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

拟议的研究涉及的数学分析的某一类微分方程描述干细胞群体动力学。干细胞可以自我更新。例如，组织或血液的损失，它们还可以分化，这意味着成为不同的细胞类型。然后发生成熟过程，直到分化的细胞取代失去的成熟细胞。因此，很明显，干细胞具有重要的重要功能。另一方面，如果乳腺干细胞暴露出癌性行为，它们可能非常危险。成熟过程的调节依赖于细胞内信号传导。在细胞水平上，像哪个成熟水平调节哪个以及如何调节这样的问题正在进行生物学研究。在早期的研究中，我们设计了一个模型，其中成熟细胞的数量调节自我更新和成熟细胞的成熟，即，祖细胞。 此外，我们允许祖细胞的行为取决于它们的成熟度。该模型可以制定为一个传输型偏微分方程，但有没有已知的分析方法，在这个配方。另一个公式，我们已经开发的是一个微分方程的右手边的时间延迟，给出了完全成熟的持续时间。由于成熟的每个时刻都是由成熟的细胞调节的，因此时滞依赖于成熟种群的历史，即系统状态的一个分量，并且我们得到具有状态依赖时滞的微分方程。通过这种延迟的简单隐式定义和额外的连续分布延迟，会产生额外的复杂性。在项目的第一阶段，我们已经证明了全局适定性，即，该模型在任何时候都有一个唯一的解决方案人口取决于初始人口。我们还证明了线性稳定性，即，对于类似于平衡种群的初始种群，系统显示类似于线性的动态，即，更加透明的系统。这可能是第一次，这些问题完全显示了人口模型的这种复杂程度。除了向平衡的收敛之外，还经常观察到细胞计数的振荡和周期性，例如，与血液疾病如周期性中性粒细胞减少症有关。尽管对所谓的特征方程进行了大量的数值分析，但在这类模型中证明振动的证据却很少。这里的一个主要目标是帮助缩小这个差距，这可能是另一个突破，也许比适定性更具挑战性。此外，我们还想研究零平衡的全局稳定性，这与种群完全灭绝的可能性有关，与持久性有关。最后，我们将继续开发数值工具，以可视化参数平面中的稳定性和振荡等属性。

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.