  Microswimmers typically operate in complex environments. In biological
systems, often diverse species are simultaneously present and interact with
each other. Here, we derive a (time-dependent) particle-scale statistical
description, namely a dynamical density functional theory, for such
multi-species systems, extending existing works on one-component microswimmer
suspensions. In particular, our theory incorporates the effect of external
potentials, but also steric and hydrodynamic interactions between swimmers. For
the latter, a previously introduced force-dipole-based minimal (pusher or
puller) microswimmer model is used. As a limiting case of our theory, mixtures
of hydrodynamically interacting active and passive particles are captured as
well. After deriving the theory, we apply it to different planar swimmer
configurations. First, these are binary pusher--puller mixtures in external
traps. In the considered situations, we find that the majority species imposes
its behavior on the minority species. Second, for unconfined binary
pusher--puller mixtures, the linear stability of an orientationally disordered
state against the emergence of global polar orientational order (and thus
emergent collective motion) is tested analytically. Our statistical approach
predicts, qualitatively in line with previous particle-based computer
simulations, a threshold for the fraction of pullers and for their propulsion
strength that lets overall collective motion arise. Third, we let driven
passive colloidal particles form the boundaries of a shear cell, with confined
active microswimmers on their inside. Driving the passive particles then
effectively imposes shear flows, which persistently acts on the inside
microswimmers. Their resulting behavior reminds of the one of circle swimmers,
though with varying swimming radii.
