  We consider the macroscopic model derived by Degond and Motsch from a
time-continuous version of the Vicsek model, describing the interaction
orientation in a large number of self-propelled particles. In this article, we
study the influence of a slight modification at the individual level, letting
the relaxation parameter depend on the local density and taking in account some
anisotropy in the observation kernel (which can model an angle of vision). The
main result is a certain robustness of this macroscopic limit and of the
methodology used to derive it. With some adaptations to the concept of
generalized collisional invariants, we are able to derive the same system of
partial differential equations, the only difference being in the definition of
the coefficients, which depend on the density. This new feature may lead to the
loss of hyperbolicity in some regimes. We provide then a general method which
enables us to get asymptotic expansions of these coefficients. These expansions
shows, in some effective situations, that the system is not hyperbolic. This
asymptotic study is also useful to measure the influence of the angle of vision
in the final macroscopic model, when the noise is small.
