Dynamical phase transition of a 1D transport process including death

Motivated by biological aspects related to fungus growth, we consider the competition of growth and corrosion. We study a modification of the totally asymmetric exclusion process, including the probabilities of injection $\alpha$ and death of the last particle $\delta$. The system presents a phase transition at $\delta_c(\alpha)$, where the average position of the last particle $<L>$ grows as $\sqrt{t}$. For $\delta>\delta_c$, a non equilibrium stationary state exists while for $\delta<\delta_c$ the asymptotic state presents a low density and max current phases. We discuss the scaling of the density and current profiles for parallel and sequential updates.