Limits of spatial branching populations. (With discussion)
The paper focuses on two kinds of limits of critical spatial branching populations, namely `large time limits' and `superprocess limits', and some of their interplay. In all the models described in the main section, the expected number of individuals in a bounded region remains finite; in addition, all the branching equilibria, which are `of Poisson type' in the sense that they consist of Poisson system of families of mutually related individuals. The `backward tree method' also turns out to be helpful in answering the question whether these equilibrium clans behave in a recurrent way in the sense that they come back to populate a bounded region again and again. There are established also the conditions on non-Poisson initial populations which guarantee the `convergence into equilibrium'. There are considered spatially homogeneous branching mechanisms and strongly inhomogeneous ones, then problems are discussed like that of spatial tail triviality of the equilibrium populations, and of recurrence/transience of ancestral lines in equilibrium. There are given some links to superprocesses [measure-valued branching processes, see \textit{E. B. Dynkin}, Ann. Probab. 21, No. 3, 1185-1262 (1993; Zbl 0806.60066); \textit{D. A. Dawson} and \textit{E. A. Perkins}, Mem. Am. Math. Soc. 454 (1991; Zbl 0754.60062) etc.]. The paper is finished by discussion with Peter Jagers about structure of probabilistic models, problems should be solved and about the context in which one should see the subject of spatial branching system.
spatially inhomogeneous critical branching
scientific article; zbMATH DE number 825069
scientific article