  We study the transport of self-propelled particles in dynamic complex
environments. To obtain exact results, we introduce a model of run-and-tumble
particles (RTPs) moving in discrete time on a $d$-dimensional cubic lattice in
the presence of diffusing hard core obstacles. We derive an explicit expression
for the diffusivity of the RTP, which is exact in the limit of low density of
fixed obstacles. To do so, we introduce a generalization of Kac's theorem on
the mean return times of Markov processes, which we expect to be relevant for a
large class of lattice gas problems. Our results show the diffusivity of RTPs
to be nonmonotonic in the tumbling probability for low enough obstacle
mobility. These results prove the potential for optimization of the transport
of RTPs in crowded and disordered environments with applications to motile
artificial and biological systems.
