Spatial models for populations with variable offspring laws

This project is concerned with the analysis of spatial stochastic particle models for which the offspring distribution is state dependent or relatively singular. For the former, the study is focused on offspring distributions that give small populations (of a particular type) an advantage. Various (non-linear) self-regulation mechanisms as well as spatial distributions of the particle population will be investigated, including versions of voter models with selection, multitype models with fixed local population size and branching particle models alongside their large population diffusion limits. One of the central questions will be to determine parameter regimes for long-term survival without explosion of the population size and -if multiple types of particles are present initially- of long-term coexistence of various or all types. Another part of this project is concerned with the description of spatially distributed particle populations with singular offspring distributions that model populations with potentially large individual families. These have attracted much recent attention as several deep connections between different classes of stochastic processes have been discovered and exploited - albeit in a non-spatial setting. The focus of this work will be on the forward evolution of particle mass as well as the backward evolution of genealogies and their diffusion limits as described by measure-valued processes, stochastic differential equations and spatial coalescent processes. Multitype models for genes with mutation, selection and recombination that have ties to both lines of research will also considered.

Understanding the behavior of large particle systems is of importance for deducing macroscopic properties from microscopic rules in a wide variety of fields such as Physics, Chemistry, Biology and also Computer Science. In this research, spatial stochastic processes are constructed and analyzed that model self-regulation phenomena which are observed in many physical systems of particle populations. The aim is to identify the underlying mechanisms and conditions for various macroscopic behaviors such as survival and coexistence of different types to occur. These questions are in particular of interest for biological populations in epidemiology, ecology and genetics.

Spatial stochastic models for reproducing populations in which the offspring of single individuals may on occasion replace a significant proportion or the population are also considered. This kind of behavior is thought to occur, for example, in populations in which certain genes carry a selective advantage. These models are therefore of particular relevance in genetics. The analysis of realistic models for the evolution of genes and the description of genealogies is of importance for the correct quantitative analysis and interpretation of the now ubiquitous gene sequence data. Applications range from reconstructing the origin and history of humans to locating and identifying genes on the genome that are causative factors for diseases.