5 to the desynchronized limit as 1 − Q ∝ δ$_{0}$$_{c}$ − δ$_{0}$ . The dis- tribution of the actual frequency P$_{Ω}$ (Ω) for δ < δ$_{0}$$_{c}$ has three distinct peaks, one at ω = 0 (the coherent group) and two symmetric peaks for ω > 0 and ω < 0 (the in- coherent group). The central peak vanishes for δ > δ$_{0}$$_{c}$ . These behaviors are qualitatively different from the nu- merical results shown in Figs. 2 and 3. In order to explain the unconventional transition behavior, it is necessary to develop a theoretical framework that incorporates spa- tial fluctuations, which is an interesting problem for the future. Conclusion. - The synchronization transition caused by long-range hydrodynamic coupling is shown to be more gradual than for the global coupling. The threshold angle for the crossover is estimated, and is not far from the mean-field estimate for the transition point. It sug- gests that a very narrow angle distribution is required to achieve correlated motion in a real bacterial carpet. We believe that this work sheds some light on the delicate is- sues involved in hydrodynamic synchronization and hope that it stimulates further theoretical and experimental studies of the rich behavior of such systems. NU thanks the hospitality at the University of Sheffield where this work was initiated, and financial support from Grant-in-Aid for Scientific Research from MEXT. RG ac- knowledges financial support from the EPSRC. ∗ Electronic address: uchida@cmpt.phys.tohoku.ac.jp † Electronic address: r.golestanian@sheffield.ac.uk [1] Y. Kuramoto, Chemical Oscillations, Waves, and Turbu- lence , (Springer, New York, 1984). [2] J. A. Acebr´ on et al., Rev. Mod. Phys. 77 , 137 (2005). [3] J. D. Crawford, and K. T. R. Davies, Physica D 125 , 1 (1999). [4] H. Sakaguchi, S. Shinomoto and Y. Kuramoto, Prog. Theor. Phys. 77, 1005 (1987) [5] Y. Kuramoto, Prog. Theor. Phys. 94 321 (1995); Y. Ku- ramoto and D. Battogtokh, Nonlin. Phenom. Complex Syst. 5 , 380 (2002). [6] D. M. Abrams and S. H. Strogatz, Phys. Rev. Lett. 93 , 174102 (2004). [7] J. L. Rogers and L. T. Wille, Phys. Rev. E 54 , R2193 (1996). [8] M. Mar´ odi, F. d’Ovidio, and T. Vicsek, Phys. Rev. E 66 011109 (2002). [9] V. E. Tarasov and G. M. Zaslavsky, Chaos 16 , 023110 (2006); N. Korabel, G. M. Zaslavsky, and V.E. Tarasov, Commun. Nonlin. Sci. Numer. Simul. 12 , 1405 (2007). [10] G.I. Taylor, Proc. R. Soc. A 209 , 447 (1951). [11] I.H. Riedel, K. Kruse, and J. Howard, Science 309 , 300 (2005). [12] M. Reichert, H. Stark, Eur. Phys. J. E 17 493 (2005). [13] Y. Yang, J. Elgeti, and G. Gompper, Phys. Rev. E 78 , 061903 (2008). [14] R.E. Goldstein, M. Polin, and I. Tuval, Phys. Rev. Lett. 103 , 168103 (2009). [15] S. Gueron and K. Levit-Gurevich, Proc. Natl. Acad. Sci.(USA) 96 , 12240 (1999). [16] M.C. Lagomarsino, B. Bassetti, P. Jona, Eur. Phys. J. B 26 , 8188 (2002); M.C. Lagomarsino, P. Jona, and B. Bassetti, Phys. Rev. E 68 , 021908 (2003). [17] P. Lenz and A. Ryskin, Phys. Biol. 3 205 (2006); T. Nie- dermayer, B. Eckhardt, and P. Lenz, Chaos 18 , 037128 (2008). [18] A. Vilfan and F. J¨ ulicher, Phys. Rev. Lett. 96 , 058102 (2006). [19] B. Guirao and J-F. Joanny, Biophys. J. 92 , 1900-1917 (2007). [20] N. Darnton et al., Biophys. J. 86 , 1863-1870 (2004); [21] M. J. Kim and K. S. Breuer, ASME J. Fluid Eng., 129 , 319 (2007); Small 4 , 111 (2008). [22] N. Uchida and R. Golestanian, preprint arXiv:0911.4253. [23] J.R. Blake, Proc. Camb. Phil. Soc. 70 , 303 (1971).