Research programme in relation to the state-of-art:
Genome evolution. We have presented a model of genome evolution [2] which explained some conjectures of Słonimski concerning the size distribution of paralog families in several microbial genomes. But this model needs modifications and further studies. It would be interesting to consider the model with rates of elementary events (mutation, duplication and gene loss) dependent on the gene location in the genome, genome size or the functional importance of a given gene.
Gene expression. Modelling of gene expression and gene regulatory networks is the topical subject of the modern biology (The Nobel Prizes in Physiology or Medicine in 2001, 2002, 2006, and 2009 are connected with this subject). It is also a challenging problem of mathematical modelling. Due to a small number of copies of molecular species involved, such as DNA, mRNA and regulatory proteins, gene expression is a stochastic phenomenon. In trying to understand observed distributions of intracellular components, the norm in computational and systems biology is to use algorithms developed initially by D. Gillespie to solve the chemical master equation for specific situations. However, these investigations demand long computer runs, are computationally expensive, and further offer little insight into the possible diversity of behaviours that different gene regulatory networks are capable of. Mathematical modelling from an analytical point of view that we are taking can give precise conditions on the long term behaviour of various dynamics. In the paper [3] we proved some results concerning stability of distribution of mRNA and protein levels and we hope to apply our technique in other cases. Mathematical tools in points a) and b) are mainly stochastic processes and semigroups of stochastic operators. The precise study of mathematical and biological properties of our models needs to use or establish new criteria for stability of such semigroups.
Chaos in structured population models. Models of this type are described by partial differential equations usually with a nonlocal reproduction term. Till now we have shown chaotic behaviour only for models without replication term [4],[11],[12]. Now it becomes clear for us that we could also prove chaos for more complex structured models. In order to establish chaotic behaviour we use tools from the ergodic theory. We construct invariant mixing measures for flows generated by our equations supported on the whole space. This implies existence of dense trajectories and instability.
Rate of convergence of individual-based models to transport equations. We have constructed and studied some individual-based models [5] of plankton dynamics and showed their connections with macroscopic ones. The convergence rate of those models is very important for numeric simulations. Mathematically, it requires rather involved tools of stochastic processes with values in the space of some generalized functions [17],[18].
