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Genealogy of catalytic branching models. (English) Zbl 1178.60057

The catalytic branching process considered has two components. Both are critical binary continuous-time branching processes. The first, the “catalyst” process, has a constant branching rate, the second, the “reactant” process, a branching rate proportional to the number of catalyst individuals alive. It is shown that the suitably rescaled reactant forests given the catalyst total mass process converge in Gromov-Hausdorff topology to a limit forest. To determine its distribution and thereby characterize a unique reactant limit forest, the authors first consider the limit of the reactant family forest cut off at a height where the catalyst falls below a certain threshold. Its contour process is given by a path of a reflected diffusion. From that path a collection of point processes is derived each describing the mutual genealogical distances between all individuals in the population alive at a certain time. Finally, the authors prove joint convergence in the product of Gromov-Hausdorff topologies of the rescaled catalyst and reactant family forests to a limiting pair of forests and discuss the key differences between the genealogical structures of these two limiting forests.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K37 Processes in random environments
60B11 Probability theory on linear topological spaces
92D25 Population dynamics (general)
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