arXiv subjects

Anton Wakolbinger

Publications and source records attributed to Anton Wakolbinger.

At least 37 records · Page 2Linked to original sources

arXiv2019

Maintenance of diversity in a hierarchical host-parasite model with balancing selection and reinfection

Inspired by DNA data of the human cytomegalovirus we propose a model of a two-type parasite population distributed over its hosts. The parasite is capable to persist in its host till the host dies, and to reinfect other hosts. To maintain type diversity within a host, balancing selection is assumed. For a suitable parameter regime we show that in the limit of large host and parasite populations the host state frequencies follow a dynamical system with a globally stable equilibrium, guaranteeing that both types are maintained in the parasite population for a long time on the host time scale.

math.PR↗

arXiv2018

Fixation probabilities and hitting times under low levels of frequency-dependent selection

In population genetics, diffusions on the unit interval are often used to model the frequency path of an allele. In this setting we derive approximations for fixation probabilities, expected hitting times and the expected site-frequency-spectrum under small frequency-dependent selection. Specifically, we rederive and extend the one-third rule of evolutionary game theory (Nowak et al., 2004) and effects of stochastic slowdown (Altrock and Traulsen, 2009). Since similar effects are of interest in other application areas, we formulate our results for general one-dimensional diffusions.

math.PR↗

arXiv2017

On the time to absorption in $Λ$-coalescents

We present a law of large numbers and a central limit theorem for the time to absorption of $Λ$-coalescents, started from $n$ blocks, as $n \to \infty$. The proofs rely on an approximation of the logarithm of the block-counting process of $Λ$-coalescents with a dust component by means of a drifted subordinator.

math.PR↗

arXiv2017

Lines of descent under selection

We review recent progress on ancestral processes related to mutation-selection models, both in the deterministic and the stochastic setting. We mainly rely on two concepts, namely, the killed ancestral selection graph and the pruned lookdown ancestral selection graph. The killed ancestral selection graph gives a representation of the type of a random individual from a stationary population, based upon the individual's potential ancestry back until the mutations that define the individual's type. The pruned lookdown ancestral selection graph allows one to trace the ancestry of individuals from a stationary distribution back into the distant past, thus leading to the stationary distribution of ancestral types. We illustrate the results by applying them to a prototype model for the error threshold phenomenon.

math.PR↗

arXiv2017

The size of the last merger and time reversal in $Λ$-coalescents

We consider the number of blocks involved in the last merger of a $Λ$-coalescent started with $n$ blocks. We give conditions under which, as $n \to \infty$, the sequence of these random variables a) is tight, b) converges in distribution to a finite random variable or c) converges to infinity in probability. Our conditions are optimal for $Λ$-coalescents that have a dust component. For general $Λ$, we relate the three cases to the existence, uniqueness and non-existence of quasi-invariant measures for the dynamics of the block-counting process, and in case b) investigate the time-reversal of the block-counting process back from the time of the last merger.

math.PR↗

arXiv2016

The fixation time of a strongly beneficial allele in a structured population

For a beneficial allele which enters a large unstructured population and eventually goes to fixation, it is known that the time to fixation is approximately $2\log(α)/α$ for a large selection coefficient $α$. For a population that is distributed over finitely many colonies, with migration between these colonies, we detect various regimes of the migration rate $μ$ for which the fixation times have different asymptotics as $α\to \infty$. If $μ$ is of order $α$, the allele fixes (as in the spatially unstructured case) in time $\sim 2\log(α)/α$. If $μ$ is of order $α^γ, 0\leq γ\leq 1$, the fixation time is $\sim (2 + (1-γ)Δ) \log(α)/α$, where $Δ$ is the number of migration steps that are needed to reach all other colonies starting from the colony where the beneficial allele appeared. If $μ= 1/\log(α)$, the fixation time is $\sim (2+S)\log(α)/α$, where $S$ is a random time in a simple epidemic model. The main idea for our analysis is to combine a new moment dual for the process conditioned to fixation with the time reversal in equilibrium of a spatial version of Neuhauser and Krone's ancestral selection graph.

math.PR↗

arXiv2016

The common ancestor type distribution of a $Λ$-Wright-Fisher process with selection and mutation

