Sander Hille
Mathematical Institute, Leiden University

Deterministic mathematical models, e.g. defined by systems of differential equations of various types, have been applied fruitfully in a diversity of fields in science in the past centuries. In population biology they appeared in the work of Lotka and Volterra on predator-prey systems at the start of the 20th century. The past decades have seen an explosion of modeling efforts e.g. in the field of Systems Biology, to gain further insight into the functioning of complicated biological systems, ranging from complete ecosystems, to hormone transport in plants and algae, to intracellular processes such as signaling and gene regulation.

For understanding particular aspects of biological systems it is necessary to include stochastic effects in the models. Two type of randomness can be identified: extrinsic noise, that originates from the interactions of an individual with its environment, e.g. due to random fluctuations or predation, and intrinsic noise, that comes e.g. from the stochasticity of molecular interactions within a biological cell at low molecular numbers. We shall argue, based upon experimental evidence for biological examples like growing bacterial colonies (infections) or gene regulation, that one must be careful in using the framework of stochastic differential equations (SDEs) in these settings. So-called piece-wise deterministic Markov processes (PDMPs) are more appropriate for the given examples.

The lectures will discuss an approach to the analysis of PDMPs of the type encountered in the mentioned biological examples that is rooted in Lasota’s fruitful and original Polish school on iterated function systems, related to (semi)fractals, the so-called „lower-bound technique” in particular. That is, we provide novel techniques for obtaining existence, uniqueness and asymptotic stability of invariant probability distributions for the PDMPs, that were obtained recently. Moreover, we shall discuss biological implications of these results and some mathematical issues that have to be resolved still to enhance the effectiveness of the applications of mathematics to more complicated biological systems.