8 the different terms to the prefactor of e ix in the r.h.s of Eq. (A10) √ ε . One could look for a more general expansion: − 2 ∂ 2 $_{x}$U$_{0}$ yields 2 A (A14) ( 3 − 2Φ Φ − 1 2 ∂$_{x}$ − 2 Φ ) U$_{0}$U$_{1}$ yields 2 A | A | 2 ( 3 − 2Φ Φ − 1 − 1 Φ$^{)}$ (A15) × ( 1 9 (4 3 − 2Φ Φ − 1 − 1 Φ ) − 2 Φ$^{)}$ − 4 3 Φ − 2 Φ − 1 ∂ 2 $_{x}$U 3 0 yields 4 Φ − 2 Φ − 1 A | A | 2 (A16) w = U$_{0}$ε α + U$_{1}$ε 2 α + U$_{2}$ε 3 α (A20) α = 1 k − 1 (A21) A $^{(}$9Φ $^{2}$(Φ − 1) $^{2}$+2 | A | $^{2}$(34Φ$^{4}$− 56Φ $^{3}$− 24Φ $^{2}$+31Φ+19) ) = 0 (A17) In this case, the expansion of equation (A5) yields two power series: $^{∑}$R$_{k}$ ε αk and $^{∑}$W$_{k}$ ε αk $^{+1}$. For the two se- ries to give terms that can balance each-other, one needs α + 1 = kα for k ≥ 2 and thus The sum of these terms vanishes only if and thus either The candidates for α are thus 1; 1 / 2; 1 / 3; . . . . Note that α ≤ 1 implies 2 α + 1 ≥ 3 α . We can therefore stop the expansion at 3 α and 2 α + 1 to get the first three terms in the expansion of equation (A5) Let us first try α = 1. The order by order the expan- sion yields Finally, the first order in the amplitude equation yields A = 0; or | A | 2 = 9Φ $^{2}$(1 − Φ) 2 2(34Φ 4 − 56Φ 3 − 24Φ 2 + 31Φ + 19) (A18) w ( x ) = 2 | A | √ ε cos( x − x$_{0}$ ) (A19) L $^{2}$U$_{0}$ = 0 , O ( ε ) (A22) L $^{2}$U$_{1}$ = − U 2 0 Φ − 3 − 2Φ Φ − 1 ∂ 2 $_{x}$U 2 0 − 2 ∂ 2 $_{x}$U$_{0}$ , O (2 ε ) (A23) where x$_{0}$ is a constant. Note that by construction | A | 2 > 0 and a non-zero solution only exists for Φ ∈ [1 . 08439 , 1 . 59237]. For these values of Φ, Eq. (A18) and (A19) work very well, as can be checked in figure 7. Outside this range the transition becomes subcritical and the standard approach does not work anymore. Al- ternative treatments have been proposed but are not as reliable (see ref [20] for more details). Interestingly, we see that the order of the transition and the amplitude of the perturbation depend on how non-linear terms in g ( w ) balance the linear growth term − 2 ε∂ 2 $_{x}$w in (A5). Since the former depends on the non-linear relation v ( ρ ), we do not expect equation (A18) to be generic, as opposed to the stability analysis which can be expressed solely in terms of D$_{e}$ ( ρ$_{0}$ ) and its derivative. Equation (A22) yields U$_{0}$ = Ae $^{ikx}$+ A $^{∗}$e− ikx but equation (A23) cannot be solved since there is a non-zero multiple of e ikx on the r.h.s. ( − 2 ∂ 2 $_{x}$U$_{0}$ ) which cannot result from the application of L 2 to any function. Thus α = 1 is not an option. For α ≤ 1 / 3, then α + 1 > 1 ≥ 3 α . There is thus no contribution of − 2 ε∂ 2 $_{x}$w to the first three orders in the expansion of (A5). In particular, the two first order are still given by (A8) and (A9), whereas the third order is given by (A10) without the term linear in U$_{0}$ . This means that the contribution (A14) is not present and the prefactor of e ikx in the r.h.s. of (A10) only contains multiples of | A | $^{2}$A . The resolvability condition (A17) is thus of the form A | A | 2 f (Φ) = 0 which implies | A | = 0. The only expansion which yields a result is thus for α = 1 / 2. In (A7), we expanded w in power series of √ ε , thus assuming that the amplitude is an analytic function of 2. What is the correct expansion? [1] Shapiro J. A. (1995) The significance of bacterial colony patterns. BioEssays 17: 597-607. [2] Harshey R. M. (2003) Bacterial motility on a surface: Many ways to a common goal. Ann. Rev. Microbiol. 57:249-273. [3] Budrene E. O., Berg H. C. (1991) Complex patterns formed by motile cells of Escherichia-coli . Nature 349: 630-633. [4] Budrene E. O., Berg H. C. (1995) Dynamics of formation of symmetrical patterns by chemotactic bacteria. Nature 376:49-53. [5] Woodward D.E. et al. (1995) Spatiotemporal patterns generated by Salmonella Typhimurium . Biophys. J. 68:2181-2189. [6] Murray J.D., Mathematical Biology , Vol. 2, Springer- Verlag, New York (2003). [7] Ben-Jacob E., Cohen, I., Levine H. (2000) Cooperative self-organization of microorganisms Adv. in Phys. 49 , 395-554. [8] Kawasaki K. et al (1997) Modeling Spatio-Temporal