Coarse-grained dynamics of alignment in animal group models

Coordinated motion in animal groups, such as bird flocks and fish schools, and their models gives rise to remarkable coherent structures. Using equation-free computational tools we explore the coarse-grained dynamics of a model for the orientational movement decision in animal groups, consisting of a small number of informed "leaders" and a large number of uninformed, nonidentical ``followers.'' The direction in which each group member is headed is characterized by a phase angle of a limit-cycle oscillator, whose dynamics are nonlinearly coupled with those of all the other group members. We identify a small number of proper coarse-grained variables (using uncertainty quantification methods) that describe the collective dynamics, and perform coarse projective integration and equation-free bifurcation analysis of the coarse-grained model behavior in these variables.