  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.
