  We consider a run-and-tumble particle whose speed and tumbling rate are
space-dependent on an infinite line. Unlike most of the previous work on such
models, here we make the physical assumption that at large distances, these
rates saturate to a constant. For our choice of rate functions, we show that a
stationary state exists, and the exact steady state distribution decays
exponentially or faster and can be unimodal or bimodal. The effect of
boundedness of rates is seen in the mean-squared displacement of the particle
that displays qualitative features different from those observed in the
previous studies where it approaches the stationary state value monotonically
in time; in contrast, here we find that if the initial position of the particle
is sufficiently far from the origin, the variance in its position either varies
nonmonotonically or plateaus before reaching the stationary state. These
results are captured quantitatively by the exact solution of the Green's
function when the particle has uniform speed but the tumbling rates change as a
step-function in space; the insights provided by this limiting case are found
to be consistent with the numerical results for the general model.
