of temperature cannot be applied.
- Making use of the concept of evolving network and information flux in a non-equilibrium system it was performed a study where the effectiveness of the isolation for the achievement of an ordered state is surveyed. In applying the concept of dynamical network of neighbours in an epidemiological problem, it was demonstrated that, under current laws of self-isolation and simply recurring to this enforcement, pandemics are impossible to quell.

Project impact

All the above results help to gather more information on the mechanistic aspect of grouping. In particular, they concur to understand the crucial link between what are the relevant rules and local coordination mechanism (i.e. what we call mutual interactions), the dynamics of the group as a multi-unit system and the nature of the collective patterns exhibited at group level.