The analytical model is in good agreement with the simulation results. (a) Population density c(ξ) (blue) and establishment probability u(ξ) profiles for a step-like antibiotic profile without external flow. Symbols are simulations, solid lines are numerical solutions to eqs. 5 and 7, dashed lines are analytical solution, eqs. S7 and S9. (b) Co-flow (ν > 0, cyan) gives rise to broader algebraic establishment probability profiles, while counter-flow (ν < 0) gives rise to an exponential decrease. Both cases are asymptotically captured by the analytical approximations (solid lines, eqs. S9, S15 and S16.