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

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Getto , P , Gyllenberg , M , Nakata , Y & Scarabel , F 2019 , ' Stability analysis of a state-dependent delay differential equation for cell maturation : analytical and numerical methods ' , Journal of mathematical biology , vol. 79 , no. 1 , pp. 281-328 . https://doi.org/10.1007/s00285-019-01357-0

Title: Stability analysis of a state-dependent delay differential equation for cell maturation : analytical and numerical methods
Author: Getto, Philipp; Gyllenberg, Mats; Nakata, Yukihiko; Scarabel, Francesca
Contributor: University of Helsinki, Department of Mathematics and Statistics
University of Helsinki, Department of Mathematics and Statistics
Date: 2019-07
Language: eng
Number of pages: 48
Belongs to series: Journal of mathematical biology
ISSN: 0303-6812
URI: http://hdl.handle.net/10138/304334
Abstract: 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. We show that the principle of linearised stability holds for this model, and develop some analytical methods for the investigation of characteristic equations for fixed delays. For a general maturation rate we resort to numerical methods and we extend the pseudospectral discretisation technique to approximate the state-dependent delay equation with a system of ordinary differential equations. This is the first application of the technique to nonlinear state-dependent delay equations, and currently the only method available for studying the stability of equilibria by means of established software packages for bifurcation analysis. The numerical method is validated on some cases when the maturation rate is independent of maturity and the model can be reformulated as a fixed-delay equation via a suitable time transformation. We exploit the analytical and numerical methods to investigate the stability boundary in parameter planes. Our study shows some drastic qualitative changes in the stability boundary under assumptions on the model parameters, which may have important biological implications.
Subject: Characteristic equation
Pseudospectral
Linearised stability
Threshold-type delay
Stem cell
Progenitor phase
STRUCTURED POPULATION-MODELS
BIFURCATION-ANALYSIS
STEM-CELLS
SELF-RENEWAL
DYNAMICS
OSCILLATIONS
GENERATION
MATCONT
CYCLES
111 Mathematics
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