  We consider a kinetic model of an active suspension of rod-like
microswimmers. In certain regimes, swimming has a stabilizing effect on the
suspension. We quantify this effect near homogeneous isotropic equilibria
$\overline{\psi} = \text{const}$. Notably, in the absence of particle
(translational and orientational) diffusion, swimming is the only stabilizing
mechanism. On the torus, in the non-diffusive regime, we demonstrate linear
Landau damping up to the stability threshold predicted in the applied
literature. With small diffusion, we demonstrate nonlinear stability of
arbitrary equilibrium values for pullers (front-actuated swimmers) and enhanced
dissipation for both pullers and pushers (rear-actuated swimmers) at small
concentrations. On the whole space, we prove nonlinear stability of the vacuum
equilibrium due to generalized Taylor dispersion.
