  Understanding the organization of collective motion in biological systems is
an ongoing challenge. In this Paper we consider a minimal model of
self-propelled particles with variable speed. Inspired by experimental data
from schooling fish, we introduce a power-law dependency of the speed of each
particle on the degree of polarization order in its neighborhood. We derive
analytically a coarse-grained continuous approximation for this model and find
that, while the variable speed rule does not change the details of the ordering
transition leading to collective motion, it induces an inverse power-law
correlation between the speed or the local polarization order and the local
density. Using numerical simulations, we verify the range of validity of this
continuous description and explore regimes beyond it. We discover, in
disordered states close to the transition, a phase-segregated regime where most
particles cluster into almost static groups surrounded by isolated high-speed
particles. We argue that the mechanism responsible for this regime could be
present in a wide range of collective motion dynamics.
