The central theme in my mathematical research is:

The use of linear and non-linear semigroup theory and functional-analytic techniques in the analysis and simulation of dynamical of mixed type,

i.e. different types of evolutionary equations (parabolic or hyperbolic PDEs, integro-partial differential equations, delay equations, systems of ODEs) are combined in a large coupled system. The main question is the study of long-term behaviour of such systems, i.e. attractors and the dynamics on these, and model reduction.

The topic of dynamical systems in spaces of measures and their perturbations has grown substantially in personal - and international - interest over the last years. In particular the application to structured population models and the relation to underlying stochastic processes I find highly interesting.

My 'pure' mathematical research is devoted to dynamical systems in spaces of measures, in particular:

1. Long term dynamics of Markov semigroups:
   Lasota and his formwer PhD students established in Poland a novel approach to the study of long-term dynamics of iterated functions systems (IFS) and semigroups of Markov operators acting on the space of finite (signed) Borel measures on a Polish space, employing so-called lower-bound techniques and the particular concepts of equicontinuous families of Markov operators. These techniques are further elaborated and applied to stochastic models for biological phenomena (see below).
2. Modelling with measures - crowd dynamics / interacting particle systems:
   The formulation of crowd dynamical models in the formalism of measures allows to study dicrete and continuum models for crowd dynamics or interacting particle systems within a single framework. Also in population dynamics, there is an increased interest in modeling directly in terms of measures, rather than using continuum descriptions. The analysis of these models within this framework requires the development of new mathematical tools that can be applied in this context.

Prototypes of systems of mixed type are taken from mathematical modelling of biological processes:

1. Transport within (plant) tissues:
   Within this broad topic research is focussed on:
   
   • The plant hormone auxin plays an important role in plant growth and development. Its transport through plant tissue is spatially organised. The Plant BioDynamics Laboratory (PBDL) in Leiden performs (a.o.) research on polar auxin transport (PAT) in infloresence stems of Arabidopsis thaliana and Chara corallina. Experimens, their modelling by systems of PDEs and subsequent fitting of the data are targetted at understanding the mechanisms of PAT in inflorescence stems.
   • Uptake of oxygen in germinating seeds.
2. Stochastic models for gene regulatory networks:
   Because in gene transcription there is necessarily a small-to-one molecular interaction between transcription factors and DNA and the inherent stochasticity in both the transcription and translation process random effects play a role in gene regulatory networks. Developing stochastic models for gene regulatory networks (combined with signalling) and mathematical tools for their analysis is required to properly assess the effects of stochasticity in cellular development and response to environmental changes. Aspects of these topics have been addressed in the  (now completed) BetNet project within the Computational Life Science programme of the Netherlands Organisation for Scientific Research (NWO), In this project the sporulation signaling network in Bacillus subtilis was studied in collaboration a.o. with the microbiology group of Oscar Kuipers at the Rijksuniversiteit Groningen.
3. Membrane receptor dynamics:
   Receptors for the chemoattractant cAMP on the cell membrane of D. discoideum are not fixed, but move diffusively along the surface of the membrane. Modelling this behaviour and the subsequent signalling network involves the coupling of diffusion of chemoattractant in the exterior, various surface reactions and diffusion of receptors over the membrane and diffusion of signalling components in the cytoplasm. Moreover, signal transduction by a single receptor, i.e. the activation of an intracellular second messenger when a ligand molecule binds the receptor is stochastic.