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

提出的研究涉及一类描述干细胞群体动力学的微分方程的数学分析。干细胞可以自我更新。例如，在失去组织或血液后，它们还可以分化，这意味着变成不同的细胞类型。然后发生成熟过程，直到分化的细胞取代丢失的成熟细胞。由此可见，干细胞具有重要的功能。另一方面，如果乳腺干细胞暴露出癌变行为，那就非常危险。成熟过程的调控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是正在进行的生物学研究的主题。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，即祖细胞的成熟。此外，我们允许祖细胞的行为取决于它们的成熟度。该模型可以表示为输运型偏微分方程，但目前还没有已知的分析方法。我们开发的另一个公式是一个微分方程，右边有一个时间延迟，它给出了完全成熟的持续时间。由于成熟的每一个时刻都是由成熟细胞调控的，延迟取决于成熟群体的历史，即系统状态的一个组成部分，我们得到了一个状态相关延迟的微分方程。通过对这种延迟的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，我们展示了全局适定性，也就是说，模型在任何时候都有一个依赖于初始种群的唯一解种群。我们还展示了线性化的稳定性，即，对于与平衡种群相似的初始种群，系统显示出与线性相似的动态，即更透明的系统。这可能是第一次在如此复杂程度的人口模型中完全显示这些问题。除了收敛到平衡之外，还经常观察到细胞计数的振荡和周期性，例如，与血液系统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，但这类模型中振荡的证明是罕见的。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，甚至可能比得体更具挑战性。此外，我们希望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，而不是持久性。最后，我们将继续开发数值工具来可视化参数平面上的稳定性和振荡等特性。

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.