7 for ε > 0 and small. The dynamics now reads where L = − (1 + ∂ 2 $_{x}$) 2 is the linear part of the evolution operator at the transition , 2 ε∂ 2 $_{x}$w gives an extra linear part due to the perturbation ( ε > 0) and g ( w ) is the non-linear part: result of chemotactic behavior [5, 6]. It is therefore re- markable that they can also arise purely from the inter- play of density-dependent diffusivity and logistic growth, without explicit reference to the dynamics (or even the presence) of a chemoattractant. This suggests that sim- ilar patterns might arise in organisms having no true chemotactic behavior at all. Such patterns could then be the result of local chemical signalling without gradi- ent detection (quorum-sensing, not chemotaxis) or even purely physical interactions (steric hindrance), either of which could in principle produce the required dependence of motility on density. Last, a motility decreasing with density is only one of the many mechanism that could lead to D ′ $_{e}$( ρ ) < 0 and our analysis would apply equally to all such cases. The simplest version of our model allows identifica- tion of just two dimensionless parameters that control the entire pattern-forming process. In both homogeneous and centrosymmetric geometries, this gives predictions for how the pattern type depends on interaction strength which are broadly confirmed by experimental data. This suggests that some of the diverse patterns formed by colonies of motile bacteria could have a relatively uni- versal origin. As usual with the amplitude equation approach, we expand w in power series of the perturbation ε and study Eq. (A5) order by order. As shown below (Eqs (21-23)), the correct expansion is 1. Amplitude equation Acknowledgments Appendix A We thank Otti Croze for discussions. We acknowledge funding from EPSRC EP/E030173. MEC holds a Royal Society Research Professorship. IP acknowledges the Spanish MICINN for financial support (FIS2008-04386). Expanding (A6) to the order ε 3 / 2 and substituting in (A5) yields order by order: Equation (A8) can be easily solved and yields The amplitude of the perturbation we are trying to derive is thus 2 | A | . Equation (A9) can also be solved directly: We show here how to derive the amplitude equa- tion (8). Let us start from the dimensionless equation of motion (6) and recall the two conditions for patterning Eq. (7): where B can de determined from higher order equations (but does not interest us here), and C and D are given by To analyze precisely the transition, we derive below the steady-state limit of the amplitude equation in 1D. By inspection one sees that the unperturbed steady-state of (A1) is given by u = 1. To characterize the ampli- tude of the perturbation around u = 1, we introduce u = 1 + w/ Φ so that w evolves with as can be checked by direct substitution in Eq. (A9). Equation (A10) does not always have a solution. In- deed, the application of L to any function U$_{2}$ cannot yield a multiple of e $^{ix}$, (since L e ix = 0 and L is linear). The r.h.s. however does contain a multiple of e ix whose prefactor must thus vanish. This gives a condition for the expansion to provide a proper steady-state solution of the problem. Let us summarize the contributions of We are interested by the vicinity of the transition where ˙ w = L w − 2 ε∂ 2 $_{x}$w + g ( w ) (A5) g ( w ) = − w 2 Φ − ∂$_{x}$ $^{[}$2(1 + ε ) $^{(}$(1 + w Φ − 1 ) e − 2 w − 1 $^{)}$∂$_{x}$ w ] (A6) w = U$_{0}$ε 1 / 2 + U$_{1}$ε + U$_{2}$ε 3 / 2 + . . . (A7) − L U$_{0}$ = 0 (A8) −L U$_{1}$ = − U 2 0 Φ − 3 − 2Φ Φ − 1 ∂ 2 $_{x}$U 2 0 (A9) −L U$_{2}$ = − 2 ∂ 2 $_{x}$U$_{0}$ − 2 U$_{0}$U$_{1}$ Φ − 3 − 2Φ Φ − 1 2 ∂ 2 $_{x}$U$_{0}$ U$_{1}$ (A10) − 4 3 Φ − 2 Φ − 1 ∂ 2 $_{x}$U 3 0 U$_{0}$ = Ae ix + A $^{∗}$e− ix (A11) ˙ u = ∇ [ Re − 2Φ $^{u}$(1 − Φ u ) ∇ u ] + u (1 − u ) − ∇ $^{4}$u (A1) U$_{1}$ = Be ix + B $^{∗}$e− ix + C + De 2 ix + D $^{∗}$e− 2 ix (A12) Φ > 1; R exp( − 2Φ)(Φ − 1) > 2 (A2) C = − 2 | A | 2 Φ ; D = A 2 9 (4 3 − 2Φ Φ − 1 − 1 Φ ) (A13) ˙ w = − ∂$_{x}$ [ Re − $^{2Φ}$(Φ − 1)(1+ w Φ − 1 ) e − 2 $^{w}$∂$_{x}$ w ] − w (1+ w Φ ) − ∂ 4 $_{x}$w (A3) Re − $^{2Φ}$(Φ − 1) = 2(1 + ε ) (A4)