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29.04.2019 14:00 Anthea Monod (Columbia University):
Tropical Statistics for Phylogenetic TreesMI 02.08.011 (Boltzmannstr. 3, 85748 Garching)

Phylogenetic trees are the fundamental mathematical representation of evolutionary processes in biology. As data objects, they are characterized by the challenges associated with "big data," as well as the complication that their discrete geometric structure results in a non-Euclidean phylogenetic tree space, which poses computational and statistical limitations. We propose a novel framework constructed from tropical geometry for the statistical analysis of evolutionary biological processes represented by sets of phylogenetic trees. Our structure allows for the definition of probability measures, expectations, variances, and other fundamental statistical quantities. In addition, our setting exhibits analytic, geometric, and topological properties that are desirable for rigorous theoretical treatment in probability and statistics, as well as increased computational efficiency over the current state-of-the-art. We demonstrate our approach and compare against the current standard on seasonal influenza data. This is joint work with Bo Lin (Georgia Tech), Qiwen Kang (University of Kentucky), and Ruriko Yoshida (Naval Postgraduate School).

29.04.2019 15:00 Maximilian Engel (TUM):
Modelling animal group size statistics: from coagulation-fragmentation equations to Markov jump processesMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

We translate a coagulation-framentation model, describing the dynamics of animal group size distributions, into a model for the population distribution and associate the evolution equation with a Markov jump process, in particular formalizing a model suggested by H.-S. Niwa [J. Theo. Biol. 224 (2003)] with simple coagulation and fragmentation rates. Based on the jump process, we develop a numerical scheme that allows us to approximate the equilibrium for the Niwa model, validated by comparison to analytical results by Degond et al. [J. Nonlinear Sci. 27 (2017)], and study the population and size distributions for more complicated rates. Furthermore, the simulations are used to describe statistical properties of the underlying jump process. We additionally discuss the relation of the jump process to models expressed in stochastic differential equations and demonstrate that such a connection is justified in the case of nearest-neighbour interactions, as opposed to global interactions as in the Niwa model.