  We study interacting active Brownian particles (ABPs) with a space-dependent
swim velocity via simulation and theory. We find that, although an equation of
state exists, a mechanical equilibrium does not apply to ABPs in activity
landscapes. The pressure difference originates in the flux of polar order and
the gradient of swim velocity across the interface between regions of different
activity. In contrast to motility-induced phase separation of ABPs with a
homogeneous swim velocity, a critical point does not exist for an
active-passive patch system, which continuously splits into a dense and a
dilute phase with increasing activity. However, if the global density is so
high that not all particles can be packed onto the inactive patch, then
MIPS-like behavior is restored and the pressure is balanced again.
