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 lo cal extinction and survival (even persistence) at late times for critical m ultitype spatially homogeneous branching particle systems in continuous rim e. Our conditions on the branching mechanism are close to the ones known fr om "classical" processes without motion component. This generalizes and com plements results of Lopez-Mimbela and Wakolbinger [LMW96] and others. Our a pproach 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 i nterest.