Les bancs de poissons sont des entités pouvant regrouper plusieurs milliers d'individus qui se déplacent de façon synchronisée, dans un environnement sujet à de multiples perturbations, qu'elles soient endogènes (e.g. le départ soudain d'un congénère) ou exogènes (e.g. l'attaque d'un prédateur). La coordination de ces bancs de poissons, décentralisée, n'est pas encore totalement comprise. Si les mécanismes sous-jacents aux interactions sociales proposés dans des travaux précédents reproduisent qualitativement les structures collectives observées dans la nature, la quantification de ces interactions et l'accord quantitatif entre ces mesures individuelles et les motifs collectifs sont encore rares dans les recherches récentes et forment l'objet principal de cette thèse. L'approche de ce travail repose sur une étroite combinaison entre les méthodes expérimentales et de modélisation dans l'objectif de découvrir les liens entre les comportements individuels et les structures observées à l'échelle collective. Nous avons caractérisé et quantifié les interactions et mécanismes à l'origine, d'abord, de la coordination des individus dans les bancs de poissons et, ensuite, de la propagation d'information, quand le groupe subit une perturbation endogène ou exogène. Ces travaux, tous réalisés en étudiant la même espèce de poisson d'eau douce, le nez-rouge (Hemigrammus rhodostomus), ont mobilisé une diversité de méthodes expérimentales, d'analyses statistique et de modélisation, à l'interface de l'éthologie, de la physique statistique et des sciences computationnelles. / Fish schools are systems in which thousands of individuals can move in a synchronised manner in a changing environment, with endogenous perturbations (e.g. when a congener leaves the group) or exogenous (e.g. the attack of a predator). The coordination of fish schools, decentralised, is not completely understood yet. If the mechanisms underlying social interactions discussed in previous studies qualitatively match the social patterns observed in nature, the quantification of these interactions and the quantitative match between individual measurements and collective patterns are still sparse in recent works and are the main focus of this thesis. This work combines closely experimental and modelling methods in order to investigate the links between the individual behaviours and the patterns observed at the collective scale. We have characterised and quantified the interactions and mechanisms at the origin of, first, the coordination of individuals in fish schools and, second, the propagation of information, when the group is under endogenous or exogenous perturbations. This thesis focuses on one freshwater fish species, the rummy-nose tetra (Hemigrammus rhodostomus), and is the result of a diversity of experimental methods, statistical analyses and modelling, at the interface of ethology, statistical physics and computational sciences.

This thesis consists of three parts: The first and second parts focus on long-time asymptotics of macroscopic and kinetic models respectively, while in the third part we connect these regimes using different scaling approaches. (1) Keller–Segel-type aggregation-diffusion equations: We study a Keller–Segel-type model with non-linear power-law diffusion and non-local particle interaction: Does the system admit equilibria? If yes, are they unique? Which solutions converge to them? Can we determine an explicit rate of convergence? To answer these questions, we make use of the special gradient flow structure of the equation and its associated free energy functional for which the overall convexity properties are not known. Special cases of this family of models have been investigated in previous works, and this part of the thesis represents a contribution towards a complete characterisation of the asymptotic behaviour of solutions. (2) Hypocoercivity techniques for a fibre lay-down model: We show existence and uniqueness of a stationary state for a kinetic Fokker-Planck equation modelling the fibre lay-down process in non-woven textile production. Further, we prove convergence to equilibrium with an explicit rate. This part of the thesis is an extension of previous work which considered the case of a stationary conveyor belt. Adding the movement of the belt, the global equilibrium state is not known explicitly and a more general hypocoercivity estimate is needed. Although we focus here on a particular application, this approach can be used for any equation with a similar structure as long as it can be understood as a certain perturbation of a system for which the global Gibbs state is known. (3) Scaling approaches for collective animal behaviour models: We study the multi-scale aspects of self-organised biological aggregations using various scaling techniques. Not many previous studies investigate how the dynamics of the initial models are preserved via these scalings. Firstly, we consider two scaling approaches (parabolic and grazing collision limits) that can be used to reduce a class of non-local kinetic 1D and 2D models to simpler models existing in the literature. Secondly, we investigate how some of the kinetic spatio-temporal patterns are preserved via these scalings using asymptotic preserving numerical methods.