Run-and-tumble particle with saturating rates

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, then 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.