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Stability and oscillations for differential equations with state-dependent delay modelling structured populations

Applicant Dr. Philipp Getto
Subject Area Mathematics
Term from 2012 to 2019
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 214819831
 
Final Report Year 2020

Final Report Abstract

A major concern of the project was the mathematical analysis of a class of so-called statedependent delay equations (SD-DDE) that model a maturing stem cell population. Recall that in ordinary differential equations (ODE), which are well-understood, the rate of change of the state at present is modelled in terms of the state at present. In our model, the addition of state-dependent delays, which changes an ODE to an SD-DDE, a type of equations that is still subject to modern analysis, can be explained as follows: there exists a maturation delay given by the time between commitment of a stem cell to maturation and the moment at which it reaches full maturity. During this time the maturation of a committed stem cell is continuously regulated by the fully mature cell population. Hence the maturation delay at present depends on the history of the mature cell population and, since the latter is a component of the state, the delay is state-dependent. Encouraged by discussions with laboratory researchers, some questions have been whether on the long term initial cell populations develop constant population sizes, in which case we speak of a stable equilibrium population, or whether their sizes will remain unstable or oscillate. Before answering these, it was instructive and necessary to investigate for which given initial populations the equations define unique future populations, i.e., to analyse when the model is well-posed. Here, in a first approach, for a comparatively small set of admissible initial populations, it was possible to combine well-posedness with the establishment of a stability theorem, which boils down stability questions to the algebraic solvability of so-called characteristic equations. In a second approach we established wellposedness for a larger set of initial populations, which allowed to show some properties that are useful to localise periodic cell populations. Due to the described nature of the delays these procedures bear considerable challenges. The incorporation of my PhD student Julia Sanchez Sanz allowed the development of computational methods for the stability analysis of ecological consumer resource and trophic models, in which the maturation delay relates to the time between birth and reproduction. In a project with Francesca Scarabel, PhD student of Mats Gyllenberg, and an international group of experts so-called pseudo-spectral methods, which allowed to use - after a discretisation process - the MATLAB package Mat-Cont, were adapted to particular classes of delay equations. For biological interpretation some emphasis was put on visualisations of stability and instability in planes of biological parameters. These showed, e.g., that in the stem cell model upon increase of a single cell parameter an equilibrium can switch first to unstable and back to stable. We hope that our formulation and analysis of challenging real world models as statedependent delay equations can encourage the further development of methods in this modern and difficult subject. We believe that state-dependent delay also plays an important role in epidemiological modelling and have plans to analyse, e.g., models of waning immunity with similar methods. Even though, other than proposed, we found a direct laboratory comparison during the running time of the project unrealistic, we hope that our biological model findings can motivate more interdisciplinary collaborations in the future.

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