Run-and-tumble motion with steplike responses to a stochastic input

We study a simple run-and-tumble random walk whose switching frequencies between run mode and tumble mode depend on a stochastic signal. We consider a particularly sharp, steplike dependence, where the run-to-tumble switching probability jumps from zero to one as the signal crosses a particular value (say y(1)) from below. Similarly, tumble-to-run switching probability also shows a jump like this as the signal crosses another value (y(2) < y(1)) from above. We are interested in characterizing the effect of signaling noise on the long-time behavior of the random walker. We consider two different time-evolutions of the stochastic signal. In one case, the signal dynamics is an independent stochastic process and does not depend on the run-and-tumble motion. In this case we can analytically calculate the mean value and the complete distribution function of the run duration and tumble duration. In the second case, we assume that the signal dynamics is influenced by the spatial location of the random walker. For this system, we numerically measure the steady state position distribution of the random walker. We discuss some similarities and differences between our system and Escherichia coli chemotaxis, which is another well-known run-and-tumble motion encountered in nature.