  Starting from recent experimental observations of starlings and jackdaws, we
propose a minimal agent-based mathematical model for bird flocks based on a
system of second-order delayed stochastic differential equations with
discontinuous (both in space and time) right-hand side. The model is
specifically designed to reproduce self-organized spontaneous sudden changes of
direction, not caused by external stimuli like predator's attacks. The main
novelty of the model is that every bird is a potential turn initiator, thus
leadership is formed in a group of indistinguishable agents. We investigate
some theoretical properties of the model and we show the numerical results.
Biological insights are also discussed.
