Self-propelled particle models for collective animal behaviour

Abstract

The numerical simulations of physical systems are a widely used instrument in scienti c investigation. When the laws governing the system are known they allow to perform synthetic experiments and study the system's behaviour; while when the object of study in unknown they help to understand how likely an intuition can be. This is particularly true in statistical mechanics where linking the behaviour of the single particles with the behaviour of the whole system can be extremely di cult. In this thesis are presented new results obtained from the study of numerical models that are strongly connected with the experiments and the theories based upon them. The analysis of the spatial anisotropy of the neighbours distribution of a bird revealed that the interaction range of starlings is topological rather than metric: each bird interacts with its rst 6 7 neighbours independently from their metric distances. This was also con rmed by the maximum entropy theory through parameters inference. This experimental nding is linked with a plau- sible biological outcome: to grant the stability of the ock. A novel topological model is introduced, showing that the topological interaction outperforms the metric one in granting the cohesion of the ock. Topological interaction, being invariant with respect to local density uctuations, makes the ock less suscep- tible to fragmentation due to noise or external obstacles when compared with metric one. A summary of a powerful statistical inference method is presented: the max- imum entropy method. It consists in nding the less structured probability distribution compatible with the experimental ndings, and inferring the distri- bution's parameter. Numerical simulations where used to test the method on active o -equilibrium systems, showing a good agreement between the simulated model and the inferred parameters A new model is introduced, deriving it's behaviour from a mechanistic inter- pretation of the theoretical probability distribution given by maximum entropy method, and limiting assumptions to the lowest possible level (ideally null). This model reproduces the experimentally measured long-range correlation of the speed of the birds; and it shows that, in order to do it, the system has to be critical, i.e. a control parameter has to be near to its critical value. One of the most important issues in collective behaviour is investigated: how information propagates through the system. Experiments revealed us that, when a ock of birds perform a turn, the turning information travels across the

ock at a constant rate with very low attenuation. This constant rate informa- tion transfer can not be described by the classic theory due to the missing of two crucial ingredients: a conservation law associated with the rotational symmetry, and a term of behavioural inertia. A new model that accounts for these two ingredients was introduced. The models exhibits two di erent regimes, propa- gating and not-propagating directional information, depending on the damping parameter, and being in the strongly overdamped regime equivalent to the Vic- sek model. The theoretically predicted dependence of the speed of propagation of the signal from the parameters, and its non-trivial relation with the alignment are correctly reproduced by the model. Finally is considered a system completely di erent from bird ocks: the midge swarms. Experiments revealed that midge swarm are strongly correlated; that their interaction is based on alignment and that their interaction range is metric. the experimental correlations were reproduced using the Vicsek model, 1 in its original de nition with periodic boundary conditions and in a variant where a global central force is applied to all the particles. Such correlations are well reproduced when the model lies in a particular region of the phase diagram: the critical region. The rst nite size scaling analysis of the three-dimensional Vicsek model is presented, and the scaling variables are compared with the ones of real swarms, supporting the criticality hypothesis. Finally a parameter tting method to infer the interaction range of midges is proposed.