  We develop a semi-analytical model of self-diffusioosmotic transport in
active pores, which includes advective transport and the inverse chemical
reaction which consumes solute. In previous work (Phys. Rev. Lett. 129, 188003,
2022), we have demonstrated the existence of a spontaneous symmetry breaking in
fore-aft symmetric pores that enables them to function as a micropump. We now
show that this pumping transition is controlled by three timescales. Two
timescales characterize advective and diffusive transport. The third timescale
corresponds to how long a solute molecule resides in the pore before being
consumed. Introducing asymmetry to the pore (either via the shape or the
catalytic coating) reveals a second type of advection-enabled transitions. In
asymmetric pores, the flow rate exhibits discontinuous jumps and hysteresis
loops upon tuning the parameters that control the asymmetry. This work
demonstrates the interconnected roles of shape and catalytic patterning in the
dynamics of active pores, and shows how to design a pump for optimum
performance.
