4 These relations, combined with the preceding linear sta- bility analysis, define a phase diagram in the ( R, Φ) plane (Figure 3) that agrees remarkably well with numerical re- sults for systems prepared in a (slightly noisy) uniform initial state. available become less reliable [26]. We emphasize again that the basic mechanism for patterning presented above does not depend on the precise form chosen for v ( ρ ). Quantitatively however, Eq. (8), and the frontier between subcriticality and supercriticality, do depend on the de- tails of the interplay between the nonlinearity in v ( ρ ) and the logistic growth term. We leave further analysis of such model-specific features to future work. Figure 3: Top : Phase diagram in the ( R, Φ) plane. The outer region corresponds to stable behavior whereas within the curve, patterning occurs. The solid line is the theoretical phase boundary – Eq (7) – which accurately fits the numerics (black squares). The blue and red sections correspond to continuous and discontinuous transitions respectively. The two magenta dots correspond to two 2D simulations which show ordered harmonic patterns close to supercriticality and amorphous patterns otherwise. Bottom-left : Transition in the supercritical regime. The blues lines correspond to the theory – Eq (8) – whereas the squares comes from simulations (Φ = 1 . 5; 1 . 35; 1 . 2 from top to bottom). Bottom-right : Transition in the subcritical regime for Φ = 1 . 06 and Φ = 1 . 7 (bottom to top). While the amplitude equation is more easily devel- oped in 1D, the stability analysis offered above is valid in higher dimensions and it is natural to ask what hap- pens in 2D, which is the relevant geometry for Petri dish studies with growing bacterial colonies. Fig. 4 shows the simulated time evolution of ρ ( r , t ) for a system started with small random fluctuations around the equilibrium density ρ$_{0}$ , with other parameters as in Fig. 1. Per- haps not surprisingly, bands are replaced by droplets of the high density phase dispersed in a low density back- ground at large times. This is the typical steady state obtained with a near-uniform starting condition. How- ever, the structure and organization of the bacterial drops in the steady state depends on the point ( R, Φ) chosen in the phase diagram. Generally, the closer the system is to the supercritical instability curve, the more ordered the patterns. For instance we have observed an essentially crystalline distribution of bacterial drops, which devel- ops defects and eventually becomes amorphous on mov- ing further away from the phase boundary (Fig. 3, insets to main panel). For particular choices of parameters, our model can also admit other steady state patterns. Close to the supercritical line, where the phase transition is continuous, we can obtain long-lived stripes, whereas for fixed large values of R and Φ close to the (right) sub- critical phase boundary, we have also observed ‘inverted droplets’ with a high density lawn punctuated by low density ‘holes’. Close to the transition, the emergent steady-state pat- tern can be studied using an amplitude equation (see Supporting Information). Introducing ε = ( R − R$_{c}$ ) /R$_{c}$ , one gets in 1D that for 1 . 08 ≤ Φ ≤ 1 . 58, the transition is supercritical (continuous) and the steady state is given by which agrees with simulations (Fig 3, bottom left). Out- side this range, the transition becomes subcritical (dis- continuous, Fig 3, bottom right) and the analytical tools u ≃ 1 + A ( ε ) cos( x ); A $^{2}$( ε ) = ε 18(1 − Φ) 2 34Φ 4 − 56Φ 3 − 24Φ 2 + 31Φ + 19 (8) In these 2D geometries initialized from a near-uniform state, droplets can coalesce in the early stages, while at late times the dynamics is governed by evaporation- condensation events (see Figs. 4 and 5). However it is already apparent from Fig. 4 that coarsening eventually stops and the droplets reach rather well-defined steady state sizes and centre-to-centre distances. This can be quantified by looking at the time evolution of the char- acteristic domain size, L ( t ), which we have computed as the inverse of first moment (times 2 π ) of the structure factor [20]. Fig. 5 suggests that L ( t ) at late times even- tually stops increasing and reaches a steady state value. (The visible steps in domain size mark discrete evapo- ration events involving smaller bacterial droplets; pre- sumably L ( t ) would become smooth for a large enough system.) These droplet patterns in steady state are very simi- lar to those observed for E. coli in a liquid medium or S. typhimurium in semi-solid agar (0.24% water-agar in Ref. [5]) when starting from a uniform distribution [6]. For the E. coli case, interactions are believed to come from chemoattractant, emitted by the bacteria them-