Persistent random walk of cells involving anomalous effects and random death

Abstract

The purpose of this paper is to implement a random death process into a persistent random walk model which produces subballistic superdiffusion (L\'{e}vy walk). We develop a Markovian model of cell motility with the extra residence variable $\tau .$ The model involves a switching mechanism for cell velocity with dependence of switching rates on $\tau $. This dependence generates intermediate subballistic superdiffusion. We derive master equations for the cell densities with the generalized switching terms involving the tempered fractional material derivatives. We show that the random death of cells has an important implication for the transport process through tempering of superdiffusive process. In the long-time limit we write stationary master equations in terms of exponentially truncated fractional derivatives in which the rate of death plays the role of tempering of a L\'{e}vy jump distribution. We find the upper and lower bounds for the stationary profiles corresponding to the ballistic transport and diffusion with the death rate dependent diffusion coefficient. Monte Carlo simulations confirm these bounds.