Biomedical subjects
Publications and source records attributed to M Gyllenberg.
Several time-discrete ecological models are formulated and analyzed. Common to these models is that a trade-off between reproductive effort and competitive ability is assumed. We investigate how behavioral adjustment (suppression of reproduction) affects the population dynamics.
A discrete model for a metapopulation consisting of two local populations connected by migration is described and analyzed. It is assumed that the local populations grow according to the logistic law, that both populations have the same emigration rate, and that migrants choose their new habitat patch at random. Mathematically this leads to a coupled system of two logistic equations. A complete characterization of fixed point and two-periodic orbits is given, and a bifurcation analysis is performed. The region in the parameter plane where the diagonal is a global attractor is determined. In the symmetric case, where both populations have the same growth rate, the analysis is rigorous with complete proofs. In the nonsymmetric case, where the populations grow at different rates, the results are obtained numerically. The results are interpreted biologically. Particular attention is given to the sense in which migration has a stabilizing and synchronizing effect on local dynamics.
A nonlinear structured cell population model of tumor growth is considered. The model distinguishes between two types of cells within the tumor: proliferating and quiescent. Within each class the behavior of individual cells depends on cell size, whereas the probabilities of becoming quiescent and returning to the proliferative cycle are in addition controlled by total tumor size. The asymptotic behavior of solutions of the full nonlinear model, as well as some linear special cases, is investigated using spectral theory of positive simigroup of operators.
Biochemical reactions can often be formulated mathematically as ordinary differential equations. In the process of modeling, the main questions that arise are concerned with structural identifiability, parameter estimation and practical identifiability. To clarify these questions and the methods how to solve them, we analyze two different second order models for anaerobic waste water treatment processes using two data sets obtained from different experimental setups. In both experiments only biogas production rate was measured which complicates the analysis considerably. We show that proving structural identifiability of the mathematical models with currently used methods fails. Therefore, we introduce a new, general method based on the asymptotic behavior of the maximum likelihood estimator to show local structural identifiability. For parameter estimation we use the multiple shooting approach which is described. Additionally we show that the Hessian matrix approach to compute confidence intervals fails in our examples while a method based on Monte Carlo Simulation works well.
In this paper we propose a mathematical model for the growth of solid tumors which employs quiescence as a mechanism to explain characteristic Gompertz-type growth curves. The model distinguishes between two types of cells within the tumor, proliferating and quiescent. Empirical data strongly suggest that the larger the tumor, the more likely it is that a proliferating cell becomes quiescent and the more unlikely it is that a quiescent cell reenters the proliferating cycle. These facts are taken as the basic assumptions of the model. It is shown that these assumptions imply diminishing of the growth fraction (i.e. proportion of proliferating cells), a phenomenon found in most tumors. Three qualitatively different cases are analyzed in detail and illustrated by examples. In the case of a tumor forming a necrotic center the model predicts that the tumor grows monotonically to its ultimate size according to a typical S-shaped Gompertz curve, and that the growth fraction tends to zero. In the case of true quiescence, where the dormant cells retain their capability of becoming proliferating, we distinguish between two types of tumors: one in which only proliferating cells can die and one in which there is mortality among quiescent cells, too. In the first case the predicted tumor growth occurs in the early stages in a way that is very similar to that of tumors forming a necrotic center; the growth fraction still tends to zero, but ultimately the tumor grows without bound. In the second case the tumor grows to a finite limit depending only on the vital rates, while the growth fraction decreases to a strictly positive value.