Fish, networks, and synchronization

Collective behavior of animal groups, such as bird flocks and fish schools, is an emergent phenomenon characterized by spatial patterns and synchronized motion of individuals. Animal groups comprising thousands of members are able to coordinate their positions and movements based on simple, local interactions among peers rather than in response to a single leader. Due to the distribution of information gathering and sharing among the group, individuals benefit from many aspects of social life, including protection from predators, efficient locomotion, and ease of foraging. Mathematical modeling of this phenomenon has garnered interest in the study of distributed systems, due to the efficiency and robustness of synchronization observed in nature. Individual-based modeling of distributed systems has yielded simulation results that bear striking resemblance to various modes of behavior observed in animal groups, including highly polarized movements and circular milling. In considering animal groups as a collection of individuals, the topology of interactions among group mates plays a pivotal role in the agreement of individuals on a common state. In fact, consensus protocols, which are updating algorithms widely used to model the synchrony characteristic in collective behavior, rely heavily on the underlying topology of interactions among individuals.;In this dissertation, we present a biologically-inspired mathematical model of collective behavior and we develop both numerical and experimental techniques to test the ability of such models to mimic biological systems. The model is inspired by psychological limitations on the perception of numbers, known as numerosity, which have been shown to play an important role in the collective behavior exhibited by animal groups. We incorporate this limitation into an individual-based behavioral model of group motion, which we explore in a simulation study. The numerosity constraint is also translated into a network model of interaction, over which we analytically derive conditions for the consensus of agents comprising homogenous peers or groups of leaders and followers. Moreover, we generalize the numerosity constraint to a network model that encompasses many relevant networks in the consensus literature, thus extending the applicability of these results. To assess complexity in a model of collective behavior, we adopt numerical methods using a machine-learning algorithm for dimensionality reduction of large-scale data sets. In addition, we conduct a variety of animal experiments using both robotic fish and computer-animated images as stimuli. Such experimental techniques are proposed as viable options for assessing the ability of a model to reproduce the complex collective behaviors observed in animal groups.