Figure 5.

In shallow gradients, counter-flow gives rise to sharp cut-offs in the population density ( ${\xi }_c^{\ast }$) and establishment probability ( ${\xi }_c^{\ast }$) profiles (see Fig. 2a). The cut-off positions are well-captured by the analytical approximations (solid lines, eqs. 15 and 16). Squares are cut-offs extracted from simulation, dots and dotted line from numerical solutions to eq. 5 and 7 without and with a growth rate cut-off (SI section 5).