88 research outputs found
Asymptotic genealogy of a critical branching process
Consider a continuous-time binary branching process conditioned to have
population size n at some time t, and with a chance p for recording each
extinct individual in the process. Within the family tree of this process, we
consider the smallest subtree containing the genealogy of the extant
individuals together with the genealogy of the recorded extinct individuals. We
introduce a novel representation of such subtrees in terms of a point-process,
and provide asymptotic results on the distribution of this point-process as the
number of extant individuals increases. We motivate the study within the scope
of a coherent analysis for an a priori model for macroevolution.Comment: 30 page
The coalescent point process of branching trees
We define a doubly infinite, monotone labeling of Bienayme-Galton-Watson
(BGW) genealogies. The genealogy of the current generation backwards in time is
uniquely determined by the coalescent point process , where
is the coalescence time between individuals i and i+1. There is a Markov
process of point measures keeping track of more ancestral
relationships, such that is also the first point mass of . This
process of point measures is also closely related to an inhomogeneous spine
decomposition of the lineage of the first surviving particle in generation h in
a planar BGW tree conditioned to survive h generations. The decomposition
involves a point measure storing the number of subtrees on the
right-hand side of the spine. Under appropriate conditions, we prove
convergence of this point measure to a point measure on
associated with the limiting continuous-state branching (CSB) process. We prove
the associated invariance principle for the coalescent point process, after we
discretize the limiting CSB population by considering only points with
coalescence times greater than .Comment: Published in at http://dx.doi.org/10.1214/11-AAP820 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A critical branching process model for biodiversity
Motivated as a null model for comparison with data, we study the following
model for a phylogenetic tree on extant species. The origin of the clade is
a random time in the past, whose (improper) distribution is uniform on
. After that origin, the process of extinctions and speciations is
a continuous-time critical branching process of constant rate, conditioned on
having the prescribed number of species at the present time. We study
various mathematical properties of this model as limits: time of
origin and of most recent common ancestor; pattern of divergence times within
lineage trees; time series of numbers of species; number of extinct species in
total, or ancestral to extant species; and "local" structure of the tree
itself. We emphasize several mathematical techniques: associating walks with
trees, a point process representation of lineage trees, and Brownian limits.Comment: 31 pages, 7 figure
Genealogy of catalytic branching models
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
-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.Comment: Published in at http://dx.doi.org/10.1214/08-AAP574 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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