Genealogy of catalytic branching models

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Genealogy of catalytic branching models

Andreas Greven, Lea Popovic, and Anita Winter

Source: Ann. Appl. Probab. Volume 19, Number 3 (2009), 1232-1272.

Abstract

We consider catalytic branching populations. They consist of a catalyst population evolving according to a critical binary branching process in continuous time with a constant branching rate and a reactant population with a branching rate proportional to the number of catalyst individuals alive. The reactant forms a process in random medium.

We describe asymptotically the genealogy of catalytic branching populations coded as the induced forest of ℝ-trees using the many individuals—rapid branching continuum limit. The limiting continuum genealogical forests are then studied in detail from both the quenched and annealed points of view. The result is obtained by constructing a contour process and analyzing the appropriately rescaled version and its limit. The genealogy of the limiting forest is described by a point process. We compare geometric properties and statistics of the reactant limit forest with those of the “classical” forest.

Primary Subjects: 60J80, 60K37, 60B11, 92D25

Keywords: Catalytic branching; random trees; contour process; genealogical point processes; R-trees; Gromov–Hausdorff topology; random evolution

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Permanent link to this document: http://projecteuclid.org/euclid.aoap/1245071025
Digital Object Identifier: doi:10.1214/08-AAP574
Zentralblatt MATH identifier: 05580239
Mathematical Reviews number (MathSciNet): MR2537365

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