  Models of many-species ecosystems, such as the Lotka-Volterra and replicator
equations, suggest that these systems generically exhibit near-extinction
processes, where population sizes go very close to zero for some time before
rebounding, accompanied by a slowdown of the dynamics (aging). Here, we
investigate the connection between near-extinction and aging by introducing an
exactly solvable many-variable model, where the time derivative of each
population size vanishes both at zero and some finite maximal size. We show
that aging emerges generically when random interactions are taken between
populations. Population sizes remain exponentially close (in time) to the
absorbing values for extended periods of time, with rapid transitions between
these two values. The mechanism for aging is different from the one at play in
usual glassy systems: at long times, the system evolves in the vicinity of
unstable fixed points rather than marginal ones.
