  The aim of the present paper is to elucidate the transition from collective
to random behavior exhibited by various mathematical models of bird flocking.
In particular, we compare Vicsek's model [Viscek et al., Phys. Rev. Lett. 75,
1226 -- 1229 (1995)] with one based on topological considerations. The latter
model is found to exhibit a first order phase transition from flocking to
decoherence, as the 'noise parameter' of the problem is increased, whereas
Viscek's model gives a second order transition. Refining the topological model
in such a way that birds are influenced mostly by the birds in front of them,
less by the ones at their sides and not at all by those behind them (because
they do not see them), we find a behavior that lies in between the two models.
Finally, we propose a novel mechanism for preserving the flock's cohesion,
without imposing artificial boundary conditions or attracting forces.
