6 Figure 6: Dynamics of formation of ”chemotactic patterns” in 2D, starting from a single small bacterial droplet in the middle of the simulation sample. Top row: Formation of ”chemotactic rings” in a system with α = 0 . 1, λ = 0 . 33, and κ = 0 . 001. The simulation box has size 40 × 40. The snapshots correspond to times equal to (in simulation units, from left to right) 10, 50, 100 and 270. Bottom row: Breakage of ”rings” into ”dots”. The four snapshots correspond to the time evolution of a system with α = 0 . 1, λ = 0 . 26, and κ = 0 . 001. We show a 40 × 40 fraction of the simulation box, with the boundaries far away and not affecting the pattern. The snapshots correspond to times equal to (in simulation units, from left to right) 10, 70, 290 and 1220. with D a many-body diffusivity and µ the correspond- ing mobility. This alone ensures that no drift velocity can arise purely from gradients in D . In nonequilbrium systems, one should expect generically to find such drift velocities, and the run-and-tumble model is merely one instance of this. Accordingly one can expect in princi- ple to find cases of negative D$_{e}$ in other microorganisms showing distinctly different forms of density-dependent self-propulsion. interactions between bacteria that are nonlocal in both space and time. It is not clear whether such nonlocal- ity is essential for the chemotactic models in [5, 6] or if fast-variables approximations and gradient expansions would reduce these models (which invole between 9 and 12 parameters) into Eq.[1]. In this case, we would still have in Eq.[1] a highly economical model for chemotac- tic pattern-formation organisms, possibly with a different interpretation of D$_{e}$ and κ . Alternatively, the success of our local model for these chemotactic organisms might be largely coincidental. But in that case, such a sparse model should be easily falsified, for instance by using the linear stability anal- ysis to relate the typical length scale of the patterns to microbial parameters. This length scale is of order 2 π/q$_{c}$ = 2 π $^{√}$| D$_{e}$| /α , with |D$_{e}$| ≃ D , a typical bacterial diffusion coefficient ( D ∼ O (100 µ m $^{2}$s − $^{1}$) for E. coli [22]). Using the previously quoted growth rate α ≃ 1 hr − $^{1}$, we get a ring separation of ∼ 1 mm, in order-of-magnitude agreement with the experimental value [5]. This test could perhaps be sharpened usefully by altering the grow- ing medium so as to change α . To summarize, we have studied the dynamics of a sys- tem of reproducing and interacting run-and-tumble bac- teria, in the case where interactions lead to a decreasing local swim speed with increasing local density. We have thereby identified a potentially generic mechanism for pattern formation in which an instability towards phase separation, caused by the tendency for bacteria to move slowly where they are numerous, is arrested by the birth and death dynamics of bacterial populations. We have shown that these two ingredients alone are enough to capture many of the patterns observed experimentally in bacterial colonies – including some that have only pre- viously been explained using far more complex models involving specific chemotactic mechanisms. Indeed, if motility decreases steeply enough with density, then a spatially homogeneous bacterial population becomes un- stable to density fluctuations leading to the formation of bands (1D) or droplets (2D). The length scale of the re- sulting pattern is set by a balance between diffusion-drift fluxes and the logistic relaxation of the population den- sity towards its fixed-point value. Starting instead from a small initial droplet of bacteria, we predict the formation of concentric rings, each of which may eventually further separate into droplets. More generally, our analysis of Eq. (1) shows that the main prerequisite for pattern formation, assuming the presence the logistic growth term, is negativity of the effective diffusion constant D$_{e}$ . For run-and-tumble dy- namics, D$_{e}$ < 0 was shown to arise for a sufficiently strong decay of swim-speed with density; it does so be- cause spatial variations in the true diffusivity D ( ρ ) cre- ate a drift flux ρV = − ρD $^{′}$( ρ ) ∇ ρ/ 2 which can overcom- pensate the true diffusive flux − D ∇ ρ [15]. Negative D$_{e}$ could, however, equally arise for any density dependent nonequilibrium diffusion process. Indeed, the principle of detailed balance, which holds only for equilibrium sys- tems, leads to the Einstein relation, that D = k$_{B}$T µ In several well studied systems, such characteristic pat- terns are (with good reason) believed to be the direct