3 D$_{e}$ can be negative although D is not. This holds for a much wider class of nonequilibrium models than the one studied here; we return to this point at the end of the paper. For the choice of v ( ρ ) made in our simulations, we have D$_{e}$ = D ( ρ )[1 − ρλ/ 2] and the flat profile will thus become unstable for ρ$_{0}$ above 2 /λ . We have confirmed this numerically, and find that upon increasing ρ$_{0}$ , the uniform state becomes (linearly) unstable, evolving in a 1D geometry into a series of “bands” of high bacterial density separated by low density regions. Depending on the parameters, this transition can be continuous (super- critical), with the onset of a harmonic profile whose am- plitude grows smoothly with ρ$_{0}$ , or discontinuous (sub- critical) with strongly anharmonic profiles (see Fig. 1). Figure 2: Three plots of Λ$_{q}$( q ) for |D$_{e}$ ( ρ$_{0}$ ) | / √ ακ = 1; 2; 3. At the transition, only one critical mode q = q$_{c}$ is unstable. The flat profile ρ = ρ$_{0}$ is thus stable if Λ$_{q}$ ≤ 0 for all q and is unstable otherwise. From the expression for D$_{e}$ ( ρ$_{0}$ ), Eq. (2), one sees that instability occurs if Φ ≡ − ρ$_{0}$D $^{′}$( ρ$_{0}$ ) 2 D ( ρ$_{0}$ ) ≥ 1 and − D$_{e}$ ( ρ$_{0}$ ) √ ακ ≥ 2 (4) Figure 1: Growth of the instability in the supercritical (left) and subcritical cases (right). The three lines correspond to three successive times. A small perturbation around ρ$_{0}$ (red line) growth toward harmonic or anharmonic patterns in the supercritical or subcritical case, respectively. Left : Super- critical case ( α = κ = 0 . 01, λ = 0 . 02, ρ$_{0}$ = 15, D$_{0}$ = 1, times: 10 $^{2}$, 10$^{3}$, 10$^{4}$). Right : Subcritical case ( α = κ = 0 . 005, λ = 0 . 02, ρ$_{0}$ = 11, D$_{0}$ = 1, times: 3 . 10 $^{2}$, 3 . 10 $^{3}$, 10$^{5}$) At the onset of the instability only one mode is unsta- ble, with wavevector q$_{c}$ = $^{√}$2 α/ |D$_{e}$ ( ρ$_{0}$ ) | , as can be seen in Figure 2. The first condition in Eq. (4), Φ ≥ 1, is equivalent to the requirement that D$_{e}$ < 0 given pre- viously. From the dispersion relation, Eq. (3), we see that the resulting destabilization is balanced by the sta- bilizing actions of bacterial reproduction and the surface tension at large and small wavelength, respectively. The unstable modes thus lie within a band q$_{1}$ < q < q$_{2}$ where q$_{1}$ ≃ q$_{α}$ ≡ $^{√}$α/ |D$_{e}$ ( ρ$_{0}$ ) | and q$_{2}$ ≃ q$_{κ}$ ≡ $^{√}$| D$_{e}$ ( ρ$_{0}$ ) | /κ set the wavelengths below and above which the stabilizing effects of bacteria reproduction and the surface tension can compete with the destabilizing effect of the negative diffusivity, respectively. For unstable modes to exist, one needs q$_{1}$ ≤ q$_{2}$ ; restoring prefactors, this yields 2 q$_{α}$ ≤ q$_{κ}$ which is the second criterion in Eq. (4). This analysis is consistent with the view that phase separation is arrested by the birth/death dynamics, which stabilizes the long wavelength modes (Λ$_{0}$ = − α ), while the phenomenolog- ical tension parameter κ primarily fixes the interfacial structure of the domains, not their separation. The transition to pattern formation arising from Eq. (1) is a fully nonequilibrium one: it is not possible to write down an effective thermodynamic free energy which would lead to this equation of motion. Nonetheless, it is possible to understand why the birth/death process effec- tively arrests the spinodal decomposition induced by the density-dependent swim speed. The latter tends to sep- arate the system into high and low density domains with densities either side of ρ$_{0}$ . (Without the logistic term, these would coarsen with time.) Bacteria thus tend to be born in the low density regions and to die in the high density regions. To maintain a steady state, they have to travel from one to the other: balancing the birth/death terms by the diffusion-drift transport flux between the domains then sets a typical scale beyond which domain coarsening can no longer progress. Were any domain to become much larger, the density at its centre would soon regress towards ρ$_{0}$ , re-triggering the spinodal instability locally. (This is closely reminiscent of what happens in a thermodynamic phase separation when the supersatura- tion is continuously ramped [25].) We now consider more closely the parameters control- ling the transition to pattern formation. For definite- ness, we address the specific case used for our simulations, D ( ρ ) = D (0) exp( − λρ ). To put Eq. (1) in dimensionless form, we define rescaled time, space and density as The equation of motion now reads ˜ t = αt ; ˜ r = ( α κ$^{)}$$^{1}$$^{/}$$^{4}$ r ; u = ρ ρ$_{0}$ (5) ˙ u = ∇ · [ Re − 2Φ $^{u}$(1 − Φ u ) ∇ u ] + u (1 − u ) − ∇ $^{4}$u (6) Φ ≥ 1; R ≥ R$_{c}$ = 2 exp(2Φ) Φ − 1 (7) ˙ δρ$_{q}$ = Λ$_{q}$ δ$_{q}$ ; Λ$_{q}$ = − α − q $^{2}$D$_{e}$ ( ρ$_{0}$ ) − κq 4 (3) where R ≡ D$_{0}$/ √ ακ and Φ = λρ$_{0}$/ 2 are the two re- maining dimensionless control parameters. Meanwhile the conditions (4) for pattern formation become To better understand the onset of the instability, let us linearize Eq. (1) around ρ ( r ) = ρ$_{0}$ and work in Fourier space. Defining ρ ( r ) = ρ$_{0}$ + ∑ q δρ$_{q}$ exp( i q · r ) yields: