5 Figure 4: Numerical results for a 2D simulation with size equal to 16 × 16, λ = 0 . 3, D$_{0}$ = 1, α = 0 . 01, κ = 0 . 001 and ρ$_{0}$ = 10. Times corresponding to the snapshots are (in simulation units, from left to right): 5, 12, 19.6 and 50.6. Figure 5: Plot of the characteristic domain size, L ( t ) as a function of time for a system in the inhomogeneous phase, with initially random density fluctuations around ρ$_{0}$ . Param- eters were: α = 0 . 01, λ = 0 . 27, κ = 0 . 001, while the system size was 20 × 20. The solid line corresponds to a single run, while the dashed line is an average over 6 runs. The steps in the single run curve correspond to evaporation-condensation events, highlighted by black squares in the snapshots shown in the figure (before and after one of the steps respectively, arrows indicate positions on the plot corresponding to the two snapshots). our simulations with a single small droplet of high den- sity ρ , we find that a similarly patterned bacterial colony structure develops. First, the bacteria spread radially (through a Fisher-like wave), forming an unstructured lawn with the highest density at the center. This back- ground density increases logistically until the onset of instability via our generic phase-separation mechanism; with circular symmetry, the instability causes concentric rings of high bacterial density to successively develop that are very stable in time (Figs. 6c and 6d). The patterns observed at later times again depend on the position of the parameters in the ( R, Φ) plane. If we fix a value of R , e.g. 100, larger values of Φ in the unstable region lead to rings being very stable. For smaller values of Φ, on the other hand, effectively corresponding to weaker interactions between the bacteria, we observe that rings initially form but rapidly destabilize through a secondary modulation of the bacterial density along them. This eventually breaks the rings into a series of drops. The inner rings destabilize first, and the system evolves even- tually to the same steady state as found starting from a uniform density, composed of drops with well defined characteristic size and separation. All this phenomenol- ogy is strikingly reminiscent of the dynamics observed by Woodward et al [5] for S. typhimirium , where rings are stable at large concentrations of potassium succinate (a ‘stimulant’ which promotes pattern formation), but break up into drops at smaller ones. Our model shows a similar morphological change when decreasing Φ, i.e. the strength of the interactions. selves, that is not degraded over time [3, 4]. The chemoattractant distribution should become more and more uniform so that these interactions decay to zero as time proceeds. In our framework this is analagous to decreasing Φ, which will turn any initially unsta- ble state into a homogeneous one, and can thus explain that the patterns observed experimentally fade with time (whereas in our simulations Φ remains constant and the pattern are stable indefinitely). E. coli in a semi-solid medium also exhibits droplet patterns of high symmetry. In our framework, such patterns result from a continuous transition, close to the supercritical line. Different views are possible concerning the ability of our generic model to reproduce the observed chemotac- tic patterns of E. coli and S. typhimurium [6]. One pos- sibility is that Eq. (1), with the interpretation we have given for it, actually does embody the important physics of pattern formation in these organisms. Indeed it is well accepted that bacteria in the high density concen- tric rings are essentially non-motile [27]. The precise mechanism leading to this observation is not well under- stood [6], but it is possible that the chemotactic mecha- nism mainly acts to switch off motility at high density. If so, by focussing solely on this aspect (with a correspond- ingly vast reduction in the parameter space from that of explicit chemotactic models [5, 6]) our model might capture the physics of these chemotactic patterns in a highly economical way. Interestingly, our model is es- sentially local, whereas chemotaxis in principle mediates The growth of bacterial colonies of S. typhimurium starting from a small inoculum of bacterial cells in semi- solid agarleads to quite specific (transient but long-lived) patterns, with the bacteria accumulating in concentric rings that can subsequently fragment into a pattern of dots [5, 6]. Once again, although these patterns are be- lieved to stem from a chemotactic mechanism [6], we find they can arise in principle without one, so long as our two basic ingredients of density-suppressed motility and logistic growth are both present. Indeed, initializing