Anton Klimovsky (Universität Duisburg-Essen)
Stochastic population models on evolving networks
Many models of complex systems can be seen as a system of many interacting (micro)variables evolving in time. We focus on the situation, where the network of interactions between the variables is complex and possibly itself evolves in time. We discuss a modeling framework for interacting particle systems on evolving networks based on such familiar ingredients as exchangeability and Markovianity. In some simple cases, we discuss the genealogies of such population models.
(Based on joint work in progress with Jiří Černý.)
Fernando Cordero (Universität Bielefeld)
On the stationary distribution of the block-counting process in populations with mutation and selection
The $\Lambda$-Wright-Fisher model is a population model subject to selection, mutation and neutral reproduction (described by a finite measure $\Lambda$ on $[0,1]$). The block-counting process traces back the number of potential ancestors of a sample of the population at present. In absence of selection and mutation the latter coincides with the $\Lambda$-coalescent. Selection and mutation translate into additional branching and pruning. Under some conditions the block-counting process is positive-recurrent and its stationary distribution is described via a linear system of equations. In this talk, we first characterise the measures $\Lambda$ leading to a geometric stationary distribution, the Bolthausen-Sznitman model being the most prominent example having this feature. For a general measure $\Lambda$, we show that the probability generating function of the stationary distribution of the block-counting process satisfies an integro-differential equation. We solve the latter for the Kingman model and the star-shaped model.
(Based on joint work with M. Möhle).