Stability analysis of a state-dependent delay differential equation for cell maturation: analytical and numerical methods.

We consider a mathematical model describing the maturation process of stem cells up to fully mature cells. The model is formulated as a differential equation with state-dependent delay, where maturity is described as a continuous variable. The maturation rate of cells may be regulated by the amount of mature cells and, moreover, it may depend on cell maturity: we investigate how the stability of equilibria is affected by the choice of the maturation rate.

Numerical Bifurcation Analysis of Physiologically Structured Populations: Consumer-Resource, Cannibalistic and Trophic Models.

With the aim of applying numerical methods, we develop a formalism for physiologically structured population models in a new generality that includes consumer-resource, cannibalism and trophic models. The dynamics at the population level are formulated as a system of Volterra functional equations coupled to ODE. For this general class, we develop numerical methods to continue equilibria with respect to a parameter, detect transcritical and saddle-node bifurcations and compute curves in parameter planes along which these bifurcations occur.

On the characteristic equation λ = α1 + (α2 + α3λ)e(-λ) and its use in the context of a cell population model.

In this paper we characterize the stability boundary in the (α1, α2)-plane, for fixed α3 with −1 < α3 < +1, for the characteristic equation from the title. Subsequently we describe a nonlinear cell population model involving quiescence and show that this characteristic equation governs the (in)stability of the nontrivial steady state. By relating the parameters of the cell model to the αi we are able to derive some biological conclusions.

Mathematical Modelling as a Tool to Understand Cell Self-renewal and Differentiation.

Mathematical modeling is a powerful technique to address key questions and paradigms in a variety of complex biological systems and can provide quantitative insights into cell kinetics, fate determination and development of cell populations. The chapter is devoted to a review of modeling of the dynamics of stem cell-initiated systems using mathematical methods of ordinary differential equations. Some basic concepts and tools for cell population dynamics are summarized and presented as a gentle introduction to non-mathematicians.

Global dynamics of two-compartment models for cell production systems with regulatory mechanisms.

We present a global stability analysis of two-compartment models of a hierarchical cell production system with a nonlinear regulatory feedback loop. The models describe cell differentiation processes with the stem cell division rate or the self-renewal fraction regulated by the number of mature cells. The two-compartment systems constitute a basic version of the multicompartment models proposed recently by Marciniak-Czochra and collaborators [25] to investigate the dynamics of the hematopoietic system.

Stability analysis of multi-compartment models for cell production systems.

We study two- and three-compartment models of a hierarchical cell production system with cell division regulated by the level of mature cells. We investigate the structure of equilibria with respect to parameters as well as local stability properties for the equilibria. To interpret the results we adapt the concept of reproduction numbers, which is well known in ecology, to stem cell population dynamics.