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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 $α$ and death of the last particle $δ$. The system presents a phase transition at $δ_c(α)$, where the average position of the last particle $<L>$ grows as $\sqrt{t}$. For $δ>δ_c$, a non equilibrium stationary state exists while for $δ<δ_c$ 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.