Models for pattern formation in biological systems

This thesis presents models for two different kinds of pattern formation in biological systems. Developmental patterning refers to pattern formation by cell differentiation during an organism's development. Cell differentiation is controlled by morphogens, signaling molecules that diffuse in a growing organism. We focus on the specific case of the activation of the epidermal growth factor receptor pathway, a highly-conserved signaling pathway across animals, that controls both the posterior-anterior and the dorso-ventral axes during Drosophila oogenesis. Not only can the diffusion of morphogens control the growth of an organism, but the diffusion itself is influenced by the changing geometry of the domain. We develop a mathematical framework enabling this double coupling of the diffusion of a signal on a time-evolving Riemannian manifold and the evolution of the manifold via a vector field depending on the diffusing signal: "Developmental Partial Differential Equations". The second kind of pattern formation, behavioral patterning, arises from local interactions between individuals that lead to a global group behavior. We focus on several of these models, also referred to as Social Dynamics systems. We examine how to control the dynamics to guide the system to a target configuration. In particular, we study the optimal control of a collective migration model to guide the system to consensus at a target velocity, as well as the controllability away from consensus of an opinion dynamics system. We analyze the influence of the state space on the dynamics by designing a general opinion dynamics model on Riemannian manifolds. Lastly, we investigate the role of the interaction network on the periodicity of the dynamics, specifically in creating a "social choreography".Ph.D.Includes bibliographical referencesby Nastassia Pouradier Dutei