17 research outputs found
Tree-valued Feller diffusion
We consider the evolution of the genealogy of the population currently alive
in a Feller branching diffusion model. In contrast to the approach via labeled
trees in the continuum random tree world, the genealogies are modeled as
equivalence classes of ultrametric measure spaces, the elements of the space
. This space is Polish and has a rich semigroup structure for the
genealogy. We focus on the evolution of the genealogy in time and the large
time asymptotics conditioned both on survival up to present time and on
survival forever. We prove existence, uniqueness and Feller property of
solutions of the martingale problem for this genealogy valued, i.e.,
-valued Feller diffusion. We give the precise relation to the
time-inhomogeneous -valued Fleming-Viot process. The uniqueness
is shown via Feynman-Kac duality with the distance matrix augmented Kingman
coalescent. Using a semigroup operation on , called concatenation,
together with the branching property we obtain a L{\'e}vy-Khintchine formula
for -valued Feller diffusion and we determine explicitly the
L{\'e}vy measure on . From this we obtain for
the decomposition into depth- subfamilies, a representation of the process
as concatenation of a Cox point process of genealogies of single ancestor
subfamilies. Furthermore, we will identify the -valued process
conditioned to survive until a finite time . We study long time asymptotics,
such as generalized quasi-equilibrium and Kolmogorov-Yaglom limit law on the
level of ultrametric measure spaces. We also obtain various representations of
the long time limits.Comment: 93 pages, replaced by revised versio
Tree-valued Fleming-Viot dynamics with mutation and selection
The Fleming-Viot measure-valued diffusion is a Markov process describing the
evolution of (allelic) types under mutation, selection and random reproduction.
We enrich this process by genealogical relations of individuals so that the
random type distribution as well as the genealogical distances in the
population evolve stochastically. The state space of this tree-valued
enrichment of the Fleming-Viot dynamics with mutation and selection (TFVMS)
consists of marked ultrametric measure spaces, equipped with the marked
Gromov-weak topology and a suitable notion of polynomials as a separating
algebra of test functions. The construction and study of the TFVMS is based on
a well-posed martingale problem. For existence, we use approximating finite
population models, the tree-valued Moran models, while uniqueness follows from
duality to a function-valued process. Path properties of the resulting process
carry over from the neutral case due to absolute continuity, given by a new
Girsanov-type theorem on marked metric measure spaces. To study the long-time
behavior of the process, we use a duality based on ideas from Dawson and Greven
[On the effects of migration in spatial Fleming-Viot models with selection and
mutation (2011c) Unpublished manuscript] and prove ergodicity of the TFVMS if
the Fleming-Viot measure-valued diffusion is ergodic. As a further application,
we consider the case of two allelic types and additive selection. For small
selection strength, we give an expansion of the Laplace transform of
genealogical distances in equilibrium, which is a first step in showing that
distances are shorter in the selective case.Comment: Published in at http://dx.doi.org/10.1214/11-AAP831 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A mixing tree-valued process arising under neutral evolution with recombination
The genealogy at a single locus of a constant size population in
equilibrium is given by the well-known Kingman's coalescent. When considering
multiple loci under recombination, the ancestral recombination graph encodes
the genealogies at all loci in one graph. For a continuous genome , we study
the tree-valued process of genealogies along the genome in
the limit . Encoding trees as metric measure spaces, we show
convergence to a tree-valued process with cadlag paths. In addition, we study
mixing properties of the resulting process for loci which are far apart.Comment: 25 pages, 3 figure
Survival and complete convergence for a spatial branching system with local regulation
We study a discrete time spatial branching system on with
logistic-type local regulation at each deme depending on a weighted average of
the population in neighboring demes. We show that the system survives for all
time with positive probability if the competition term is small enough. For a
restricted set of parameter values, we also obtain uniqueness of the nontrivial
equilibrium and complete convergence, as well as long-term coexistence in a
related two-type model. Along the way we classify the equilibria and their
domain of attraction for the corresponding deterministic coupled map lattice on
.Comment: Published in at http://dx.doi.org/10.1214/105051607000000221 the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org