research partners    
Ph.D. students    
consortium members    

Piotr Gwiazda
Institute of Applied Mathematics and Mechanics
University of Warsaw
Banacha 2,
02-097 Warsaw
phone: +48 22 5544551
fax: +48 22 5544700

Structured population models in metric spaces
Piotr Gwiazda (MIM, University of Warsaw) and Anna Marciniak-Czochra (IWR, University of Heidelberg)
The project is carried out by Agnieszka Ulikowska at University of Warsaw
This project is devoted to the development of mathematical tools for analysis and simulations of dynamics of structured populations. Since modelling of spread, growth and differentiation of cell populations requires taking into account biological and mechanical properties of individual cells, it is necessary to study models describing the dynamics of structured populations governed by processes on the level of individuals. Models describing the time evolution of physiologically structured populations have been extensively studied, see e.g., [1, 3, 4]. However, mathematical theory for systems of equations modelling the spread by diffusion and transport in flows and chemical reactions is still not developed to the extent necessary for application in biosciences.

Recently, a novel approach considering distributional solutions in the spaces of positive Radon measures with flat metric has been developed [2]. This approach was applied to study dependence of solutions on the initial conditions and parameters in a nonlinear structured population model given in the form of a nonlocal first order hyperbolic system. Lipschitz continuous dependence with respect to the modelling ingredients is important in the context of numerical approximation and experimental data, for example, for proper calibration of the model.

The aim of this project is to generalize the new approach and apply it to the structured population models arising in the modelling of hierarchical proliferating and interacting cell systems. Such models involve usually growth terms and coupling of several differential equations. New numerical schemes will be designed and implemented using this framework. Numerical work will be performed in cooperation with the group of Prof. Rolf Rannacher (University of Heidelberg).

In addition, the aim of this project is to explore biological meaning of the stability results obtained using different metrics such as the flat metric or the modified Wasserstein metric. The project will lead to the development of new mathematical tools to study properties of interacting cell systems and therefore also will contribute to the understanding of the role of internal physiological structures and heterogeneity.

[1] O. Diekmann and P. Getto, (2005) J. Differ. Equations 215: 268-319.
[2] P. Gwiazda, T. Lorenz and A. Marciniak-Czochra, submitted.
[3] H.R. Thieme, (2003) Mathematics in population biology, Woodstock Princeton University Press, Princeton.
[4] G. F. Webb, (1985) Nonlinear Age-Dependent Population Dynamics, Dekker.