In our eight research projects, we intend to prove mathematical assertions about probabilistic models motivated by population biology, extended with simulations whenever needed. We expect that these projects can yield relevant evolutionary predictions for epidemiological models, evolutionary experiments, and cancer biology. We also plan to develop new methods to extend the mathematical scope and applications of stochastic population biology.
In our projects, we typically investigate the fate of beneficial mutations arising in a large resident population. The total population size is assumed  either constant or regulated by logistic competition. Some of our models include multiple mutations. In that case, we derive interesting (deterministic or random) scaling limits of the individual-based model under a suitable scaling of time, mutation frequency, and selective advantages of mutants. One such scaling limit is the so-called system of Poissonian interacting trajectories. This system can be used for predicting the speed of adaptation in microbial evolutionary experiments such as the Lenski experiment and to test prior heuristics for this speed.
In other models, we study the fate of a single mutation arising at one individual of the large resident population. We want to compute the probability of a successful mutant invasion and the time until reaching a macroscopic mutant population size. We achieve this via combining proof methods related to probabilistic models (e.g., branching processes) and deterministic ones (e.g., systems of ordinary differential equations). Such methods are based on the seminal work by N. Champagnat on the three phases of a mutant invasion, which we also want to extend to situations where initially, the resident population behaves periodically. This will allow us to investigate new, biologically relevant examples such as predator-prey or host-virus models and competitive Lotka-Volterra systems. 
In three models, we study the effects of dormancy, i.e. an inactive, reversible state of individuals or cells, which is an evolutionary trait that can be found at many positions of the tree of life. Dormancy can be advantageous in the case of harsh environmental conditions, but it often comes with reproductive trade-offs. We investigate this phenomenon in a microbial host-virus model in one project. Dormancy and stemness of cancer cells represent a major obstacle in modern oncology and a possible source of metastases. We study a basic model for the efficiency of variants of chemotherapy in the presence of these phenomena, combining rigorous proofs with simulations. Thirdly, we study a variant of the contact process, a classical spatial epidemic models, with long dormancy periods of the pathogen.
One of our projects is related to the Fisher-KPP equation, which is a classical spatial model in population genetics, modelling the spread of a beneficial allele. We study a variant of this equation where instead of a continuous and weak selective force, rare selective events and deleterious mutation events occur, which yield random jumps in the solution of the equation.
All projects have international collaborators belonging to prominent schools of stochastic modelling motivated by population biology. (...) We plan to hire a postdoc on a 50% position for two years for some of the research projects. We also plan to organize a one-week workshop on stochastic models for the spread of beneficial mutations at Erdős Center. We are in regular contact with evolutionary biologists and cancer biologists in order to improve and control our models and present them to the biological community.