3 FIG. 1: (color online) Phase diagram in the ( v 0 , ρ 0 ) plane. At v 0 = 0 the system exhibits a continuous mean field transition at ρ 0 = ρ$_{c}$ from an isotropic (I) to a homogeneous polarized (HP) state. The isotropic phase survives at finite v 0 in the region ρ < ρ$_{c}$ bounded by the vertical dashed line (blue online). For ρ 0 > ρ$_{c}$ there is a critical v$_{c}$ ( ρ 0 ) separating a polarized moving state with large anomalous fluctuations, named the “fluctuating flocking state”, at low self-propulsion speed from a high-speed phase of traveling stripes. The circles denote the values of v$_{c}$ ( ρ 0 ) obtained numerically with the error bars indicating the step size used in the computation. The dashed-dotted line (purple online) is the longitudinal instability boundary v L c $_{1}$( ρ 0 ) obtained in Section III. The dashed line (red online) is the splay instability boundary v S c ( ρ 0 ), given by Eq. (A11). The top right panel is a real space snapshot of the density profile in the striped phase. The stripes travel in the direction of the white arrow, that also denotes the direction of mean polarization in the high density regions. The bottom right panel shows a real space snapshot of the density profile in the “coarsening” transient leading to the fluctuating flocking state at v 0 < v$_{c}$ . Density values from low (blue) to high (yellow) are indicated in the side bar. The value of the density in the blue stripes is well below the critical value ρ$_{c}$ = 0 . 5, while the red stripes are well into the polarized phase. fusive current − D ∇ ρ that drives the system to a ho- mogeneous state. The anisotropy of the diffusion coeffi- cient relative to the direction of mean motion is neglected here for simplicity. The various terms in Eq. (2) (other than the one proportional to a$_{4}$ ) are obtained from the microscopic hard rod model and have a simple physi- cal interpretation. The polarization field P plays a dual role in self-propelled systems. On one hand, it represents the vector order parameter associated with the sponta- neous breaking of rotational symmetry in the polarized state. Its dynamics is then in the class of that of equi- librium polar liquid crystals and X-Y spin systems. On the other hand, v$_{0}$ P is also the mean velocity of the flock with which particles are convected. The interplay be- chemotaxis, may be needed to obtain a polar state. Non- physical or external mechanisms of this type are embod- ied in Bertin et al in an alignment rule, but are absent in the work by Baskaran and Marchetti that aimed at identifying the role of purely physical interactions in con- trolling the large scale behavior of self-propelled systems. The goal of the present paper is to study the stability of the homogeneous polar state. For this reason we have added phenomenologically the term proportional to a$_{4}$ to the equations derived in [23]. Both a$_{2}$ and a$_{4}$ will then treated as phenomenological parameters. The density satisfies a conservation law, with a flux controlled by two terms: ρv$_{0}$ P describing convection along the mean self-propulsion velocity, v$_{0}$ P , and a dif-