  The evolution of leadership in migratory populations depends not only on
costs and benefits of leadership investments but also on the opportunities for
individuals to rely on cues from others through social interactions. We derive
an analytically tractable adaptive dynamic network model of collective
migration with fast timescale migration dynamics and slow timescale adaptive
dynamics of individual leadership investment and social interaction. For large
populations, our analysis of bifurcations with respect to investment cost
explains the observed hysteretic effect associated with recovery of migration
in fragmented environments. Further, we show a minimum connectivity threshold
above which there is evolutionary branching into leader and follower
populations. For small populations, we show how the topology of the underlying
social interaction network influences the emergence and location of leaders in
the adaptive system. Our model and analysis can describe other adaptive network
dynamics involving collective tracking or collective learning of a noisy,
unknown signal, and likewise can inform the design of robotic networks where
agents use decentralized strategies that balance direct environmental
measurements with agent interactions.
