  We present a detailed study of the large-scale collective properties of
self-propelled particles (SPPs) moving in two-dimensional heterogeneous space.
The impact of spatial heterogeneities on the ordered, collectively moving phase
is investigated. We show that for strong enough spatial heterogeneity, the
well-documented high-density, high-ordered propagating bands that emerge in
homogeneous space disappear. Moreover, the ordered phase does not exhibit
long-range order, as occurs in homogeneous systems, but rather quasi-long range
order: i.e. the SPP system becomes disordered in the thermodynamical limit. For
finite size systems, we find that there is an optimal noise value that
maximizes order. Interestingly, the system becomes disordered in two limits,
for high noise values as well as for vanishing noise. This remarkable finding
strongly suggests the existence of two critical points, instead of only one,
associated to the collective motion transition. Density fluctuations are
consistent with these observations, being higher and anomalously strong at the
optimal noise, and decreasing and crossing over to normal for high and low
noise values. Collective properties are investigated in static as well as
dynamic heterogeneous environments, and by changing the symmetry of the
velocity alignment mechanism of the SPPs.
