Emergent Behavior in Flocks

We provide a model (for both continuous and discrete time) describing the evolution of a flock. Our model is parameterized by a constant beta capturing the rate of decay-which in our model is polynomial-of the influence between birds in the flock as they separate in space. Our main result shows that when beta<1/2 convergence of the flock to a common velocity is guaranteed, while for betages1/2 convergence is guaranteed under some condition on the initial positions and velocities of the birds only