Using graphical methods based on a `lookdown' and pruned version of the {\em ancestral selection graph}, we obtain a representation of the type distribution of the ancestor in a two-type Wright-Fisher population with mutation and selection, conditional on the overall type frequency in the old population. This extends results from Lenz, Kluth, Baake, and Wakolbinger (Theor. Pop. Biol., 103 (2015), 27-37) to the case of heavy-tailed offspring, directed by a reproduction measure $Λ$. The representation is in terms of the equilibrium tail probabilities of the line-counting process $L$ of the graph. We identify a strong pathwise Siegmund dual of $L$, and characterise the equilibrium tail probabilities of $L$ in terms of hitting probabilities of the dual process.

math.PR↗

arXiv2016

Radix sort trees in the large

The trie-based radix sort algorithm stores pairwise different infinite binary strings in the leaves of a binary tree in a way that the Ulam-Harris coding of each leaf equals a prefix (that is, an initial segment) of the corresponding string, with the prefixes being of minimal length so that they are pairwise different. We investigate the {\em radix sort tree chains} -- the tree-valued Markov chains that arise when successively storing infinite binary strings $Z_1,\ldots, Z_n$, $n=1,2,\ldots$ according to the trie-based radix sort algorithm, where the source strings $Z_1, Z_2,\ldots$ are independent and identically distributed. We establish a bijective correspondence between the full Doob--Martin boundary of the radix sort tree chain with a {\em symmetric Bernoulli source} (that is, each $Z_k$ is a fair coin-tossing sequence) and the family of radix sort tree chains for which the common distribution of the $Z_k$ is a diffuse probability measure on $\{0,1\}^\infty$. In essence, our result characterizes all the ways that it is possible to condition such a chain of radix sort trees consistently on its behavior "in the large".

math.PR↗

arXiv2016

An individual-based model for the Lenski experiment, and the deceleration of the relative fitness

The Lenski experiment investigates the long-term evolution of bacterial populations. Its design allows the direct comparison of the reproductive fitness of an evolved strain with its founder ancestor. It was observed by Wiser et al. (2013) that the relative fitness over time increases sublinearly, a behaviour which is commonly attributed to effects like clonal interference or epistasis. In this paper we present an individual-based probabilistic model that captures essential features of the design of the Lenski experiment. We assume that each beneficial mutation increases the individual reproduction rate by a fixed amount, which corresponds to the absence of epistasis in the continuous-time (intraday) part of the model, but leads to an epistatic effect in the discrete-time (interday) part of the model. Using an approximation by near-critical Galton-Watson processes, we prove that under some assumptions on the model parameters which exclude clonal interference, the relative fitness process converges, after suitable rescaling, in the large population limit to a power law function.

math.PR↗

arXiv2015

A path-valued Markov process indexed by the ancestral mass

A family of Feller branching diffusions $Z^x$, $x \ge 0$, with nonlinear drift and initial value $x$ can, with a suitable coupling over the {\em ancestral masses} $x$, be viewed as a path-valued process indexed by $x$. For a coupling due to Dawson and Li, which in case of a linear drift describes the corresponding Feller branching diffusion, and in our case makes the path-valued process Markovian, we find an SDE solved by $Z$, which is driven by a random point measure on excursion space. In this way we are able to identify the infinitesimal generator of the path-valued process. We also establish path properties of $x\mapsto Z^x$ using various couplings of $Z$ with classical Feller branching diffusions.

math.PR↗

arXiv2015

The Kingman tree length process has infinite quadratic variation

In the case of neutral populations of fixed sizes in equilibrium whose genealogies are described by the Kingman $N$-coalescent back from time $t$ consider the associated processes of total tree length as $t$ increases. We show that the (càdlàg) process to which the sequence of compensated tree length processes converges as $N$ tends to infinity is a process of infinite quadratic variation; therefore this process cannot be a semimartingale. This answers a question posed in Pfaffelhuber et al. (2011).

math.PR↗

arXiv2015

Looking down in the ancestral selection graph: A probabilistic approach to the common ancestor type distribution

In a (two-type) Wright-Fisher diffusion with directional selection and two-way mutation, let $x$ denote today's frequency of the beneficial type, and given $x$, let $h(x)$ be the probability that, among all individuals of today's population, the individual whose progeny will eventually take over in the population is of the beneficial type. Fearnhead [Fearnhead, P., 2002. The common ancestor at a nonneutral locus. J. Appl. Probab. 39, 38-54] and Taylor [Taylor, J. E., 2007. The common ancestor process for a Wright-Fisher diffusion. Electron. J. Probab. 12, 808-847] obtained a series representation for $h(x)$. We develop a construction that contains elements of both the ancestral selection graph and the lookdown construction and includes pruning of certain lines upon mutation. Besides being interesting in its own right, this construction allows a transparent derivation of the series coefficients of $h(x)$ and gives them a probabilistic meaning.

