  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.
