2 rial run-and-tumble motion, involving a swim speed that depends (via unspecified interactions) on local bacterial density. This gives pattern similar to those observed in experiments [5, 6]. However, the basic mechanism —density-dependent motility coupled to logistic popu- lation growth— is not limited to this example. Our work demonstrates that chemotaxis per se is not a prerequisite for observing what are sometimes colloquially referred to as ‘chemotactic patterns’. ever, a non-uniform swimming speed v ( r ) also results in a mean drift velocity V = − vτ ∇ v [21] which here gives V = − D $^{′}$( ρ ) ∇ ρ/ 2 [15]. This contribution is crucial to phase separation [15] and will again play a major role here. However this term has no counterpart in ordinary Brownian motion (even if particles have variable diffusiv- ity) and was accordingly overlooked in previous studies which relied on phenomenological equations involving a density-dependent diffusivity and no drift [8]. Coupling the diffusion-drift equation for run-and- tumble bacteria, as derived in [15], with the logistic growth term, the full dynamics is then given by: It is indeed remarkable that density-dependent swim speed and logistic growth alone are sufficient to create some of the specific pattern types previously identified with specific chemotaxis mechanisms. In mechanistic terms, we find that the logistic growth dynamics effec- tively arrests a spinodal phase separation that is known to follow from a density-dependent swim speed [15]. Put differently, an initially uniform bacterial population with small fluctuations will aggregate into droplets, but these will not coarsen further once a characteristic length scale is achieved, at which aggregation and birth/death effects come into balance. Starting instead from a small inocu- lum, we predict formation of concentric rings which, un- der some conditions, at least partially break up into spots at late times [5]. where the ‘effective diffusivity’ is To exemplify our generic mechanism we will start from a minimal model of run-and-tumble bacteria, that can run in straight lines with a swim speed v and randomly change direction at a constant tumbling rate τ − 1 [21, 22]. To this we add our two key ingredients: a local density- dependent motility, and the birth/death of bacteria, the latter accounted for through a logistic growth model. Of course, bacteria can interact locally in various ways, ranging from steric collisions [15] to chemical quorum- sensing [6]. (Indeed a nonspecific dependence of motility on bacterial density was previously argued to be central to bacterial patterning by Kawasaki [8].) Here we fo- cus on the net effect of all such interactions on the swim speed v ( ρ ), which we assume to decrease with density ρ . This dependence might include the local effect of a secreted chemoattractant (such as aspartate [3–5] which causes aggregation, effectively decreasing v ) but does not assume one. This results from the summed effects of the true diffu- sive flux − D ( ρ ) ∇ ρ and the non-linear drift flux ρV . In Eq.(1) we have also introduced a phenomenological sur- face tension κ > 0, which controls gradients in the bacte- rial density. Such a contribution has been shown to arise when the speed of a bacterium depends on the average density in a small local region around it, rather than a strictly infinitesimal one [15]. Eq. (1) neglects noise, both in the run-and-tumble dynamics and in the birth/death process. The former noise source conserves density and should become irrelevant at the experimental time scale of days. On the other hand, the non-conservative noise in the birth and death dynamics may be more important, and we have verified that our results are robust to its introduction at small to moderate levels. Numerical sim- ulations of Eq. (1) have been performed with standard finite difference methods (although noise does require careful treatment, as in [23, 24]), with periodic bound- ary conditions used throughout. For definiteness, all our simulations have been carried out with v ( ρ ) = v$_{0}$e − λρ/ $^{2}$, where v$_{0}$ > 0 is the swim speed of an isolated bacterium and λ > 0 controls the decay of velocity with density. The precise form of v ( ρ ) is however not crucial for the phenomenology presented here, and the instability anal- ysis offered below does not assume it. In addition to their run-and-tumble motion, real bac- teria continuously reproduce, at a medium-dependent growth rate which ranges from about one reciprocal hour in favourable environments such as Luria broth to sev- eral orders of magnitude lower for ‘minimal’ media such as M9. In bacterial colonies patterns may evolve on timescales of days [6], over which such population growth dynamics can be important. We now derive continuum equations for the local den- sity ρ ( r , t ) in a population of run-and-tumble bacteria, with swim speed v ( ρ ), growing at a rate α (1 − ρ/ρ$_{0}$ ). The latter represents a sum of birth and death terms, in balance only at ρ = ρ$_{0}$ . At large scales in a uniform sys- tem, the motion of individual bacteria is characterized by a diffusivity D ( ρ ) = v ( ρ ) $^{2}$τ /d , where τ − 1 is the tumbling rate and d the dimensionality [21, 22]. Crucially how- The logistic population dynamics alone would cause the bacterial density to evolve toward a uniform den- sity, ρ ( r ) = ρ$_{0}$ , which constitutes a fixed point for the proposed model. Although this homogeneous configu- ration is stable in the absence of bacterial interactions, it has been shown [15] that, without logistic growth, a density-dependent swim speed v ( ρ ) leads to phase sepa- ration via a spinodal instability whenever dv/dρ < − v/ρ . By Eq. (2) this equates to the condition D$_{e}$ < 0, and it is indeed obvious that the diffusive part of Eq. (1) is unstable for negative D$_{e}$ . It is important, clearly, that ∂ρ ( r , t ) ∂t = ∇ · [ D$_{e}$ ( ρ ) ∇ ρ ( r , t )] (1) + αρ ( r , t )$^{(}$1 − ρ ( r , t ) ρ$_{0}$ ) − κ ∇ $^{4}$ρ ( r , t ) D$_{e}$ ( ρ ) = D ( ρ ) + 1 2 ρD $^{′}$( ρ ) (2)