An integral test for a critical multitype spatially homogeneous branching particle process and a related reaction-diffusion system

An integral test (Theorem 5) is established for the dichotomy concerning local extinction and survival (even persistence) at late times for critical multitype spatially homogeneous branching particle systems in continuous rime. Our conditions on the branching mechanism are close to the ones known from "classical" processes without motion component. This generalizes and complements results of Lopez-Mimbela and Wakolbinger [LMW96] and others. Our approach is based on some genealogical tree analysis combined with the study of the long-term behavior of L-1-norms of solutions of related systems of reaction-"diffusion" equations. which is perhaps also of some independent interest.