  We study the behavior of an infinite system of ordinary differential
equations modeling the dynamics of a metapopulation, a set of (discrete)
populations subject to local catastrophes and connected via migration under a
mean field rule; the local population dynamics follow a generalized logistic
law. We find a threshold below which all the solutions tend to total extinction
of the metapopulation, which is then the only equilibrium; above the threshold,
there exists a unique equilibrium with positive population, which, under an
additional assumption, is globally attractive. The proofs employ tools from the
theories of Markov processes and of dynamical systems.
