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

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The proposed research deals with the mathematical analysis of a certain class of differential equations describing stem cell population dynamics. Stem cells can self-renew. After, e.g., a loss of tissue or blood, they additionally can differentiate which means become a different cell type. A maturation process then occurs until the differentiated cells replace the lost mature cells. It is thus clear that stem cells have essential vital functions. On the other hand, if e.g. mammary stem cells expose cancerous behaviour they can be very dangerous. The regulation of the maturation process relies on intracellular signalling. At the cellular level questions like which level of maturity regulates which and how are subject to ongoing biological research. In earlier research we have designed a model in which the quantity of mature cells regulates self-renewal and the maturation of the maturing, i.e., progenitor, cells. Additionally we allow the progenitor cells behaviour to depend on their maturity. The model can be formulated as a transport type partial differential equation but there are no known methods of analysis in this formulation. An alternative formulation we have developed is a differential equation with a time delay on the right hand side that gives the duration of the full maturation. As each moment of the maturation is regulated by the mature cells the delay depends on the history of the mature population, i.e. a component of the state of the system, and we get a differential equation with state-dependent delay. Additional complications arise through the mere implicit definition of this delay and additional continuously distributed delays. In the first phase of the project we have shown global well-posedness, i.e., that the model has for all times a unique solution population depending on the initial population. We have also shown linearised stability, i.e., that for initial populations similar to an equilibrium population the system displays similar dynamics as a linear, i.e., a much more transparent system. This may have been the first time that these issues were shown completely for population models of this degree of complexity. Apart from convergence to equilibria also oscillations and periodicity in cell counts are frequently observed, e.g., in relation to hematological disorders like cyclical neutropenia. In spite of an abundance of numerical analysis of so called characteristic equations proofs of oscillations in this type of models are rare. A major objective here is to help in closing this gap, which could be another breakthrough perhaps even more challenging than well-posedness. Additionally we would like to investigate global stability of the zero equilibrium, which relates to possibilities of total extinction of the population, versus persistence. Finally we will continue our development of numerical tools to visualize properties like stability and oscillations in parameter planes.