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

具有状态相关延迟建模结构化总体的微分方程的稳定性和振荡

基本信息

批准号：
214819831
负责人：
Dr. Philipp Getto
金额：
--
依托单位：
Facultad de Ciencias Sociales
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2012
资助国家：
德国
起止时间：
2011-12-31 至 2018-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/214819831?language=en
关键词：
Stability oscillations differential equations state

项目摘要

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.

拟议的研究涉及描述干细胞群体动态的某一类微分方程的数学分析。干细胞可以自我更新。例如，在组织或血液损失后，它们还可以区分哪些细胞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，干细胞具有重要的生命功能。另一方面，如果例如乳腺干细胞暴露出癌症行为，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，诸如哪种成熟水平调节哪些因素以及如何调节等问题需要持续进行的生物学研究。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（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，但该公式中没有已知的分析方法。我们开发的另一种公式是右侧带有时间延迟的微分方程，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细胞调节的，因此延迟取决于成熟群体的历史，即系统状态的一个组成部分，我们得到了一个具有状态依赖延迟的微分方程。由于该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。在该项目的第一阶段，我们展示了全局适定性，即模型始终具有取决于初始群体的唯一解决方案群体。我们还展示了线性化稳定性，即，对于类似于平衡群体的初始群体，系统显示出与线性类似的动态，即更加透明的系统。对于这种复杂程度的人口模型来说，这可能是第一次完整地展示这些问题。除了趋于平衡之外，还经常观察到细胞计数的振荡和周期性，例如与周期性中性粒细胞减少症等血液疾病有关。尽管对所谓的特征方程进行了大量的数值分析，但此类模型中振荡的证据却很少。这里的一个主要目标是帮助缩小这一差距，这可能是另一个突破，甚至可能比适定性更具挑战性。此外，我们想研究零平衡的全球稳定性，这与种群完全灭绝的可能性与持久性有关。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。