  We studied metastability and extinction time of a finite system with a large
number of interacting components in discrete time by means of analytical and
numerical investigation. The system is markovian with respect to the potential
profile of the components, which are subject to leakage and gain effects
simultaneously. We show that the only invariant measure is the null
configuration, that the system ceases activity almost surely in a finite time
and that extinction time presents a cutoff behavior. Moreover, there is a
critical parameter determined by leakage and gain below which the extinction
time does not depend on the system size. Above such critical ratio, the
extinction time depends on the number of components and the system tends to
stabilize around a unique metastable state. Furthermore, the extinction time
presents infinitely many scales with respect to the system size.
