On Stability of V-Like Formations

Group behavior has received much attention as a test case of self-organization. There has been much written in recent years to investigate interactions within groups of agents. These agents can be animals moving in an interactive way, such as birds, but can also refer to situations such as people driving in traffic. The models that describe these interactions are able to reproduce different structures and patterns relating to the movement and interaction of the agents involved. The advantages and necessities of this type of analysis in any complex biological, technological, economic, or social system are important and far-reaching. Each model that we will discuss describes interaction between agents. Focusing on animal groups, such systems received attention not only from biologists and ethologists, but also from physicists, mathematicians, and engineers. This interest has produced a huge amount of literature, which is well documented and reviewed: see, for instance, [7, 8, 11] and [3]. Loosely speaking, the basic idea behind these works is that complex collective behavior arises from simple interactions among close animals. This paper considers a simple case of five birds in V formation, flying at constant velocity. The aim is to study the feasibility of this solution and its stability for three models proposed in the literature. The first model we will analyze was proposed in [4]. It is an agent-based model, in the sense that each agent is singularly considered. The model is also leaderless, meaning that all agents behave following the same set of rules and their behavior is not imposed by others. The model is purely based on attraction-repulsion intractions between group mates. Attraction allows the group to be formed and stay tight, while repulsion allows for coliision avoidance between group mates and keeps them well spaced. This model does not combine attraction and repulsion with velocity alignment, unlike many other models that have been proposed. The model also assumes that each agent interacts with a limited number of group mates that are in a suitable pre-defined sensitivity zone. The authors were able to produce cluster formations, line formations and V-like formations by altering the attraction and repulsion parameters of their model. The second model was proposed by Cucker and Smale in [6] and additional results are available in [10]. The model is of second order, in the sense that the dynamics is given for position and velocity of each animal. More precisely,