Permanence
Permanence is a stability measure which specifies that species in an assemblage can recover when brought close to extinction. Whenever any species or set of species comes close to extinction, they are able to reinvade the system and reestablish themselves. Permanence puts no constraint on the dynamics: they may have a stable steady state, periodic behaviour, or chaos.
I have done a some work measuring the properties of model food webs assembled using permanence as a constraint. A talk and a paper on this topic may be found in the bar to the right of this page.