  Microswimmer suspensions self-organize into complex spatio-temporal flow
patterns, including vortex lattices and mesoscale turbulence. Here we explore
the consequences for the motion of passive tracers, based on a continuum model
for the microswimmer velocity field. We observe two qualitatively different
regimes distinguished via the dimensionless Kubo number $K$. At advection
strengths right above the transition to turbulence, the flow field evolves very
slowly ($K \gg 1$) and the spatial vortex structures lead to dominant trapping
effects. In contrast, deep in the turbulent state, much faster dynamics ($K \ll
1$) consistent with the so-called sweeping hypothesis leads to transport
properties completely determined by the temporal correlations. In between ($K
\approx 1$), we observe a regime of optimal transport, signaled by a maximum of
the diffusion coefficient.
