  Certain Markov processes, or deterministic evolution equations, have the
property that they are dual to a stochastic process that exhibits extinction
versus unbounded growth, i.e., the total mass in such a process either becomes
zero, or grows without bounds as time tends to infinity. If this is the case,
then this phenomenon can often be used to determine the invariant measures, or
fixed points, of the process originally under consideration, and to study
convergence to equilibrium. This principle, which has been known since early
work on multitype branching processes, is here demonstrated on three new
examples with applications in the theory of interacting particle systems.
