Research Centre for Mathematical Modelling (RCM2)

Seminars (Colloquium)


Friday, 12.01.2018, 16ct, V3-204: Roland Langrock (Bielefeld University)
Spline-based nonparametric inference in general state-switching models

Hidden Markov models (HMMs) and their various extensions have been successfully applied in various disciplines, including biology, speech recognition, economics/finance, climatology, psychology and medicine. They combine immense flexibility with relative mathematical simplicity and computational tractability, and as a consequence have become increasingly popular as general-purpose models for time series data. In this talk, I will demonstrate how the HMM machinery can be combined with penalised splines to allow for flexible nonparametric inference in HMM-type classes of models. The focus of the presentation will lie on practical aspects of nonparametric modelling in these frameworks, with the methods being illustrated in ecological and economic real data examples.

Friday, 27.10.2017, 16ct, U2-228: Meike Wittmann (Bielefeld University)
Fluctuating balancing selection and its effects on neutral genetic diversity

For organisms with several generations per year, seasonally fluctuating selection can be a powerful mechanism to maintain genetic polymorphism. For example, an allele favored during summer may stably coexist with an allele favored during winter, a form of balancing selection. Despite intense debate over decades, it is still unclear how much of the variation observed in the genomes of natural populations is due to balancing selection. In recent years, evolutionary biologists have started scanning genomes for genetic footprints of balancing selection (e.g. regions of increased diversity). However, these scans have generally assumed the simplest form of balancing selection where alleles are maintained at constant frequencies over time. There is currently insufficient theory to tell us what genetic footprint to expect under seasonally fluctuating selection, and how to distinguish it from neutrality but also from other forms of balancing selection. In this talk I will present results from coalescent models and stochastic simulations to characterize the impact of fluctuating balancing selection on neutral genetic diversity at various scales: at closely linked sites, at the scale of the chromosome, and at the genomic scale.

Tuesday, 13.06.2017, 16ct, V3-201: Mike Steel (University of Canterbury, New Zealand)
Phylogenetic questions inspired by the theorems of Arrow and Dilworth

Biologists frequently need to reconcile conflicting estimates of the evolutionary relationships between species by taking a ‘consensus’ of a set of phylogenetic trees. This is because different data and/or different methods can produce different trees. If we think of each tree as providing a ‘vote’ for the unknown true phylogeny, then we can view consensus methods as a type of voting procedure. Kenneth Arrow’s celebrated ‘impossibility theorem’ (1950) shows that no voting procedure can simultaneously satisfy seemingly innocent and desirable properties. We adopt a similar axiomatic approach to consensus and asks what desirable properties can be jointly achieved.
In the second part of the talk, we consider phylogenetic networks (which are more general than trees as they allow for reticulate evolution).The question ‘when is a phylogenetic network merely a tree with additional links between its edges?’ is relevant to biology and interesting mathematically. Such ‘tree-based’ networks can be efficiently characterized.We describe these along with new characterization results related to Dilworth’s theorem for posets (1950), and matching theory on bipartite graphs.In this way, one can obtain fast algorithms for determining when a network is tree-based and, if not, to calculate how ‘close’ to tree-based it is.

Monday, 10.04.2017, 16 ct, U10-146: Reinhard Bürger (Vienna)
Two-locus clines on the real line
A population-genetic migration-selection model will be investigated which is continuous in space and time. The model assumes that two diallelic, recombining loci are under selection caused by an abrupt environmental change. The habitat is linear and unbounded, and dispersal occurs by diffusion. Selection is modeled by step functions such that in one region one allele at each locus is advantageous and in the other deleterious. Environmentally independent, intermediate dominance at both loci is admitted. The nonconstant stationary solutions of the resulting system of PDEs are called clines. First, an explicit expression for the single-locus cline with dominance is derived, thus generalizing classical results by Haldane and others. Interestingly, the slope of the cline in the center turns out to be independent of dominance. Second, under the assumption of strong recombination, the first-order approximation of the allele-frequency cline at each of the loci is derived, as is the linkage disequilibrium. Therefore, we obtain the quasi-linkage-equilibrium approximation of the two-locus cline. Its asymptotic properties are characterized explicitly. The consequences of dominance and linkage for the shape of the two-locus cline are explored for arbitrary recombination rates. Analogous models on a bounded habitat will be discussed briefly.

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