Forschungsschwerpunkt Mathematische Modellierung (FSPM2)

Seminars (Colloquium)


Freitag, 27.10.2017, 16ct, U2-228: Meike Wittmann (Bielefeld University)


Dienstag, 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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