Systematic design of decentralized feedback for coordinated control of multi-agent systems has much to gain from the rigorous examination of the nonlinear dynamics of collective animal behavior. Animals in groups, from bird ocks to fish schools, employ decentralized strategies and have limitations on sensing, computation, and actuation. Yet, at the level of the group, they are known to manage a variety of challenging tasks quickly, accurately, robustly and adaptively in an uncertain and changing environment. In this paper we review recent work on models and methods for studying the mechanisms of collective migration and collective decision-making in high-performing animal groups. Through bifurcation analyses we prove systematically how behavior depends on parameters that model the system and the environment. These connections lay the foundations for proving systematic control design methodologies that endow engineered multi-agent systems with the remarkable features of animal group dynamics.