Collective behavior of flocks in a marginalized ordering state

We develop a model for the rich dynamics of a flock in a marginalized ordering state. The aim is to present an inter-individual coordination mechanism that keeps a flock constantly ready to respond to perturbations naturally present in biological systems. We extend the generalized Cucker-Smale model with the coupling of acceleration, and introduce adaptive reaction times of each bird. We regard two key factors in the reaction times: (1) the local ordering state of each bird and (2) reaction sensitivity of a flock to the neighbor's momentum change with 1/kappa. We show that our model displays innate fluctuations that lead to rich dynamics as a reminiscent of natural flocks due to the adaptive reaction delay. This happens without relying on stochastic variables. We compute the correlation lengths of the fluctuations and find that the correlation of velocity and speed is scale-free, indicating some criticality of a flock. Surprisingly, at a large value of 1/kappa (i.e., reaction sensitivity is high), the transition occurs from the standard diffusion to the super-diffusive Levy flights as we increase the strength of the velocity alignment. Our results indicate that the emergence of long-term behaviors such as Levy flights can also be explained in terms of the inter-individual interaction that makes the system in a marginalized ordering state