Alexander Lorz: Mathematical models for population dynamics and resistance development

Abstract: We are interested in the Darwinian evolution of a population structured by a phenotypic trait. In the model, the trait can change by mutations and individuals compete for a common resource e.g. food. Mathematically, this can be described by non-local Lotka-Volterra equations. They have the property that solutions concentrate as Dirac masses in the limit of small diffusion. We review results on long-term behaviour and small mutation limits. Moreover, we show connections to a free-boundary problem.
A promising application of these models is that they can help to quantitatively understand how resistances against treatment develop.
In this case, the population of cells is structured by how resistant they are to a therapy. We describe the model, give first results and discuss optimal control problems arising in this context.