  We consider a critical branching particle system in $\R^d$, composed of
individuals of a finite number of types $i\in\{1,...,K\}$. Each individual of
type $i$ moves independently according to a symmetric $\alpha_i$-stable motion.
We assume that the particle lifetimes and offspring distributions are
type-dependent. Under the usual independence assumptions in branching systems,
we prove extinction theorems in the following cases: (1) all the particle
lifetimes have finite mean, or (2) there is a type whose lifetime distribution
has heavy tail, and the other lifetimes have finite mean. We get a more complex
dynamics by assuming in case (2) that the most mobile particle type corresponds
to a finite-mean lifetime: in this case, local extinction of the population is
determined by an interaction of the parameters (offspring variability,
mobility, longevity) of the long-living type and those of the most mobile type.
The proofs are based on a precise analysis of the occupation times of a related
Markov renewal process, which is of independent interest.
