In this Thesis I present simulation- and theory-based studies of pattern formation and growth in collections of micro-organisms, in particular bacterial colonies. The aim of these studies is to introduce simple models of the 'micro-scale' behaviour of bacterial cells in order to study the emergent behaviour of large collections of them. To do this, computer simulations and theoretical techniques from statistical physics, and in particular non-equilibrium statistical physics, were used, as the systems under study are far from thermodynamic equilibrium, in common with most biological systems. Since the elements making up these sytems - the micro-organisms - are active, constantly transducing energy from their environment in order to move and grow, they can be viewed as `active matter' systems. First, I describe my work on a generalization of an archetypal model of active matter - the Vicsek model of flocking behaviour - in which the speed of motion of active particles depends on the local density of particles. Such an interaction had previously been shown to be responsible for some forms of pattern formation in bacterial colonies grown on agar plates in the laboratory. Simulations and theory demonstrated a variety of pattern formation in this system, and these results may be relevant to explaining behaviour observed in experiments done on collections of molecular motors and actin fibres. I then go on to describe work on modelling pattern formation and growth in bacterial biofilms - dense colonies of cells growing on top of solid surfaces. I introduce a simple simulation model for the growth of non-motile cells on a flat surface, whereby they move only by growing and pushing on each other as they grow. Such colonies have previously been observed experimentally to demonstrate a transition from round to 'branched' colonies, with a pattern similar to diffusion-limited aggregation. From these simulations and analytical modelling, a theory of the growth of such colonies is developed which is quite different from previous theories. For example, I find that the colony cannot grow at a constant speed if the cells are not compressible. Finally, I present some results on genetic drift and evolution in growing bacterial colonies. Genetic drift is greatly enhanced in colonies which are expanding in space, as only a few individuals at the edge of the population are able to pass on their genes onto their progeny. The individual-based simulations of biofilms described above are used to analyse which factors - such as the shape of the colony, the thickness of the growing layer of cells, and the interactions between the cells - affect the rate of genetic drift and the probability of fixation of beneficial mutations. This has implications, for example, for the evolution of antibiotic resistance in such colonies.

abstract: Swarms of animals, fish, birds, locusts etc. are a common occurrence but their coherence and method of organization poses a major question for mathematics and biology.The Vicsek and the Attraction-Repulsion are two models that have been proposed to explain the emergence of collective motion. A major issue for the Vicsek Model is that its particles are not attracted to each other, leaving the swarm with alignment in velocity but without spatial coherence. Restricting the particles to a bounded domain generates global spatial coherence of swarms while maintaining velocity alignment. While individual particles are specularly reflected at the boundary, the swarm as a whole is not. As a result, new dynamical swarming solutions are found. The Attraction-Repulsion Model set with a long-range attraction and short-range repulsion interaction potential typically stabilizes to a well-studied flock steady state solution. The particles for a flock remain spatially coherent but have no spatial bound and explore all space. A bounded domain with specularly reflecting walls traps the particles within a specific region. A fundamental refraction law for a swarm impacting on a planar boundary is derived. The swarm reflection varies from specular for a swarm dominated by kinetic energy to inelastic for a swarm dominated by potential energy. Inelastic collisions lead to alignment with the wall and to damped pulsating oscillations of the swarm. The fundamental refraction law provides a one-dimensional iterative map that allows for a prediction and analysis of the trajectory of the center of mass of a flock in a channel and a square domain. The extension of the wall collisions to a scattering experiment is conducted by setting two identical flocks to collide. The two particle dynamics is studied analytically and shows a transition from scattering: diverging flocks to bound states in the form of oscillations or parallel motions. Numerical studies of collisions of flocks show the same transition where the bound states become either a single translating flock or a rotating (mill). / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2015

Le phénomène de mouvement collectif est présent parmi beaucoup de systèmes biologiques comme dans les volées d'oiseaux ou des bancs de poissons. Dans ces systèmes, le mouvement collectif apparait sans aucun leader ni force extérieure et est exclusivement dû à l'interaction parmi les individus et à la nature hors-équilibre de tout le système. Nous voulons étudier des modèles simples de mouvement collectif afin d'établir des classes d'universalité parmi la matière active sèche, c'est-à-dire des individus interagissant sans l'aide d'un fluide. Beaucoup de ces systèmes ont déjà été étudiés microscopiquement. Nous voulons obtenir des équations hydrodynamiques de ces systèmes pour confirmer les résultats microscopiques et pour prédire des propriétés nouvelles. Nous effectuons une dérivation d'équations hydrodynamiques en utilisant l'approche Boltzmann-Ginzburg-Landau introduit dans cette thèse. Quatre modèles de type Vicsek sont considérés. Un modèle polaire simple identique au modèle de Vicsek, un modèle mixte avec des particules polaires avec interactions nématiques, un modèle avec des particules polaires et interactions nématiques et finalement un modèle avec des particules polaires avec des interactions non-métriques. Dans chaque cas les équations obtenues sont étudiées de façon analytique et numérique. Nous trouvons que les équations obtenues reproduisent de façon fidèles les propriétés qualitatives des modèles microscopiques considérées, comme les différentes phases observées et la nature de transition entre ces phases. Dans certains cas des phases nouvelles sont trouvées, qui n'ont pas été reportées auparavant dans les modèles microscopiques. Beaucoup d'entre elles ont été confirmées a posteriori dans les simulations numériques de ces modèles.