q-bio.PE↗

arXiv2015

Feller's Contributions to Mathematical Biology

This is a review of William Feller's important contributions to mathematical biology. The seminal paper [Feller1951] "Diffusion processes in genetics" was particularly influential on the development of stochastic processes at the interface to evolutionary biology, and interesting ideas in this direction (including a first characterization of what is nowadays known as "Feller's branching diffusion") already shaped up in the paper [Feller 1939] (written in German) "The foundations of a probabistic treatment of Volterra's theory of the struggle for life". Feller's article "On fitness and the cost of natural selection" [Feller 1967] contains a critical analysis of the concept of "genetic load".

math.HO↗

arXiv2014

The evolving beta coalescent

In mathematical population genetics, it is well known that one can represent the genealogy of a population by a tree, which indicates how the ancestral lines of individuals in the population coalesce as they are traced back in time. As the population evolves over time, the tree that represents the genealogy of the population also changes, leading to a tree-valued stochastic process known as the evolving coalescent. Here we will consider the evolving coalescent for populations whose genealogy can be described by a beta coalescent, which is known to give the genealogy of populations with very large family sizes. We show that as the size of the population tends to infinity, the evolution of certain functionals of the beta coalescent, such as the total number of mergers, the total branch length, and the total length of external branches, converges to a stationary stable process. Our methods also lead to new proofs of known asymptotic results for certain functionals of the non-evolving beta coalescent.

math.PR↗

arXiv2013

The spatial Lambda-Fleming-Viot process: an event-based construction and a lookdown representation

We construct a measure-valued equivalent to the spatial Lambda-Fleming-Viot process (SLFV) introduced in [Eth08]. In contrast with the construction carried out in [Eth08], we fix the realization of the sequence of reproduction events and obtain a quenched evolution of the local genetic diversities. To this end, we use a particle representation which highlights the role of the genealogies in the attribution of types (or alleles) to the individuals of the population. This construction also enables us to clarify the state-space of the SLFV and to derive several path properties of the measure-valued process as well as of the labeled trees describing the genealogical relations between a sample of individuals. We complement it with a look-down construction which provides a particle system whose empirical distribution at time t, seen as a process in t, has the law of the quenched SLFV. In all these results, the facts that we work with a fixed configuration of events and that reproduction occurs only locally in space introduce serious technical issues that are overcome by controlling the number of events occurring and of particles present in a given area over macroscopic time intervals.

math.PR↗

arXiv2013

From Brownian motion with a local time drift to Feller's branching diffusion with logistic growth

We give a new proof for a Ray-Knight representation of Feller's branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion $H$ with a drift that is affine linear in the local time accumulated by $H$ at its current level. In \cite{LPW}, such a representation was obtained by an approximation through Harris paths that code the genealogies of particle systems. The present proof is purely in terms of stochastic analysis, and is inspired by previous work of Norris, Rogers and Williams \cite{NRW}.

math.PR↗

arXiv2013

Trees under attack: a Ray-Knight representation of Feller's branching diffusion with logistic growth

We obtain a representation of Feller's branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion $H$ with a drift that is affine linear in the local time accumulated by $H$ at its current level. As in the classical Ray-Knight representation, the excursions of $H$ are the exploration paths of the trees of descendants of the ancestors at time $t=0$, and the local time of $H$ at height $t$ measures the population size at time $t$ (see e.g. \cite{LG4}). We cope with the dependence in the reproduction by introducing a pecking order of individuals: an individual explored at time $s$ and living at time $t=H_s$ is prone to be killed by any of its contemporaneans that have been explored so far. The proof of our main result relies on approximating $H$ with a sequence of Harris paths $H^N$ which figure in a Ray-Knight representation of the total mass of a branching particle system. We obtain a suitable joint convergence of $H^N$ together with its local times {\em and} with the Girsanov densities that introduce the dependence in the reproduction.

math.PR↗

arXiv2012

The total external branch length of Beta-coalescents

For $1<α<2$ we derive the asymptotic distribution of the total length of {\em external} branches of a Beta$(2-α, α)$-coalescent as the number $n$ of leaves becomes large. It turns out the fluctuations of the external branch length follow those of $τ_n^{2-α}$ over the entire parameter regime, where $τ_n$ denotes the random number of coalescences that bring the $n$ lineages down to one. This is in contrast to the fluctuation behavior of the total branch length, which exhibits a transition at $α_0 = (1+\sqrt 5)/2$.

math.PR↗