Planar swarming motion under single leader as nash equilibrium

Two dimensional foraging swarms are modeled as a dynamic noncooperative game played by swarm members, each one of which minimizes its total effort during the journey by controlling its velocity. It is assumed that each member monitors its distance only to the member that starts the journey up front, called the leader. The leader is only concerned with minimizing its total control effort. The foraging location is assumed to be known by all members. It is shown that a unique Nash equilibrium exists under certain assumptions on the nature of relative weighing between the motions along the two coordinates in the plane. The Nash equilibrium displays a number of observed characteristics of biological swarms; for instance, a V-shape formation is preserved during the whole journey.