9 Figure 7: We simulated equation (A1) for systems of size L = 400 with periodic boundary conditions, using several values of Φ (From left to right, top to bottom, Φ = 1 . 06; 1 . 07; 1 . 12; 1 . 2; 1 . 35; 1 . 5; 1 . 7; 1 . 95; 2 . 5). The steady-state w ( x ) was then decomposed in Fourier series w ( x ) = a$_{0}$ + ∑ N/ 2 n a$_{n}$ cos(2 πnx/L ) + b$_{n}$ sin(2 πnx/L ), where N is the number of data points and the cut-off when n → ∞ is given by the Nyquist frequency. The blue points correspond to the amplitude of the largest mode: w$_{q}$ = max$_{n}$ √ a 2 n + b 2 $_{n}$. When the transition is continuous, we compare these points with the results of the amplitude equation w$_{q}$ = 2 √ ε | A | , where | A | is solution of (A18) (red lines) and the agreement is excellent. For Φ > 1 . 58 or Φ < 1 . 08, the transition is clearly discontinuous. and rectification of dilute bacteria. Europhys. Lett. 86 :60002. [9] Tyson R., Lubkin, S.R., Murray, J.D. (1999) A minimal mechanism for bacterial pattern formation. Proc. Roy. Soc. Lon. B 266:299-304. [19] Angelani L, Di Leonardo R, Ruocco G (2009) Self- Starting Micromotors in a Bacterial Bath. Phs. Rev. Lett. 102:048104. [20] Chaikin P.M., Lubenski T.C, Principles of condensed matter physics , Cambridge University Press, Cambridge (1995). [10] Brenner, M.P., Levitov, L.S., Budrene, E.O. (1998) Phys- ical mechanisms for chemotactic pattern formation by bacteria. Biophys. J. 74:1677-1693. [11] Espiov S.E., Shapiro J.A. (1998) Kinetic model of Pro- teus mirabilis swarm colony. J. Math. Biol. 36:249-268. [21] Schnitzer M.J. (1993) Theory of continuum random walks and application to chemotaxis. Phys. Rev. E 48:2553-2568. [12] Toner J., Tu Y.H., Ramaswamy S. (2005) Hydrodynam- ics and phases of flocks. Ann. Phys. (N.Y.) 318: 170-244. [13] Ramaswamy S, Simha R.A., Toner J. (2003) Active ne- matics on a substrate: giant number fluctuations and long-time tails. Europhys. Lett. 62: 196-202. [22] Berg H.C., E. coli in Motion, Springer (2003). [23] Dickman R. (1994) Numerical study of a field theory for directed percolation. Phys. Rev. E 50:4404-4409. [24] Dornic I., Chate H., Munoz M.A. (2005) Integration of Langevin equations with multiplicative noise and the vi- ability of field theories for absorbing phase transitions. Phys. Rev. Lett. 94: 100601. [25] Cates M.E., Vollmer J., Wagner A., Vollmer D. (2002) Phase separation in binary fluid mixtures with contin- uously ramped temperature. Phil. Trans. Roy. Soc. A 361:793-804. [26] P. Becherer, A.N. Morozov, W. van Saarloos, (2009) Probing a subcritical instability with an amplitude ex- pansion: An exploration of how far one can get. Physica D 238:1827-1840. [27] Mittal N., Budrene E.O., Brenner M.P., van Oudernaar- [14] Narayan V., Ramaswamy S., Menon N. (2007) Long- lived giant number fluctuations in a swarming granular nematic. Science 317: 105-108. [15] Tailleur J, Cates M.E. (2008) Statistical mechanics of in- teracting run-and-tumble bacteria. Phys. Rev. Lett. 100: 218103. [16] P. Galajda, J. Keymer, P. Chaikin, R. Austin (2008) J. Bacteriol. 189:8704-8707. [17] P. Galajda et al. (2008) Funnel ratchets in biology at low Reynolds number: choanotaxis. J. Modern Optics 55:3413-3422. (2008) [18] Tailleur J, Cates M-E (2009) Sedimentation, trapping, Patterns Generated by Bacillus subtilis . J. theor. Biol. 188:177-185.