Fig. 5: Cell motility as a factor that limits bacterial survival in sink-like environments.

a Formation of the wild-type profile. The formation of a wild-type profile corresponding to a fixed resistance state R is described by mean-field equations, which explain the flow of the initial profile in the profile space of cell numbers (N₁, …, N_L). This flow admits two fixed points: a trivial fixed point N_x = 0 (red, population extinction if stable) and a non-trivial fixed-point (yellow, population survival if stable). b Phase portraits of the wild-type profile formation in the staircase model with L = 2, R = 1 (the so-called source-sink model²⁵). Phase portraits are shown for different motility rates ν and different environment types (source-like/sink-like), depending on the average wild-type fitness \(\overline{f}=r/2-\delta\). Only two topologically distinct types of flow are possible: the non-trivial fixed point globally attracts all possible wild-type profiles with N_x > 0 (population survival), or the trivial fixed point globally attracts all possible wild-type profiles with N_x > 0 (population extinction). Extinction occurs in sink-like environments (\(\overline{f} < \, 0\)) with motility above the critical motility ν > ν_c. c Critical motility ν_c corresponds to a bifurcation of this dynamical system. When motility ν is varied, the non-trivial fixed point moves through the space of possible profiles. At low motility, the non-trivial fixed point is stable and corresponds to a wild-type profile that predominantly occupies spatial positions x ≤ R where it can divide. As motility increases, the wild-type profile gets increasingly levelled across all spatial compartments, and its dynamics becomes governed by the average fitness \(\overline{f}\). Precisely when the environment is sink-like \(\overline{f} < \, 0\), the stable non-trivial fixed point (surviving wild-type) collides with the unstable trivial fixed point (extinct wild-type), and they exchange their stability at the critical motility ν_c. In short, highly motile populations experience an average environment, which can drive their extinction if the environment is sink-like. Parameters: L = 8, K = 10⁵, r = 1/h, δ = 0.75/h for source-like environment, δ = 0.25/h for sink-like environment.